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Bardula is a pseudonym created by a Belgian artist who currently lives and works in France.
Bardula creates luminous paintings including the hypnotic paintings "Blue Interferences" and "Blue ice" (see Figures 1 and 2 above).
Sources :
Light ZOOM Lumière.
Bardula.
Here is a selection of the best motion illusions (see above Figure 1 cf [GomboDigital], Figures 2 to 5 cf [Sélection.ca] and Figure 6 cf [Akiyoshi Kitaoka]) :
1. Rotating vortex (Vectordivider image via Getty Images)
2. Rotating spirals (Vectordivider image via Getty Images)
3. Mesmerizing effect (Mark Grenier image via Shutterstock)
4. Scroll (Guten Tag Vector image via Shutterstock)
5. Glitter Grid (Mark image via Shutterstock)
6. "Expanding pupils" (Image from Akiyoshi Kitaoka)
Sources :
Sélection du Reader's Digest (Canada) - 24 illusions d'optique complètement étourdissantes.
GomboDigital - 5 illusions d'optique qui vont vous scotcher/.
Akiyoshi Kitaoka - Anomalous motion illusions 35.
The Author of this site has made four wind sculptures installed in his garden in Berrac (Gers).
Description :
Design :
These sculptures are made with recycled products (aluminum rails for thermal insulation frame, PVC camping bowls, plastic jerrycan sides, tennis ball, rebar, etc.).
All moving parts are carried on ball bearings.
All the fixed elements are assembled by stainless steel screws.
The sculptures are fixed to the ground by a vertical mast (galvanized steel fence post or old steel water pipe).
Pierre Luu is a French sculptor who has created sculptures with unpredictable movements, driven by wind or water, including the "wind turbine with random motion" (see Figures above, and video "Mobile eolien art cinétique" in [Pierre Luu]).
Description :
Figure 1 above : general view (cf [Art et Eau]).
Figure 2 above : zoom on blades twist angle (video clip 0:12).
Figure 3 above : zoom on blades lenght (video clip 0:22).
The "wind turbine with random motion" is made up of 5 moving parts in unstable equilibrium (cf [Pierre Luu - Quelque chose ne tourne pas rond][Art et Eau - Quelque chose ne tourne pas rond]).
The balance is all the more unstable as there is no weather vane to orient the sculpture in the wind direction. The blue ball is aesthetic and symbolizes the Earth (cf my email of March 5, 2023 from Pierre Luu to Régis Petit).
The two blades are of different size with a secondary rotation nested within the primary rotation (cf [Pierre Luu - Eolide].
The sculpture slowly comes to life and changes shape thanks to the wind action. The movement is maintained by inertia due to the balance of the masses (cf [Art et Eau - Quelque chose ne tourne pas rond]).
The sculpture unfolds in an enigmatic choreography and only finds temporary stability when the blades reach a certain speed (cf [Pierre Luu - Quelque chose ne tourne pas rond]).
Design :
The whole is designed in a search for balance between the masses, the gravity centers, the surfaces exposed to the wind and the relative angles of the surfaces (cf [Pierre Luu - Quelque chose ne tourne pas rond]).
The mobile elements are fixed by ball bearings for all sculptures in project version (cf email of March 7, 2023 from Pierre Luu to Régis Petit). This combination allows fluid rotations and movements even in light winds (cf [Pierre Luu - Fragments mobile éolien).
Material : stainless steel and composite materials (cf [Art et Eau - Quelque chose ne tourne pas rond]).
Height : 3 m 50 (cf [Art et Eau - Quelque chose ne tourne pas rond]).
Sources :
Pierre Luu - Mobile eolien art cinétique (YouTube, 01:57).
Pierre Luu - Un art en mouvement - Sculptures éoliennes et mobiles.
Pierre Luu - Un art en mouvement - Quelque chose ne tourne pas rond.
Pierre Luu - Un art en mouvement - Fragments mobile éolien.
Pierre Luu - Un art en mouvement - Eolide.
Pierre Luu - Un art en mouvement - Solaris : sculpture éolienne et solaire autonome en énergie.
Art et Eau - Ellipse, quelque chose ne tourne pas rond.
Jeff Kahn is an American sculptor who created kinetic sculptures, titled "Invisible Forces", from aluminum and stainless steel.
These sculptures explore balance and gravity and how almost imperceptible air currents interact with them. They are extremely sensitive to the surrounding environment (light breezes, sun heat, weight of the morning dew). See above Figures 1, 2 and 3 showing three particular sculptures : "Astrolabe", "Naked Alien" and "I Ching".
Jeff Kahn's studio is located in Lenhartsville, Pennsylvania, USA.
Sources :
Jeff Kahn - Bio.
Jeff Kahn - Catalog.
Jeff Kahn - Videos.
Anthony Howe is an American sculptor who has created hypnotic mobile sculptures including "Di-Octo" in 2014 (see Figures 1, 2 and 3 above, and "Di-Octo" video in [Anthony Howe] and [KULTT]).
Anthony Howe currently lives in Eastsound, Orcas Island, San Juan County, Washington State (USA).
Description :
Di-Octo is a half-octopus, half-star, wind-driven and near-silent mobile sculpture.
The original Di-Octo, designed and made by Anthony Howe, has been industrialized in two identical copies by Show Canada Inc (Laval steelworks in Quebec) as follows (cf email of March 10, 2023 from David Boulay (Show Canada Inc) to Régis Petit) :
Design :
Di-Octo is 8 meters high, 3 meters in diameter, weighs 725 kilograms and requires only 2 km/h of wind for its moving parts to activate (cf [Concordia University]).
Di-Octo is composed of 36 arms each carrying 16 very thin steel domes and rotating around a vertical circular ring. The inter-arm connections are of the intermediate wheel type with drive fingers. See detail in Figure 3 above (cf [Show Canada]).
The arms always turn in the same direction, regardless of the wind direction. This is due to the domes shape (cf email of March 19, 2023 from David Boulay to Régis Petit).
Di-Octo is entirely made of 316 stainless steel, which gives it better corrosion resistance as well as non-magnetic properties (cf [Show Canada]).
Other similar sculptures :
Anthony Howe designed and made other sculptures similar to Di-Octo (cf [Anthony Howe, https://www.howeart.net/about]) :
Sources :
Anthony Howe.
Anthony Howe - Shindahiku (Fern pull).
The DC Blike Blogger - Shindahiku (Fern Pull).
KULTT - Les sculptures hypnotiques d'Anthony Howe.
Anthony Howe - Di-Octo (Youtube 1:10).
Anthony Howe - Di-Octo (long version) (Youtube 1:33).
Université Concordia - Di-Octo : captivant, cinétique et unique.
Show Canada.
JuanG3D : Di-Octo 3D Model.
What's on - Check out these alien-esque kinetic sculptures in Dubai.
UAE - Famous American artist brings kinetic sculptures to Dubai.
reddit - "Octo II", Anthony Howe, stainless steel, 2013..
Jennifer Townley is a Dutch artist who has created hypnotic mobile sculptures including "Asinas" in 2015 (see Figure 1 above, and video "Asinas").
When viewed at a standstill from the front, it looks like a double helix like the usual representation of DNA.
Description (cf [Jennifer Townley]) :
"Asinas" is a mobile sculpture composed of two helixes that intertwine and slide into each other, producing a fluid and natural movement.
The two helixes slowly rotate in opposite directions and at slightly different speeds, gradually transforming the sculpture.
A demonstration of how this sculpture works helps to better understand this description (see video "Asinas Working Demonstration" in [Amogh Jadhav] and video "SolidWorks Mechanical Sculpture" in [tecnoloxia.org]).
Design :
The sixty-five white wooden bricks that form the two helixes increase in size towards the middle of the sculpture, giving it a conical shape.
Each brick has the shape of a Z with 90 degree angles. The bricks of a helix are fixed on the rotation axis. The bricks of the other helix are connected to one another through small spacers (see Figure 2 above from [Amogh Jadhav]).
The bricks are made from painted wood. The frame is made of steel as well as all the parts connecting the gears to their axes, the bearings to the frame, etc.
Then there are all the other parts : an electric motor, heavy steel spur gears and sprockets, two belts and lots of bearings (cf [The Plus Paper]).
Sources :
Asinas - Jennifer Townley - 2015 - Kinetic art (Youtube 2:31).
Jennifer Townley - Asinas.
L'Usine Nouvelle - Hypnotiques, ces sculptures cinétiques vous étonneront.
Amogh Jadhav - Asinas.
Amogh Jadhav - Asinas Working Demonstration (Youtube 2:14).
tecnoloxia.org - As esculturas cinéticas de Jennifer Townley.
MadCadSkills : Jennifer Townley - SolidWorks Mechanical Sculpture (Youtube 3:43).
The Plus Paper - Asinas : Fluent Movement ( http://www.thepluspaper.com/2015/03/23/asinas-fluent-movement/ ).
Theo Jansen is a Dutch sculptor who in 1991 created strange creatures including the walking robot (see Figures 1 and 2 above).
Working :
This walking robot is a mechanism with very light legs which can move on a horizontal plane under the wind action or on an inclined plane under the action of its own weight (see video cf [Jansen, Plaudens Vela]).
The only actuator in the robot is a central crankshaft making the connection between the legs and the robot body (see red while on Figure 2, and also [Exergia]).
For a robot with three pairs of legs, the crankshaft has three cranks offset successively by 120° to have a constant movement of the robot during the propulsion phase (see Figure 2).
Body description :
The robot body consists of a horizontal platform (length 2a) and vertical fixed supports (length l) carrying the crankshaft (eccentricity m). See Figure 3 above.
The double length (a) of the platform is calculated to ensure non-collision between the front legs and the rear legs.
The length (l) of the supports can be modified to ensure an overall horizontal movement of the robot. Increasing or decreasing the length (l) amounts to pivoting all the bars of each leg around each fixed point F.
Legs description :
Each leg consists of ten articulated bars (bars b to k) of which two form a rigid link (bars e and h)). See Figure 3 above.
The two legs of the same pair are identical and mirror each other on each side of the crankshaft.
The foot of each leg describes an ovoid curve whose lower part is almost flat and horizontal, thus allowing the foot to be in contact with the ground during the propulsive phase.
In the return phase, the foot lifts off the ground and the robot can step over small obstacles without lifting its body too much.
The table of Figure 3 gives the length of each bar according to different authors :
Sources :
Jansen - Plaudens Vela.
Jansen - plaudens vela 1 (Youtube 0:53).
Wikipedia - Mécanisme de Jansen.
Exergia - Simulation von Theo Jansen's Strandbeest.
Giesbrecht Daniel - Design and optimisation of a one-degree-offreedom eight-bar leg mechanism for a walking machine.
The following files describe the monumental and architectural heritage of 140 municipalities located less than 20 km from the towns of Lectoure or Condom in Gers (France), and including Gers Lomagne and its surroundings.
List of municipalities :
The municipalities are listed alphabetically, each followed by the department number: Gers (32 by default), Lot-et-Garonne (47), Tarn-et-Garonne (82).
Each pdf file weighs approximately 500 KB, the heaviest being Lectoure (3.3 MB).
Sources :
- Wikipedia, Descriptif de chaque commune dont département, toponymie, histoire, maire, nombre d'habitants, altitude, lieux et monuments.
- Ministère de la Culture, Immeubles protégés au titre des Monuments Historiques, par département et par commune. N'inclut pas les sites protégés.
- Ministères Ecologie Energie Territoires, Liste des servitudes des sites et monuments du Gers jusque janvier 2015, par commune et incluant la protection des sites et des monuments au titre des Monuments Historiques.
- SDAP renommé STAP (Services Territoriaux de l'Architecture et du Patrimoine), Liste des monuments historiques et des sites du Lot-et-Garonne, par commune et jusqu'en 2006.
- DREAL Midi-Pyrénées (Direction Régionale de l'Environnement, de l'Aménagement et du Logement Midi-Pyrénées), Bilan des sites classés et inscrits du Tarn-et-Garonne, avril 2013, par commune.
- Ministère de la Culture, Base Mérimée du patrimoine monumental français, par commune et par monument incluant date d'origine, lieu, descriptif et propriété.
- Comet Anaïs Villages et bourgs de la Gascogne gersoise à la fin du Moyen Age (1250-1550), par commune, Thèse d'histoire, 2017, Volume 1 : Synthèse (405 p), Volume 2 : Figures (442 p), Volume 3 : Notices (680 p), Volume 4 : Atlas (391 p).
- Google, Recherche par commune (histoire, origine du nom, bastide, castelnau, castrum, fortification, rempart, château, fossé, vestige) ou par monument (protection récente des monuments et des sites au titre des Monuments Historiques)
- Google Images et Google Vidéos, Recherche par commune (monument, "carte postale", vidéo Youtube).
- IGN (Institut Géographique National, renommé Institut National de l'information Géographique et forestière), Géoportail, par commune (situation graphique des lieux-dits et des rues).
- Google, Google Maps, par commune (situation GPS des lieux-dits, rues principales, photos par Street View).
- Google, Recherche par commune (cadrans solaires, moulins, pigeonniers, puits, fontaines, lavoirs).
- Mapio, Photos d'internautes avec titre et géolocalisation précise. Recherche par Région, Département, Arrondissement, Commune.
See detail.
See detail.
Constellations, apparent groupings of stars forming imaginary figures in the sky, have fascinated humanity for millennia.
Today, 88 official constellations are used to map the sky. Some, such as The Little Bear or Cassiopeia, are visible all year round, while others, such as The Swan in summer or Orion in winter, are only revealed in certain seasons.
The zodiacal constellations, crossed by the Sun during the year, are part of these 88 official ones and play a special role in astrology.
Observing the sky also reveals remarkable stars, such as Sirius or Vega, real landmarks in the celestial vault. To make the most of them, it is then important to follow certain practical observation tips.
C4.1. Official constellations [CHA][PER] :
A constellation is an apparent grouping of stars in the night sky as seen from Earth.
The International Astronomical Union (IAU) defined 88 official constellations in 1922 [IAU1]. They cover the entire celestial sphere, divided between the northern and southern hemispheres.
Please note :
- The North Star (Polaris) is an excellent representative of the celestial north pole through which the Earth's own rotation axis passes. The angular deviation between the two (about 0°38' in 2025) is in fact negligible to the naked eye. Given the Earth's own rotation, each constellation makes a complete turn in 24 hours around Polaris in a counterclockwise direction.
- The relative geometric position between constellations, as well as between stars in the same constellation, does not change significantly during the rotation of the Earth (diurnal motion) and its revolution around the Sun (seasonal motion). This is due to the very large distance between the Earth and the stars of these constellations. Only the portion of the sky visible from a given place on Earth at a specific time changes.
- Warning : On a sky map, the east and west directions are reversed to correspond to the point of view of the observer looking towards the sky.
- Stars twinkle. This phenomenon is due to the turbulence of the Earth's atmosphere that disrupts the light coming from these very distant point sources.
- The planets, on the other hand, do not twinkle or twinkle very little. They are in fact much closer to the Earth and appear in the form of small stable disks.
- The brightest objects in the Earth's night sky for the northern hemisphere are, in descending order of brightness :
C4.2. List of constellations :
The 88 official constellations are distributed as follows :
* 54 constellations visible totally or partially from Metropolitan France
- 7 constellations visible all year round
- 20 seasonal constellations visible during the extended summer (May to October)
- 15 seasonal constellations visible during the extended winter (November to April)
- 12 constellations that are difficult to see with the naked eye
* 34 constellations not visible from Metropolitan France
The 54 constellations visible totally or partially from Metropolitan France are as follows, listed in alphabetical order :
These constellations are described below, classifying them by period of year and then by position in the sky, according to the following definitions :
C4.3. Constellations visible all year round
:
The constellations visible all year round (circumpolar constellations) are as follows (see Figures below [IST][LES]) :
To find these constellations in the sky, the simplest method is as follows [DAR][CHA][PER], by referring to the map below :
- Map from July 25, 2025 at 00:00 for mainland France or latitudes from 40 to 55°N, with zenith at the center of the portion of visible sky [STE].
|
1a. Find The Great Bear : Large dipper located in north, at medium height in sky. Find Nord Star Polaris : Bright star located near The Great Bear extending towards the dipper top five times the distance between the two stars on the outer edge of the dipper (see Figure 1 above). 1b. Find The Little Bear : Small dipper with three bright stars : Polaris at the handle end and two stars on the outer edge of the dipper (see Figure 1 above). 2. Find Cassiopeia : W or M, located on a line passing through The Great Bear with Polaris in the middle. 2. Find The Dragon : The head forming a triangle of three bright stars (β, γ, ξ) located above the large dipple perpendicular to the handle end, the body and tail forming a large S of six bright stars which partially wraps between the Great Bear and the Little Bear. 3. Find The Dragon : The head forming a triangle of three bright stars (β, γ, ξ) located above the large dipple perpendicular to the handle end, the body and tail forming a large S of six bright stars which partially wraps between the Great Bear and the Little Bear. 4. Find Cepheus : Polygon with seven bright stars, located halfway between Cassiopeia and The Dragon's head. 5. Find The Giraffe : Group of nine stars including a bright one located in the extension of the handle of the Little Bear. 6. Find The Lynx : Arc of three or four bright stars visible mainly in winter and spring, the brightest star (α) being located near The Lion on a line passing through Regulus with The Lion's head (ε) in the middle. |
C4.4. Summer seasonal constellations :
The seasonal constellations visible during the extended summer (May to October) are the following (see Figures below [IST][LES]) :
To find these constellations in the summer sky, the simplest method is the following [CHA][PER], by referring to the maps below :
- Summer Triangle.
- Map of July 25, 2025 at 00:00 for mainland France or latitudes from 40 to 55°N, with zenith at the center of the portion of visible sky [STE].
|
Find the Summer Triangle near the zenith around midnight in summer (July to September) : Quasi-isosceles triangle formed by three super-bright stars : Vega (Lyra), Altair (Eagle) and Deneb (Swan). Find the star Vega : Brightest star of the Summer Triangle, blue-white in color. 1. Find The Lyre : Small parallelogram attached to Vega. 2. Find The Eagle : Large X ended by Altair, southernmost star of the Summer Triangle, white in color. 3. Find The Swan : Large cross ended by Deneb, northernmost star of the Summer Triangle, white in color. 4. Find The Herdsman : Group of seven bright stars including a super-bright one (Acturus, orange), located near the Great Bear extending the handle of the dipper and also on a line passing through Deneb with Vega in the middle. 5. Find Hercules : Group of fourteen bright stars located just in front of the head of The Dragon ((triangle β, γ, ξ). 6. Find The Northern Crown : Group of seven stars including two bright ones, located halfway between The Herdsman and Hercules. 7. Find The Scales : Polygon with six bright stars, located on a line passing through The Great Bear with The Herdsman in the middle. 8. Find The Scorpion : Large S with nineteen bright stars including a super-bright one (Antares, red), located near The Scales to the east. 9. Find Sagittarius : Group of fifteen bright stars located near the broken tail of The Scorpion to the east. 10a. Find Ophiuchus : Polygon of twelve bright stars located halfway between Antares and Vega. 10b. Find The Serpent : Group of bright stars located on either side of Ophiuchus (eight for The Serpent's Head forming a Y and five almost aligned for The Serpent's Tail). 11. Find Andromeda : Group of nine bright stars located on a line passing through Polaris with Cassiopeia in the middle. 12. Find Pegasus : Group of eleven bright stars, four of which form a square and one is super-bright (Alpheratz, blue) bordering Andromeda. 13. Find The Fishes : Large V with three bright stars, located just next to Pegasus. 14. Find The Whale : Polygon of six bright stars extended by a line of three other bright ones, located south of The Fishes. 15. Find The Sea Goat : Polygon with eight bright stars, located on a line passing through Vega with Altair in the middle. 16. Find The Water Bearer : Group of ten bright stars, located southeast of Cygnus by extending the axis of its wings. 17. Find The Dolphin : Small diamond with additional tail, with five stars including two bright ones, located just in front of the Eagle's eye (Altair). 18. Find The Arrow : Arrow with four stars including two bright ones, located just in front of the Eagle's eye (Altair). 19. Find The Shield : Small diamond with five stars including a bright one, located just behind the Eagle's tail. |
C4.5. Winter seasonal constellations :
The seasonal constellations visible during the extended winter (November to April) are the following (see Figures below [IST][LES]) :
To find these constellations in the winter sky, the simplest method is the following [CHA][PER], by referring to the maps below :
- Winter triangle and Winter hexagon.
- Map of February 14, 2025 at 00:00 for mainland France or latitudes from 40 to 55°N, with zenith at the center of the portion of visible sky [STE].
- The "Big G".
|
Find the Winter Triangle located south around midnight in winter (December to February) : Quasi-isosceles triangle formed by three super-bright stars : Sirius (Canis Major), Procyon (Canis Minor) and Betelgeuse (Orion). Find the star Sirius (brightest star in the Winter Triangle, blue-white in color. 1. Find The Canis Major : Group of ten bright stars including Sirius. 2. Find The Canis Minor : Group of two bright stars including Procyon, easternmost star of the Winter Triangle, white in color. 3. Find Orion : Hourglass with twelve bright stars including a belt of three aligned stars and Betelgeuse, westernmost star of the Winter Triangle, red in color. 4. Find The Hare : Group of eight bright stars, located just south Orion. 5. Find The Bull : Large Y with eleven bright stars including a super-bright one (Aldebaran, orange), located just above the arc of Orion, on a line passing through Sirius with Betelgeuse in the middle. 6. Find The Charioteer : Polygon with nine bright stars including a super-bright one (Capella, yellow) and a bright one bordering The Bull (Elnath, blue). 7. Find Perseus : Group of nine bright stars including a super-bright one (Mirfak, white), located on a line passing through Orion with The Bull in the middle. 8. Find The Ram : Curved line of four bright stars, located on a line through Betelgeuse with Aldebaran in the middle. 9. Find The Triangle : Elongated triangle of three bright stars, located just next The Ram. 10. Find The Twins : Group of eleven bright stars including two super-bright ones (Pollux, orange, and Castor, white), located in front of the two horns of The Bull. 11. Find The Crab : Large Y with four bright stars, located on a line through Sirius with Procyon in the middle. 12. Find The Lion : Elongated polygon with nine bright stars including a super-bright one (Regulus, blue), located near The Great Bear extending towards the dipper underside five times the distance between the two stars on the outer edge of the dipper. 13. Find The Virgin : Group of nine bright stars including a super-bright one (Spica, blue), located east of The Lion. 14. Find The Crow : Group of five bright stars, located southeast of The Lion. 15. Find The Cup : Polygon with four stars including a bright one, located south of The Lion. Notice the Winter Hexagon near the zenith around midnight in winter (December to February) : Symmetrical hexagon with six super-bright stars : Sirius (Canis Major), Procyon (Canis Minor), Pollux (The Twins), Capella (The Charioteer), Aldebaran (The Bull), Rigel (Orion). Notice the "Big G" near the zenith around midnight in winter (December to February) : Big G with nine super-bright stars : Betelgeuse, Bellatrix and Rigel (Orion), Sirius (Canis Major), Procyon (Canis Minor), Pollux and Castor (The Twins), Capella (The Charioteer), Aldebaran (The Bull). |
C4.6. Other constellations :
Other constellations are the following :
12 constellations that are difficult to see with the naked eye :
* Constellations too close to the horizon :
- The Dove (Columba, Col)
- The River (Eridanus, Eri)
- The Southern Fish (Piscis Austrinus, PsA)
- The Water Snake (Hydra, Hya)
* Constellations too faint in brightness (no star with a magnitude less than 4.0) :
- The Berenices's Hair (Coma Berenices, Com)
- The Little Fox (Vulpecula, Vul)
- The Little Lion (Leo Minor, LMi)
- The Lizard (Lacerta, Lac)
- The Sextant (Sextans, Sex)
* Constellations drowned in star-dense regions :
- The Hunting Dogs (Canes Venatici, CVn)
- The Little Horse (Equuleus, Equ)
- The Unicorn (Monoceros, Mon)
34 constellations not visible from Metropolitan France :
- The Air Pump (Antlia, Ant)
- The Altar (Ara, Ara)
- The Bird of Paradise (Apus, Aps)
- The Centaur (Centaurus, Cen)
- The Chameleon (Chamaeleon, Cha)
- The Chisel (Caelum, Cae)
- The Clock (Horologium, Hor)
- The Compass (Pyxis, Pyx)
- The Crane (Grus, Gru)
- The Drafting Compass (Circinus, Cir)
- The Fly (Musca, Mus)
- The Flying Fish (Volans, Vol)
- The Furnace (Fornax, For)
- The Goldfish (Dorado, Dor)
- The Indian (Indus, Ind)
- The Keel (Carina, Car)
- The Male Water Snake (Hydrus, Hyi)
- The Microscope (Microscopium, Mic)
- The Octant (Octans, Oct)
- The Painter (Pictor, Pic)
- The Peacock (Pavo, Pav)
- The Phoenix (Phoenix, Phe)
- The Reticle (Reticulum, Ret)
- The Sails (Vela, Vel)
- The Sculptor (Sculptor, Scl)
- The Southern Cross (Crux, Cru)
- The Southern Crown (Corona Australis, CrA)
- The Southern Triangle (Triangulum Australe, TrA)
- The Square (Norma, Nor)
- The Stern (Puppis, Pup)
- The Table Mountain (Mensa, Men)
- The Telescope (Telescopium, Tel)
- The Toucan (Tucana, Tuc)
- The Wolf (Lupus, Lup)
C4.7. Zodiacal constellations :
The astronomical zodiac is a band in the sky that extends about 8° on either side of the ecliptic (the plane of the Earth's orbit around the Sun).
It includes thirteen official constellations, which are the only ones that the Sun obscures during its annual path, as seen from Earth.
These constellations are as follows, listed from 1 to 12 in the order in which the Sun passes through them :
Their visibility from the northern hemisphere is as follows :
* Constellations visible all year round : none.
* Summer seasonal constellations (May to October) : The Scales, The Scorpion, Ophiuchus, Sagittarius, The Sea Goat, The Water Bearer, The Fishes.
* Winter seasonal constellations (November to April) : The Ram, The Bull, The Twins, The Crab, The Lion, The Virgin.
C4.8.
Tips for good observation :
To properly observe the stars, constellations, planets and satellites in the sky, it is advisable to [CHA][PER] :
C4.9. Color and magnitude :
The apparent color of stars (seen with the naked eye) depends mainly on their surface temperature according to the following simplified classification :
- Blue : Very hot stars ( > 10 000 K approximately), such as Spica.
- White : Hot stars (from 6 000 to 10 000 K approximately), such as Sirius.
-
Yellow : Stars of average temperature (from 5 200 to 6 000 K approximately), such as the Sun.
- Orange : Cold stars (from 3 700 to 5 200 K approximately), such as Aldebaran.
- Red : Very cold stars ( < 3 700 K approximately), such as Betelgeuse.
However, factors significantly influence the apparent color :
- Brightness (whitish effect for very bright stars)
-
Earth's atmosphere (reddish effect near the horizon due to the light diffusion in the air)
-
Interstellar dust (accentuation of red by absorption of short wavelengths (blue))
- Interstellar gas clouds (absorption and diffusion of certain wavelengths depending on their composition)
- Sensitivity of the human eye (attenuation of blue and red in the dark)
The apparent magnitude (M) of a star corresponds to its brightness state as perceived from Earth :
M = -2.5 log10[F/F0]
with :
F = luminous flux received from the star (in W/m2)
F0 = reference luminous flux corresponding to M = 0 (historically that of Vega, before the current more precise measurements).
M is a standardized measure that takes into account four factors :
- Intrinsic luminosity of the star. It corresponds to the total power of light (L in Watt) emitted at its surface, then diffused uniformly in all directions across a spherical surface of increasing radius r.
- Distance between the star and the Earth. The apparent luminosity (I in W/m2), perceived at the distance r from the star, decreases in fact according to the inverse square law : I = L/(4 π r2).
- Extinction (absorption and diffusion of light by the Earth's atmosphere, interstellar dust and gas clouds between the star and the Earth)
- Sensitivity of the human eye (which perceives the apparent luminosity according to an inverse logarithmic scale)
Warning : The lower the numerical value (M) of the apparent magnitude, the brighter the star.
C4.10. Sources relative to constellations :
[CHA] ChatGPT, le moteur d'Intelligence Artificielle développé par OpenAI.
[DAR] Découvrir le ciel à l'oeil nu, Bertrand D'Armagnac et Carine Souplet, Stelvision.
[IAU1] IAU, Les constellations.
[IAU2] IAU, Comment sont nommées les étoiles ?.
[IMA] Imago Mundi, La Girafe.
[IMA] Imago Mundi, Le Lion.
[IMA] Imago Mundi, Le Lynx.
[IMA] Imago Mundi, Ophiuchus.
[IMA] Imago Mundi, Sagittaire.
[IST] iStock, Constellations.
[LES] Les Astronautes, Comment reconnaître les constellations dans le ciel ?.
[PER] Perplexity, le moteur d'Intelligence Artificielle développé par Perplexity AI.
[STE] Stelvision, Carte du ciel du jour (pour France métropolitaine ou latitudes de 40 à 55°N, avec zénith au centre de la portion de ciel visible).
See detail.
Purchasing an electric car, whether 100 % electric, hybrid or hydrogen, requires careful consideration.
These different types of vehicles have the following advantages and disadvantages compared to a conventional car with a thermal engine (petrol or diesel).
D3.1. The 100 % electric car
D3.2. The hybrid car
D3.3. The hydrogen car
D3.4. Synthesis :
|
Purchasing a 100 % electric or hybrid vehicle : Many buyers focus on the purchase price, short-term fuel savings, and tax or environmental benefits. But you also need to consider : - Costs beyond 8 years, including possible battery replacement. - Underestimated expenses, such as as additional costs (insurance, tire replacement, repairs, software updates) and cost of recharging the battery, particularly at fast charging stations. - The financial risk that a minor accident could damage the battery (for example, a simple collision with a sidewalk) and result in the vehicle being classified as a wreck when the cost of replacing the battery is too high in relation to the market value of the vehicle (economic classification VEI) or when the vehicle presents a technically irreparable safety risk (technical classification VGE). - The concerns about the autonomy of the 100 % electric car and the network of charging stations. - The few mechanics authorized to work on electric and hybrid vehicles. For low-income households, two major obstacles remain : the high purchase price and the risk of being classified as a wreck after even a minor collision. Thus, the appeal of a 100 % electric or hybrid car is based more on environmental concerns, particularly the reduction of CO2 emissions, than on strictly short- or long-term economic logic. An informed decision therefore requires a complete analysis of all these aspects in order to allow buyers to make an objective choice adapted to their situation. Purchasing a hydrogen vehicle : Purchasing a hydrogen vehicle is now more relevant for professional fleets than for individuals due to a market that is still developing. The main obstacles include : - a high purchase price, - an insufficient network of hydrogen refueling stations, - as for electric or hybrid cars, the financial risk that a accident could damage the battery, or even the hydrogen tank or fuel cell, and result in the vehicle being classified as a wreck. European regulation : From 2035, by European regulation published in the EU Official Journal on 25 April 2023 and entered into force on 15 May 2023 [EUR] : - The sale of new cars with combustion engines (petrol, diesel and current hybrids), such as passenger cars or light commercial vehicles, will be banned in the European Union. Only "zero-emission" vehicles, such as 100 % electric cars or those using synthetic fuels (e-fuels) or hydrogen (FCEV), will be allowed to be sold. - The ban does not apply to the second-hand market. - Combustion-engine cars already in circulation are not affected and may continue to be used and resold. The hybrid car market, despite its current success, is therefore set to gradually disappear by 2035, unless they achieve exceptional performance in electric mode. The hybrid car thus appears to be a simple transitional stage towards all-electric. First aid : The additional actions to take on an accident-damaged electric, hybrid or hydrogen vehicle are as follows : - Identify and report the type of vehicle to the emergency services because these vehicles present specific risks, particularly concerning the high-voltage battery, the high-pressure hydrogen tank and fire management. - Maintain a safety distance of at least 30 metres around the accident-damaged vehicle to avoid any danger of electric shock, explosion or fire. - Do not use water to extinguish a battery fire without the advice of the fire department. In the rare event of a lithium-metal battery fire, the use of water may make the fire worse. |
D3.5. Sources relative to electric cars :
[AUJ1] L'auto-journal, Voitures électriques : quel est le coût des réparations ?.
[AUJ2] L'auto-journal, Véhicules électriques et entretiens : un vrai cauchemar ?.
[AUM] L'Automobile Magazine, Les voitures électriques sont-elles 3 fois plus dangereuses pour les piétons ?.
[AUT] Auto, Attention, les voleurs s'attaquent aux batteries des véhicules hybrides.
[CAP1] Capital, Accidents de la route : dans quels cas votre voiture est-elle jugée irréparable ?.
[CAP2] Capital, Automobile : découvrez pourquoi réparer un véhicule électrique coûte plus cher !.
[CEN] La Centrale, Quand la batterie flanche : dites adieu à votre voiture électrique et direction la casse !.
[CHA] ChatGPT, le moteur d'Intelligence Artificielle développé par OpenAI.
[CNR] CNRS Le journal, Les défis de la voiture à hydrogène.
[ECO] Ecoconso, Voiture électrique : ses avantages et inconvénients.
[EDM] Les éditions du moteur, Voiture électrique : Le désamour inattendu qui touche 30 % des propriétaires.
[EUR] EUR-Lex, Règlement (UE) 2023/851 du Parlement européen et du Conseil du 19 avril 2023 modifiant le règlement (UE) 2019/631 en ce qui concerne le renforcement des normes de performance en matière d'émissions de CO2 pour les voitures particulières neuves et les véhicules utilitaires légers neufs conformément à l'ambition accrue de l'Union en matière de climat.
[MAC] Machines Production, Voitures électriques : l'épineux problème des batteries après un accident.
[NUM] Numerama, Au moindre accident votre voiture électrique peut finir au rebut à cause de sa batterie.
[PER] Perplexity, le moteur d'Intelligence Artificielle développé par Perplexity AI.
[PRE] Pressecitron, Une voiture électrique coûte-t-elle vraiment moins cher à l'usage qu'une thermique ?.
[QUE] Que choisir, Comment choisir une voiture hybride.
D7.1. Introduction :
Quantum physics is a fundamental branch of physics that describes the universe at the microscopic level of atoms, molecules, and elementary particles, taking into account their dual nature of wave and particle.
It differs from classical physics in its counterintuitive concepts such as energy quantization, wave-particle duality, superposition of states, quantum indeterminism, and quantum nonlocality.
These concepts have given rise to many modern technologies such as electronics, quantum computers, medical imaging, and nanotechnology.
Albert Einstein played a major role in the development of quantum physics, notably through his explanation of the photoelectric effect (1905), the quantization of atomic oscillators (1907), wave-particle duality for light (1909), stimulated emission (1917) and the EPR paradox (1935).
Although he always recognized the validity and effectiveness of quantum formalism, he opposed the probabilistic interpretation of Niels Bohr and the Copenhagen School. Einstein defended a simultaneously deterministic and local view of the quantum world, believing in the existence of "hidden variables" that could explain quantum phenomena more completely.
However, subsequent experiments, notably those of Alain Aspect in 1982 [ASP], contradicted this view by demonstrating that quantum mechanics violates quantum locality regardless of whether it is deterministic or not.
Quantum mechanics is the specific branch that formalizes the phenomena of quantum physics in the form of fundamental principles and mathematical equations.
D7.2. Standard model of particles physics (reminder) :
See Relativity - Lexicon : Standard model of particles physics.
D7.3. Fundamental principles :
The fundamental principles governing quantum physics are as follows [CHA][PER], listed in chronological order of discovery.
1. Quantization of Energy (Max Planck, 1900, Albert Einstein, 1905-1917)
Energy can only take discrete values, called quanta, and not continuous values.
This principle was gradually established and extended as follows :
- Blackbody radiation (Max Planck, 1900, and Nobel Prize in Physics in 1918) : For an atom, the energy difference E between two energy levels is given by : ΔE = h ν, where h is Planck's constant and ν is the frequency associated with the transition.
- Photoelectric effect (Albert Einstein, 1905, and Nobel Prize in Physics in 1921) : Einstein extends Planck's quantization to light itself.
- Quantization of atomic oscillators (Albert Einstein, 1907) : Einstein applies quantization to the vibrations of atoms in a solid.
- Stimulated emission (Albert Einstein, 1917) : Einstein completes the description of the interaction between light and matter in which an incident photon causes an atom to de-excite, emitting a second photon of the same direction, frequency, and polarization.
Confirmation of the quantization introduced by Planck (1900) : Franck and Hertz's experiment (1914) showed that accelerated electrons lost energy in discrete amounts during collisions with mercury atoms.
2. Wave-Particle Duality (Albert Einstein, 1909, Louis de Broglie, 1924)
Quantum objects, such as electrons and photons, exhibit both particle and wave properties depending on the experimental conditions.
The wavelength λ associated with a particle with momentum p is given by : λ = h/p
For a non-relativistic particle with mass m and velocity v, we have: p = m v.
For a photon with energy E, we have : p = E/c
This principle was established in two steps as follows :
- Einstein lays the foundations of wave-particle duality for light (1909).
- De Broglie generalizes this concept to all matter (1924, and Nobel Prize in Physics in 1929).
Confirmation : Davisson-Germer's experiment (1927) demonstrated the diffraction of electrons by a nickel crystal, thus confirming their wave nature.
3. Statistics of Bosons and Fermions (Satyendra Nath Bose and Albert Einstein, 1924, Enrico Fermi and Paul Dirac, 1926)
Bosons (with integer spin, like the photon) follow Bose-Einstein statistics for their collective behavior.
Fermions (with half-integer spin, like the electron) follow Fermi-Dirac statistics for their collective behavior.
Confirmations :
- Bose-Einstein condensate (Eric Cornell and Carl Wieman, 1995, and Nobel Prize in Physics in 2001) by cooling rubidium atoms (bosons).
- Pauli blockade (MIT, 2021) by cooling a lithium-6 gas cloud (fermions).
4. Pauli Exclusion (Wolfgang Pauli, 1925, and Nobel Prize in Physics in 1945)
Two identical fermions, like electrons, protons, or neutrons, cannot simultaneously occupy the same quantum state in the same system.
This principle is fundamental to explaining the electronic structure of atoms, particularly the distribution of electrons in atomic shells and subshells, which leads to the structure of the periodic table of elements.
Confirmations :
- Analysis of the Zeeman effect (1927-1930).
- Study of electronic band structure and nuclear physics (1930-1940).
5. Quantum Superposition (collective, 1926)
A quantum system exists in an indeterminate global state as a superposition of several states simultaneously until a measurement is made. After the measurement, the system collapses into a single state corresponding to the observed result.
Confirmation : Clinton Davisson and Lester Germer's experiment (1927) on the diffraction of electrons by a crystal.
|
Formalization [PER][CHA] :
It applies to any quantum phenomenon, such as the Stern-Gerlach experiment (see below) for the discrete case and the calculation using the continuous Schrödinger method in the continuous case. The wave function Ψ(x, t)is an abstract representation containing all possible information about the state of a particle or quantum system at a given instant. It allows us to calculate the probabilities of different measurement outcomes but does not determine with certainty the outcome of a single measurement. It is expressed mathematically in a standard way as the superposition of eigenstates Ψj in the form : - Discrete case : Ψ(x, t) = ∑j [cj Ψj(x, t)] where j is the index that numbers the different eigenstates Ψj - Continuous case : Ψ(x, t) = ∫R [c(k) Ψk(x, t) dk] where k is a continuous variable. where the variable x can be a position, a volume, a momentum, an energy, etc. Given the normalization condition (see below), the unit of Ψ(x, t) is dimensionless in the discrete case and the square root of the inverse of the unit of x in the continuous case (for example, Ψ(x,t) has the unit of m-1/2 for x expressed in meters). The eigenstates Ψj are characteristic modes of the system for which the measurement of an observable always gives the same result with a probability of 100 %. These modes are analogous to : - Discrete case : the specific tuning frequencies of a radio, where the observable is the radio frequency, - Continuous case : the specific vibration modes of a taut string, where the observable is the vibration frequency. Each eigenstate Ψj(x, t) is expressed in the complex form : Ψj(x) e-i Ej t/h' with a non-temporal part Ψj(x) that is a solution to the stationary Schrödinger equation and a phase part that depends on time t. Ej is the energy associated with the eigenstate Ψj and h' is the reduced Planck constant. The coefficients cj are the probability amplitudes associated with each eigenstate. They are expressed in the complex form : |cj| ei θj where |cj| is the module of coefficient and θj is the initial phase angle in the complex plane. The determination of the cj coefficients is essentially experimental. The |cj| moduli are obtained by statistical measurements on many copies of the system. Relative phases require more advanced techniques such as quantum tomography. More generally, regardless of the type of quantum state (eigenstate or superimposed), the probability amplitude is a complex coefficient associated with each eigenstate of the wave function, allowing the calculation of the probability of observing a specific outcome during a measurement. This concept was introduced in 1926 by Max Born [BOU], who showed that the wave function Ψ could not be interpreted directly as a probability because this would violate the rules for calculating probability for compound events. He proposed that the probability of observing a given state be given by the square of the modulus of the probability amplitude, i.e. |Ψ|2 Hence the following four cases : 1. Discrete case of an eigenstate Ψj : The state of the system is Ψj, with a probability amplitude of 1 for Ψj and a probability of 1 of observing state Ψj 2. Continuous case of an eigenstate Ψj : The state of the system is Ψj(x, t), with a probability amplitude Ψj(x, t) and a probability density ρ(x, t) = |Ψj(x, t)|2 giving the probability of finding at time t the value x of the measured variable. 3. Discrete case of a superposition of eigenstates : The state of the system is a superposition Ψ(x, t) = ∑j cj Ψj(x, t), with a probability amplitude cj for each eigenstate Ψj and a probability Pj = |cj|2 of observing state Ψj 4. Continuous case of a superposition of eigenstates : The state of the system is a superposition Ψ(x, t) = ∫R c(k) Ψk(x, t) dk, with a probability amplitude density c(k) for each eigenstate Ψk and a probability density ρ(x, t) = |Ψ(x, t)|2 giving the probability of finding at time t the value x of the measured variable taking into account the effects of quantum interference between the different eigenstates. Note that the probability amplitude is the wave function itself only in the continuous case of an eigenstate (case 2). The normalization condition guaranteeing that the sum of the probabilities is equal to 1 also imposes that : 1. Discrete case of an eigenstate Ψj : ∑i |Ψj(xi)|2 = 1 where xi are the discretization points. 2. Continuous case of an eigenstate Ψj : ∫R |Ψj(x, t)|2 dx = 1 3. Discrete case of a eigenstates superposition : ∑j |cj|2 = 1 4. Continuous case of a eigenstates superposition : ∫R |c(k)|2 dk = 1 or ∫R |Ψ(x, t)|2 dx = 1 Concrete example (Stern-Gerlach experiment, 1922) : The wave function of a silver atom whose spin is in a superposition of states along the z axis can be written : Ψ = c1 |+z> + c2 |-z> where the eigenstates |+z> and |-z> represent the "up" and "down" orientations of the spin along the z axis, respectively. These eigenstates are also the eigenvectors of the spin operator Sz which allows us to measure the projection of the angular momentum on the z axis. This operator is defined by the following matrix : Sz = h'/2 (1 0) (0 -1) where h' is the reduced Planck constant. The eigenvectors associated with this matrix are |+z> = (1, 0)T for the eigenvalue +h'/2 (spin "up"), and |-z> = (0, 1)T for the eigenvalue -h'/2 (spin "down"). These eigenvalues represent the quantization of the angular momentum projections on the z axis, expressed in units of h' The complex coefficients c1 = 1/(21/2) and c2 = i/(21/2) are the probability amplitudes, taking into account both the moduli and the relative phase between the two eigenstates. Therefore, if we perform a spin measurement on the z axis, there is a 50 % chance (= |c1|2) of obtaining the value + h'/2 ("up" state) and a 50 % chance (= |c2|2) of obtaining the value -h'/2 ("down" state). |
6. Quantum Indeterminism (Max Born, 1926, and Nobel Prize in Physics in 1954)
Measurement results are not deterministic, but probabilistic, when the system evolves freely between measurements or is reset to its initial state before each new experiment.
On the other hand, repeated measurements under the same experimental conditions, at very short intervals, give the same result because the system remains in the measured state after the first observation.
Confirmation : Repeated experiments on Young's slits.
Interpretations :
- In the Copenhagen interpretation (Bohr, Heisenberg), this principle calls into question the idea that the properties of a quantum system have an objective reality independent of measurement.
- But other interpretations, such as the de Broglie-Bohm (hidden variable theory) or the Everett (many-worlds) interpretation, challenge this idea while remaining compatible with experimental predictions. To this end, the de Broglie-Bohm interpretation preserves determinism while accepting the principle of quantum nonlocality, and the Everett interpretation avoids the nonlocality problem by proposing that all possible outcomes of a measurement occur simultaneously in distinct universe branches.
7. Quantum Uncertainty (Werner Heisenberg, 1927, and Nobel Prize in Physics in 1932)
There is a fundamental limit to the precision with which certain pairs of physical quantities can be measured, including position and momentum, energy and time, spin and orientation.
For example, the relationship between the uncertainties between position (x) and momentum (p) is given by :
Δx Δp ≥ h'/2
where h' is the reduced Planck constant.
Indirect confirmation : Davisson-Germer's (1927) experiment on electron diffraction by a nickel crystal.
8. Tunneling Effect (George Gamow, 1928)
A particle can penetrate an energy barrier even without the classically required energy. The particle's wave function does not vanish at the barrier but attenuates within it, allowing a non-zero probability of crossing it.
Confirmation : George Gamow's experiment (1928) on the alpha decay of radioactive nuclei.
9. Quantum Entanglement (EPR paradox, by Albert Einstein, Boris Podolsky, Nathan Rosen, 1935)
Two particles are said to be entangled when they form a global quantum system such that measuring the state of one instantly determines the state of the other, regardless of the distance between them.
Although entanglement allows for instantaneous correlations, it does not allow for faster-than-light transmission of information, thus preserving relativistic causality.
Confirmation : Experiments by Alain Aspect (1982, and 2022 Nobel Prize in Physics) that confirmed the predictions of quantum mechanics in violation of Bell's inequalities, thus confirming quantum entanglement [ASP].
10. Path Integral (Richard Feynman, 1948)
A particle traveling from one point to another simultaneously follows all possible paths to reach its destination. Each path is associated with a complex amplitude (modulus and phase), and the probability of finding the particle at a given location is determined by the sum of the amplitudes (with their phases) of all possible trajectories.
This approach is an elegant and powerful reformulation of the Schrödinger equation.
Confirmation : Hitachi's (1989) experiment on single electron interference.
11. Quantum Decoherence (David Bohm, 1951)
A quantum system can lose its coherence (superposition of states) by interacting with its environment, leading to seemingly classical behavior.
Confirmation : Experiments demonstrating the quantum-classical transition (Serge Haroche, 1996, Anton Zeilinger, 2003).
12. Quantum nonlocality (Bell's theorem, by John Bell, 1964)
Quantum objects can exhibit instantaneous correlations over distance. Nonlocality encompasses quantum entanglement by including correlations over time, complex multi-particle systems, interactions between particles and macroscopic objects, and delocalized phenomena without requiring direct interaction.
Confirmation : Experiments by Alain Aspect (1982) confirmed the predictions of quantum mechanics in violation of Bell's inequalities, thus confirming nonlocality [ASP].
13. Quantum Field Theory (collective)
Quantum field theory (QFT) is an extension of quantum mechanics incorporating restricted relativity. It models particles as excitations of quantum fields that fill all of space-time, making it possible to explain the creation and annihilation of particles. Only quantum entanglement, a non-local phenomenon, departs from the traditional relativistic framework based on the notion of locality and the propagation of information at finite (and not instantaneous) speed.
The following contributions are worth mentioning :
- Quantum theory of the electromagnetic field (Paul Dirac, 1927), confirmed in 1930 by the analysis of atomic spectra (notably Alfred Lande and Otto Stern).
- Quantum electrodynamics (QED), describing the interaction between light and matter (Richard Feynman, Julian Schwinger, Sin-Itiro Tomonaga, Freeman Dyson, 1940, and Nobel Prize in Physics in 1965), confirmed in 1947 by Polykarp Kusch and Henry Foley through experiments on the electron's magnetic moment.
- Non-Abelian gauge theories (Yang Chen-Ning and Robert Mills, 1954), providing a theoretical framework for describing non-Abelian interactions such as the weak and strong forces.
- Unification of electromagnetism and the weak force in a standard model (Steven Weinberg, Abdus Salam, Sheldon Glashow, 1960, and Nobel Prize in Physics in 1979), confirmed in 1983 at CERN by the discovery of the W+, W-, and Z0 bosons.
- Higgs boson (Peter Higgs and François Englert, 1964, and Nobel Prize in Physics in 2013) confirmed in 2012 at CERN by the ATLAS and CMS experiments at the LHC (Large Hadron Collider).
- Quantum chromodynamics (QCD), describing the strong interaction between quarks and gluons (David Gross, Frank Wilczek, David Politzer, 1973, and Nobel Prize in Physics in 2004, Kenneth G. Wilson, 1974), confirmed in 1979 at the DESY (Deutsches Elektron SYnchrotron) center by observing hadronic jets from high-energy particle collisions.
D7.4. Quantum state :
The quantum state of an elementary particle is a complete mathematical description of its observable and probabilistic aspects, distinct from its intrinsic properties.
It also extends to compound systems where interactions, correlations, and entanglement phenomena between particles play a fundamental role.
It is formalized by a state vector |ψ> in an abstract Hilbert space, associated with a wave function ψ(x,t) in a concrete basis, generally the position or momentum basis.
Intrinsic properties are invariant characteristics that define the fundamental nature of the particle, in particular the mass m and the intrinsic quantum numbers.
Observables are measurable physical quantities (such as position x or momentum p) represented mathematically by Hermitian operators (such as X or P).
- Before measurement, the observable is a statistical prediction obtained by calculating the weighted average of the possible outcomes of the measurement, the weights being determined by the probabilities associated with the initial quantum state of the system.
- After measurement, an observable provides a value that corresponds to one of the eigenvalues of the operator associated with the observable.
Operators act on the quantum state of the system (wave function or state vector in a Hilbert space) to calculate information, such as the possible values of observables (eigenvalues) and the probabilities associated with each measurement result. For example :
- the position operator X is the multiplication by x in the form : X(Ψ(x)) = x Ψ(x), allowing the calculation of the average value of the position for this state.
- the kinetic energy operator T acts on the wave function in the form : T(Ψ(x)) = -h'2/(2 m) (d2Ψ(x, t)/dx2)
- the momentum operator P acts by differentiation in the form : P(Ψ(x)) = -i h' dΨ(x)/dx, allowing the calculation of the average value of the momentum for this state.
Note that P can also be written : P = -i h' d/dx
- the Hamiltonian operator H acts on the wave function in the form : H(Ψ(x, t)) = -h'2/(2 m) (d2Ψ(x, t)/dx2) + V(x) Ψ(x, t), allowing the calculation of the total energy of the system.
Note that H can also be written : H = -h'2/(2 m) (d2/dx2) + V(X) = P2/(2 m) + V(X)
- The angular momentum operator J acts on the wave function in the form of a vector product : J(Ψ(x, t)) = r x P(Ψ(x, t)), where r is the position vector and P is the quantized motion operator, allowing the system's angular momentum to be calculated.
- The parity operator π acts on the wave function in the form : π(Ψ(x)) = Ψ(-x), allowing the symmetry of the wave function to be determined with respect to spatial inversion.
- The scaling operators (a+ creation and a annihilation) act on quantum states by adding or removing an excitation particle or quantum.
- The spin operator S acts on spin wave functions in the form of 2x2 Pauli matrices, allowing the intrinsic angular momentum of particles to be described. These matrices are as follows :
σ1 = σx =
(0 1)
(1 0)
σ2 = σy =
(0 -i)
(i 0)
σ3 = σz =
(1 0)
(0 -1)
These matrices verify the property : σ1 σ2 σ3 = i I where I is the Identity matrix.
The eigenvalue is a possible and specific outcome obtained when measuring an observable on a quantum system. It is obtained with certainty if the system is in a corresponding eigenstate, and with a given probability if the system is in a superposition of eigenstates. See Stern-Gerlach experiment.
The eigenvector represents the state of the system when the measurement of an observable yields an eigenvalue. See Stern-Gerlach experiment.
Eigenvectors form a basis for the space of quantum states, allowing all possible configurations of the system to be described.
Measurement probabilities describe how these observables are distributed across their possible value domains in terms of probability density.
They follow Born's rule according to which the probability density ρ is |<Ψj|Ψ>|2, where Ψj is the eigenstate associated with an eigenvalue of the measured observable, that is, the state the system is in immediately after the measurement.
The bra-ket notation <.|.> denotes the Hermitian scalar product which generalizes the classical scalar product to complex vector spaces in the form :
< u|v > = v+.u
where :
u and v = any two column vectors
v+ = adjoint of v = transposed complex conjugate such that v+ = (v*)T
* = complex conjugate operator without transposition
Example : if u = (1 + 2 i, 3)T and v = (4 + 5 i, 6 i)T, then v* = (4 - 5 i, -6 i)T, v+ = (v*)T = (4 - 5 i, -6 i) and < u|v > = v+.u = (4 - 5 i)(1 + 2 i) + (-6 i)(3) = 14 - 15 i
Examples of elementary particles with intrinsic/observable distinction :
Characteristics of the Electron e- :
| Classification : Lepton
| Composition : N/A (Elementary particle)
| Mass (m) = 0.511 MeV/c2
| Intrinsic magnetic moment (μ) = -9.284 10-24 J/T
| Chirality = Right and Left (mixing possible)
| CPT (Charge, Parity, Time) Symmetry : Yes
| Fundamental interactions : electromagnetic, weak and gravitational forces
| Lifetime (τ) = perfectly stable (6.6 1028 years)
| Anti-particle : positron
| Intrinsic quantum numbers :
| | Spin (S) = 1/2
| | Isospin (T3) = -1/2
(for weak isospin)
| | Electric charge (Q) = -1e
| | Color Charge : N/A (particle other than Quark and Gluon)
| | Flavor (Le) : electronic
| | Leptonic number (L) = +1
| | Baryonic number (B) = 0
| Observable quantum numbers :
| | Principal quantum number (n) = strictly positive integer
| | Secondary or azimuthal quantum number (l) = integer from 0 to n
- 1
| | Magnetic quantum number (ml) = integer from -l to +l
| | Magnetic spin quantum number or spin projection (ms) = +1/2 ("up") or -1/2 ("down")
| | Total angular momentum quantum number (j) = |l ± s|
| | Quantum number or projection of the total angular momentum (mj) = from -j to +j by integer steps
| | Projection of the magnetic moment (μz) = ±9,284 10-24 J/T
| | Parity (P) = accordind to contexte
| | Position, momentum and total energy
Characteristics of the Positron e+ = Anti-electron :
| Classification : Anti-Lepton
| Composition : N/A (Elementary particle)
| Mass (m) = 0.511 MeV/c2
| Intrinsic magnetic moment (μ) = +9.284 10-24 J/T
| Chirality = Right and Left (mixing possible)
| CPT (Charge, Parity, Time) Symmetry : Yes
| Fundamental interactions : electromagnetic, weak and gravitational forces
| Lifetime (τ) = perfectly stable in isolation and short in the presence of matter
| Anti-particle : electron
| Intrinsic quantum numbers :
| | Spin (S) = 1/2
| | Isospin (T3) = +1/2
(for weak isospin)
| | Electric charge (Q) = +1e
| | Color Charge : N/A (particle other than Quark and Gluon)
| | Flavor (Le) : electronic
| | Leptonic number (L) = -1
| | Baryonic number (B) = 0
| Observable quantum numbers :
| | Principal quantum number (n) = strictly positive integer
| | Secondary or azimuthal quantum number (l) = integer from 0 to n - 1
| | Magnetic quantum number (ml) = integer from -l to +l
| | Magnetic spin quantum number or spin projection (ms) = +1/2 ("up") or -1/2 ("down")
| | Total angular momentum quantum number (j) = |l ± s|
| | Quantum number or projection of the total angular momentum (mj) = from -j to +j by integer steps
| | Projection of the magnetic moment (μz) = ±9,284 10-24 J/T
| | Parity (P) = accordind to context
| | Position, momentum and total energy
D7.5. Quantum calculation methods :
All quantum calculation methods aim to calculate the possible values of observables and their measurement probabilities, but they do so in different ways depending on the application domains. These include :
Warning : It is common to omit certain universal constants from equations, including Planck's constant (h), reduced Planck constant (h'), light speed (c), Identity matrix (I) and gravitational constant (G).
Calculation methods include :
D7.5.1. Non-relativistic Schrödinger-type methods :
These approaches study the time evolution of the quantum state |Ψ> (i.e. wave function Ψ(x, t)), while the observables remain fixed unless the potential V(x, t) is explicitly time-dependent.
They are used particularly in quantum chemistry and follow the Schrödinger equation (1926, Nobel Prize in Physics 1933).
In particular, for a spinless particle of mass m, moving in one-dimensional space and subject to a potential V(x), this equation is written as :
(L1) i h' dΨ(x, t)/dt = -(1/2)(h'2/m)(d2Ψ(x, t)/dx2) + V(x) Ψ(x, t)
where h' is the reduced Planck constant (or Dirac constant = h' = h/(2 π))
and h is Planck's constant (h = 6.626 10-34 J.s).
See example of calculation using the continuous Schrödinger method and example of calculation using the discretized Schrödinger method.
D7.5.2. Relativistic Schrödinger-type methods :
The time evolution of the quantum state is described by wave equations generalizing the Schrödinger equation for particles moving at speeds close to that of light. We can cite :
1. Dirac equation for relativistic particles with spin 1/2 (examples : electron, neutrino).
In particular, for a free particle, this equation is written :
(D1) (i h' γμ dμ - m c I) Ψ = 0
where :
Ψ is the four-component Dirac wavefunction, called spinor, which simultaneously encodes the two spin eigenstates ("up"/"down"), the positive/negative energy solutions (particle/antiparticle), and their relativistic coupling, as follows :
- When momentum p is zero, the four components decompose into two distinct pairs. The first (Φ) encodes the probability amplitudes associated with the spin eigenstates of the particle (e.g., electron). The second (Χ) encodes those of the corresponding antiparticle (e.g., positron).
- As soon as p ≠ 0, the four components describe either a particle (positive energy) or an antiparticle (negative energy). Each individual solution remains a four-component state where spin (described by the Pauli matrices σ) and momentum (vector p) are coupled. One of the pairs encodes the spin eigenstates inherited from the rest frame, modulated by the motion. The other pair encodes their relativistic modification by connecting the two pairs via the total energy E.
m is the mass of the particle at rest.
I is the 4x4 Identity matrix
γμ are specific 4x4 matrices introduced by Dirac, with the index μ ranging from 0 to 3 (0 corresponding to time).These matrices are as follows in standard Pauli-Dirac representation :
γ0 =
(I 0)
(0 -I)
γi =
(0 σi)
(-σi 0)
γ5 = i γ0 γ1 γ2 γ3
(0 I)
(I 0)
in which σi are the 2x2 Pauli matrices and I is the 2x2 identity matrix.
The γμ matrices satisfy the Lorentz group anticommutation relation : {γμ, γν} = γμ γν + γν γμ = 2 gμν I avec gμν = Minkowski metric and I = 4x4 Identity matrix.
dμ are the partial derivatives with respect to the coordinates xμ = (ct, x1, x2, x3) in Minkowski spacetime.
γμ dμ implies a summation over the index μ according to the Einstein summation convention.
The Dirac equation therefore relates the dynamical properties of the particle (energy and momentum via the differential operator i h' γμ dμ) to its proper mass m.
See example of calculation using the Dirac method.
2. Klein-Gordon equation for relativistic particles with spin 0 (example : Higgs boson).
3. Proca equation for relativistic particles with spin 1 (examples : photon, W and Z bosons).
4. Rarita-Schwinger equation for relativistic particles with spin 3/2 (example : gravitino).
5. Generalizations of the Rarita-Schwinger equations for relativistic particles with half-integer spin greater than 3/2.
6. Linearized Einstein equations for the massless spin-2 graviton.
All these equations are invariant under the Lorentz-Poincaré transformation, which makes them compatible with restricted relativity.
However, the interpretation of these equations within the framework of a single-particle theory leads to certain inconsistencies. This is why Quantum Field Theory is essential as a more general and coherent framework.
D7.5.3. Heisenberg-type methods :
These approaches study the temporal evolution of observables, while the quantum state remains fixed.
They are used particularly in quantum operator calculations and quantum field theory. They follow the Heisenberg equation :
(H1) dA/dt = (i/h') [H, A] + DA/Dt
where :
A is the operator A(t) that represents the observable to be studie (e.g., position x(t) or momentum p(t))
H is the Hamiltonian operator of the system
[H, A] is the commutator between H and A, defined by : [H, A] = H A - A H
DA/Dt is the explicit derivative of the operator A in Schrödinger representation, obtained by differentiating only its time part. For example, if A(t) = cos(ω t) X, then DA/Dt = -ω sin(ω t) X
See example of calculation using the Heisenberg method.
D7.5.4. Dirac-type methods :
These approaches combine the time evolution of the quantum state and observables by describing the interactions between relativistic particles within the framework of quantum field theory, particularly in quantum electrodynamics through perturbative calculations and Feynman diagrams.
These approaches use an extended version of the relativistic Dirac equation for a free spin-1/2 particle.
D7.5.5. Advanced Algebraic Methods :
These general methods, based on algebraic tools (matrices, operators, Lie algebras), are used when it is difficult or even impossible to exactly solve the equations of motion, whether differential (Schrödinger-type), matrix (Heisenberg-type) or interactional (Dirac-type).
They allow the solution of simple systems (atoms, small molecules) or more complex systems (large molecules, materials) by providing exact or approximate solutions.
These include :
- Perturbation theory, which is used when the system can be considered a modification of a solvable case.
- Variational method, which provides an estimate of the energies of the bound states by minimizing a trial function.
- Ab initio approaches, which approximately solve the Schrödinger equation from first principles, without resorting to empirical parameters.
D7.5.6. Numerical methods :
These methods are used when it is impossible to obtain analytical solutions to equations.
These include the Monte Carlo method, the finite difference method, and the Lanczos method.
D7.6. Example of calculation using the continuous Schrödinger method :
The following example illustrates quantum computations using the Schrödinger equation.
Assumptions :
We consider a free, non-relativistic, massless, spinless particle located in a one-dimensional infinite potential well.
The time evolution of the wave function Ψ is then given by the Schrödinger equation (relation L1).
(L1) i h' dΨ(x, t)/dt = -(1/2)(h'2/m)(d2Ψ(x, t)/dx2) + V(x) Ψ(x, t)
When we are looking for the eigenstates Ψj of the system (numbered by the index j), the wave function Ψj(x,t) can be decomposed into a stationary spatial part and an oscillating temporal part, in the form :
(L2) Ψj(x,t) = Ψj(x) T(t)
The relation (L1) then becomes :
(L3) i h' (1/T) dT/dt = -(1/2)(h'2/m) (1/Ψj(x)) (d2Ψj(x)/dx2) + V(x)
Since the left-hand side depends only on t and the right-hand side only on x, they are equal to a constant Ej (total energy of the particule in the eigenstate Ψj).
Spatial equation :
Consequently, relation (L3) becomes :
(S1) -(1/2)(h'2/m) (d2Ψj(x)/dx2) + V(x) Ψj(x) = Ej Ψj(x)
If the infinite potential well is of width L, we also have :
(S2) V(x) = 0 for 0 < x < L and V(x) = ∞ for x ≤ 0 or x ≥ L
Inside the well, equation (S1) then becomes :
(S3) d2Ψj(x)/dx2 + kj2 Ψj(x) = 0
with : kj = (2 m Ej)1/2 / h'
The general solution is therefore :
(S4) Ψj(x) = A sin(kj x) + B cos(kj x)
where A and B are arbitrary constants.
For x = 0, Ψj(x) = 0, hence B = 0
For x = L, Ψj(x) = 0, hence A sin(kj L) = 0, which requires discrete values for kj and Ej such that :
(S5) kj = j π/L
(S6) Ej = (1/2)(h' kj)2 /m = (1/2)(j π h'/L)2 /m
with j being a non-zero positive integer (Ψj cannot be zero everywhere in the well).
Finally, we obtain :
(S7) Ψj(x) = A sin(j π x/L)
It remains to calculate A taking into account the normalization condition of Ψj(x, t) :
(S8) ∫0 to L [|Ψj(x)|2 dx] = 1
Given relation (S7) and the trigonometric identity : sin2(θ) = (1/2) (1 - cos(2 θ)), we then obtain :
1 = ∫0 to L [(1/2) A2 (1 - cos(2 j π x/L)) dx] = (1/2) A2 (∫0 to L [dx] - ∫0 to L [cos(2 j π x/L) dx])
The first integral is [x]0 to L = L
The second integral is (L/(2 j π)) [sin(2 j π x/L)]0 to L = 0
Hence : A = ±(2/L)1/2
We conventionally take the positive value of A because a wave function can always be multiplied by a global phase factor (such as -1 or even eiθ) without changing the physical predictions.
Relation (S7) then becomes :
(S9) Ψj(x) = (2/L)1/2 sin(j π x/L) with j a non-zero positive integer
Time equation :
Taking into account the constant Ej, relation (L3) becomes :
(T1) dT(t)/dt + i Qj T(t) = 0
with :
(T2) Qj = Ej/h' = (j π/L)2 h'/(2 m)
which has the following solution :
(T3) T(t) = Cj e-i Qj t
with Cj being an arbitrary complex constant equal to any phase factor of modulo 1 (Cj = ei θj with θj being a real number) in order to satisfy the normalization condition for Ψj(x, t) (Relation S8).
To simplify the calculations, we conventionally take Cj = 1. But this choice is arbitrary and must be revised if interferences or superpositions of states are studied, because the relative phases between different states play a crucial role in these situations.
Complete wave function :
Given relations (L2)(S9)(T3), the complete normalized wave function is therefore :
(C1) for 0 < x < L : Ψj(x,t) = Ψj(x) T(t) = (2/L)1/2 sin(j π x/L) ei θj e-i Qj t with j a non-zero positive integer ; otherwise : Ψj(x,t) = 0
For any value of j, there are always two nodes at the ends of the well (x = 0 and x = L) and (j - 1) additional nodes inside the well.
The ground state (j = 1) corresponds to : Ψ1(x,t) = (2/L)1/2 sin(π x/L) ei θ1 e-i Q1 t with two nodes at the ends of the well.
The first excited state (j = 2) corresponds to : Ψ2(x,t) = (2/L)1/2 sin(2 π x/L) ei θ2 e-i Q2 t with an additional node.
etc.
Probability amplitudes :
The probability density ρ is then written :
(D1) ρ = |Ψj(x,t)|2 = (2/L) sin2(j π x/L)
Note that this probability density is independent of time, which characterizes a stationary state.
Measurements of the observable :
Given relation (T2), the energy Ej of the particle in the eigenstate Ψj(x) is written :
(O1) Ej = (j π h'/L)2/(2 m) = (j h/L)2/(8 m) with j a non-zero positive integer
Conclusion :
- If the particle is in an eigenstate Ψj(x), its measured energy will always be Ej.
- If the particle is in a superposition of eigenstates Ψj(x) = ∑j [cj Ψj(x)], its measured energy will be Ej with a probability |cj|2 where cj is the superposition coefficient.
D7.7. Example of calculation using the discretized Schrödinger method :
The following example is the same as the previous one assuming that the wave function Ψ(x) is discretized on a set of N points {x1, ..., xi, ..., xN} inside the interval [0, L].
Discretized Schrödinger equation :
Let j be the index which numbers the different eigenstates Ψj of the quantum system.
Be careful not to confuse the indices i and j : The index j, although limited to N in this discretized representation, is not intrinsically linked to the index i of the spatial discretization points and could, in principle, extend to infinity in a continuous quantum system. This distinction is fundamental to avoid any confusion between the physical nature of quantum states (indexed by j) and their discrete digital representation (indexed by i).
The second-order Taylor expansion of Ψj(xi+1) and Ψj(xi-1) is written :
Ψj(xi+1) = Ψj(xi) + Δ dΨj(xi)/dx + (1/2) Δ2 d2Ψj(xi)/dx2
Ψj(xi-1) = Ψj(xi) - Δ dΨj(xi)/dx + (1/2) Δ2 d2Ψj(xi)/dx2
where Δ is the spacing between the discretized points.
By adding these two relations, we obtain the second derivative d2Ψj(x)/dx2 in discretized form as follows :
d2Ψj(xi)/dx2 = (Ψj(xi+1) - 2 Ψj(xi) + Ψj(xi-1)) / Δ2
In the example, this gives :
- spacing Δ = L/(N + 1)
- left limit : Ψj(x0) = Ψj(0) = 0
- right limit : Ψj(xN+1) = Ψj(L) = 0
The Schrödinger equation (relation S1) in its discretized form then becomes a matrix equation for any eigenstate Ψj :
f H Ψj = Ej Ψj
with :
f = multiplicative factor = -(h'/Δ)2/(2 m)
Ψj = column vector of components Ψj(xi)
Ej = eigenenergy associated with the eigenstate Ψj
H = hermitian matrix =
(-2 1 0 0 ... 0 0 0)
( 1 -2 1 0 ... 0 0 0)
( 0 1 -2 1 ... 0 0 0)
( 0 0 1 -2 ... 0 0 0)
( . . . . . . .)
( 0 0 0 0 ... -2 1 0)
( 0 0 0 0 ... 1 -2 1)
( 0 0 0 0 ... 0 1 -2)
It now remains to find the N eigenvalues λj of the matrix H and the N associated eigenvectors Ψj. Two methods exist :
Resolution by diagonalization of H :
This consists of finding, generally using numerical methods, two matrices D and P such that : H = P D P-1
in which :
- D is a diagonal matrix containing the eigenvalues of H on its diagonal
- P is a matrix whose columns are the eigenvectors associated with the eigenvalues
- P-1 is the inverse matrix of P
- Note that for a real symmetric matrix like H, then P is orthogonal (P-1 = PT).
Resolution by direct calculation :
The direct calculation of the N eigenvalues λj of the matrix H, generally using numerical methods, consists of solving the characteristic equation : det(H - λ I) = 0 where I is the identity matrix.
More quickly, since the matrix H is tridiagonally symmetric with a = -2 on the main diagonal and b = 1 on the two side diagonals, we obtain the solution : λj = a + 2 b cos(j π/(N + 1)). See demonstration below.
The eigenenergies Ej are then related to the eigenvalues λj by the relation : Ej = f λj
These energies represent the possible outcomes of a measurement of the system's energy.
The direct calculation of the N eigenvectors Ψj then consists of solving the system : (H - λ I) Ψj = 0, for example using the Gaussian method.
More quickly, since the matrix H is tridiagonally symmetric, the components i of the jth normalized eigenvector are given by : Ψji = (2/(N + 1))1/2 sin(j i π/(N + 1)). See demonstration below.
These eigenvectors describe the eigenstates of the particle, each eigenvector representing the spatial probability distribution of the particle for the corresponding energy. This means that |Ψji|2 = (2/(N + 1)) sin2(j i π/(N + 1)) gives the probability of finding the particle at position xi for the energy state Ej.
The normalization condition : ∑i |Ψji|2 = 1 for each eigenstate Ψj is then automatically verified.
Comparison with the continuous case solution :
Replacing i with x/Δ = x (N + 1)/L in the expression for Ψji, we obtain :
Ψj(x) = (2/(N + 1))1/2 sin(j π x/L)
which is equivalent to the spatial part of Ψj(x,t) in relation C1 to within a normalization factor ((2/(N + 1))1/2 instead of (2/L)1/2).
This difference reflects the different nature of the normalization involving a probability density in the continuous case, and point probabilities in the discrete case.
Concerning the energy Ej, it is written : Ej = f λj = f (-2 + 2 cos(j π/(N + 1)))
When N is large, given the limited expansion of the cosine around 0 : cos(α) = 1 - (1/2)α2 + o(α2), the expression Ej becomes : Ej = -f (j π/(N + 1))2
Given Δ = L/(N + 1), we also have : f = -(h'/Δ)2/(2 m) = -(h' (N + 1)/L)2/(2 m)
Replacing f in Ej, we then obtain : Ej = (j π h'/L)2/(2 m) which is equivalent to the energy Ej of the continuous case (relation O1).
These two equivalences show the consistency between the continuous and discrete approaches for this example of calculation.
Proof of : λj = a + 2 b cos(j π/(N + 1)) :
To simplify the notation, we set vi = Ψji
The equation (H - λ I) Ψj = 0 gives the following relation (R1) for each component i such that 2≤ i ≤ N - 1 :
(R1) b vi-1 + a vi + b vi+1 - λ vi = 0
We assume the solution is of the form vi = sin(i θj) where θj is a parameter to be determined.
Given the trigonometric identity : sin((i ± 1) θj) = sin(i θj) cos(θj) ± cos(i θj) sin(θj), relation (R1) then simplifies to :
(R2) λ = a + 2 b cos(θj)
The boundary conditions : v0 = vN + 1 = 0 then give :
v0 = 0 = sin(0 θj), which is satisfied.
vN + 1 = 0 = sin((N + 1) θj), which requires :
(R3) θj = j π/(N + 1)
Hence the complete solution : λj = a + 2 b cos(j π/(N + 1))
Proof of : Ψji = (2/(N + 1))1/2 sin(j i π/(N + 1))
Let Cj be a positive normalization factor such that : Ψji = Cj sin(i θj)
The norm of the vector Ψj must be equal to 1, which imposes :
1 = ∑i Ψji2 = Cj2 ∑i sin2(i θj)
It remains to calculate : ∑i sin2(i θj)
Given the trigonometric identity : sin2(α) = (1/2) (1 - cos(2 α)) and Euler's formula for the complex exponential : exp((-1)1/2 α) = cos(α) + (-1)1/2 sin(α), we obtain :
(R4) ∑i sin2(i θj) = (1/2) (N - S)
with : S = Real_part[∑i exp(2 i θj (-1)1/2)]
The sum of the exponentials forms a geometric series with common ratio r = exp(2 θj (-1)1/2) and first term r, which is written :
S = Real_part[r (1 - rN)/(1 - r)]
We also have :
r = exp(2 θj (-1)1/2) = exp(2 j π/(N + 1) (-1)1/2)
Let us assume that r = 1. This would imply that its argument is an integer multiple of 2 π, i.e. : 2 j π/(N + 1) = 2 k π with k integer, or after simplification : j = k (N + 1) which is impossible (since 0 < j < N + 1). The denominator (1 - r) is therefore never zero.
We also have :
rN = exp(2 θj (-1)1/2)N = exp((-1)1/2 2 j π N/(N + 1)) = exp((-1)1/2 2 j π (1 - 1/(N + 1)) = exp((-1)1/2 2 j π) / exp((-1)1/2 2 j π/(N + 1)) = 1/r
Hence finally : S = Real_part[-1] = -1
By transferring this value into relation (R4), we obtain :
∑i sin2(i θj) = (1/2) (N + 1)
Cj2 = 2/(N + 1)
Hence the normalized form of the vector Ψj :
Ψji = (2/(N + 1))1/2 sin(j i π/(N + 1))
D7.8. Example of calculation using the Heisenberg method :
The following example illustrates quantum computations using the Heisenberg equation.
Consider a one-dimensional quantum harmonic oscillator of mass m and angular frequency ω (relative to the spring stiffness).
The time evolution of any observable a(t) is then given by the Heisenberg equation (relation H1) in the form :
(H1) dA/dt = (i/h') [H, A] + DA/Dt
with :
A = operator A(t) which represents the observable a(t)
X = position operator
P = -i h' d/dx = momentum operator
(H2) H = P2/(2 m) + V(X) = Hamiltonian H of the system
V(X) = (1/2) m ω2 X2 = potential for a harmonic quantum oscillator
[H, A] = H A - A H = commutator between H and A
DA/Dt = explicit derivative of the operator A in Schrödinger representation
We are trying to calculate the two operators A(t) = X(t) and A(t) = P(t).
These are two basic operators whose mathematical definition does not explicitly depend on time, so : DA/Dt = 0
Calculation of [P, X] :
[P, X] Ψ = P (X Ψ) - X (P Ψ) = -i h' d(x Ψ)/dx - x (-i h' dΨ/dx) = -i h' (Ψ + x dΨ/dx) + i h' x dΨ/dx = -i h' Ψ
So [P, X] = -i h'
Calculation of [P2, X] :
Given the commutator property : [AB, C] = (AB)C - C(AB) = (ABC - ACB) + (ACB - CAB) = A[B, C] + [A, C]B, we can write :
(H3) [P2, X] = P[P, X] + [P, X] P = -2 i h' P
Calculation of [X2, P] :
(H4) [X2, P] = X [X, P] + [X, P] X = 2 i h' X
Calculation of [H, A] for A = X :
Given relation (H2), this can be written :
[H, X] = (1/(2 m)) [P2, X] + (1/2) m ω2 [X2, X]
Given relation (H3) and the property [X2, X] = 0, this can finally be written :
[H, X] = -i h' P/m
And the relation (H1) then becomes :
(H5) dX/dt = P/m
Calculation of [H, A] for A = P :
Given relation (H2), this is written :
[H, P] = (1/(2 m)) [P2, P] + (1/2) m ω2 [X2, P]
Given the property [P2, P] = 0 and relation (H4), this is finally written :
[H, P] = i h' m ω2 X
And relation (H1) then becomes :
(H6) dP/dt = -m ω2 X
Relations (H5)(H6) constitute a system of coupled differential equations whose solution is as follows :
X(t) = X(0) cos(ω t) + (1 /(m ω)) P(0) sin(ω t)
P(t) = P(0) cos(ω t) - m ω X(0) sin(ω t)
The operators X and P therefore oscillate harmonically at frequency ω depending on the initial conditions X(0) and P(0).
Note that we find the classical relation P(t) = m dX(t)/dt which is valid in the case of the example and in certain simple systems in Heisenberg representation, but which is not universal for all quantum problems. This relation is notably valid for any H of the form : P2/(2 m) + V(X) with V(X) combination of polynomial terms (V(X) = a + b X + c X2 + ...), given that [Xn, X] = 0 for any positive or zero integer n.
D7.9. Example of calculation using the Dirac method :
The following example illustrates quantum computations using the Dirac equation.
We consider a free Dirac fermion (without external interaction), described by a relativistic wave function.
The time evolution of the wave function Ψ is then given by the Dirac equation (relation D1) in the form :
(D1) (i h' γμ dμ - m c I4) Ψ(ct, x) = 0
which is explained as follows :
(D1*) i h' γ0 dΨ/d(ct) = (-i h' γk dk + m c I4)Ψ(ct, x)
with :
k = 1, 2, 3 (spatial components)
m = mass of the particle at rest.
I4 = 4x4 Identity matrix
γμ = specific 4x4 matrices introduced by Dirac.
x = 3D space vector = (x1, x2, x3)
dμ = partial derivatives with respect to the coordinates xμ = (ct, x1, x2, x3) in Minkowski spacetime.
In the case where we are looking for the eigenstatesΨj of the system (numbered by the index j), the wave function Ψj(ct, x) can be decomposed into a stationary spatial part and an oscillating temporal part, in the form :
(D2) Ψj(ct, x) = Ψj(x) T(ct)
The relation (D1*) then becomes :
(D3) i h' (1/T(ct)) dT(ct)/d(ct) γ0 Ψj(x) = (-i h' γk dk + m c I4) Ψj(x)
Multiplying both sides on the left by the matrix γ0, and taking into account the property (γ0)2 = I4, the relation D3 becomes :
(D4) i h' (1/T(ct)) dT(ct)/d(ct) I4 Ψj(x) = γ0 (-i h' γk dk + m c I4) Ψj(x)
For this relation to be true regardless of t and x, it is necessary that both sides be matrix operators acting identically on Ψj(x), therefore proportional to the Identity matrix (I4) with a common scalar factor Ej/c (the total relativistic energy of the particle in the eigenstate Ψj).
Spatial equation :
Consequently, relation (D4) becomes :
(DS1) γ0 (-i h' γk dk + m c I4) Ψj(x) = (Ej/c) I4 Ψj(x)
We then seek a solution in the form of a traveling plane wave :
(DS2) Ψj(x) = uj(pj) exp(i pj.x/h')
where :
pj = (p1, p2, p3) is the relativistic momentum vector in the inertial reference frame, expressed in 3D space = m γj vj where γj is the Lorentz factor (γj = (1 - vj2/c2)-1/2).
uj(pj) is a spinor associated with pj
pj.x is the 3D spatial scalar product = pjk xk
Given the spatial derivative dkexp(i pj.x/h') = (i pk/h') exp(i pj.x/h'), relation (DS1) simplifies to :
(DS3) γ0 (γk pk + m c I4) uj(pj) = (Ej/c) I4 uj(pj)
We then decompose the spinor uj(pj) into two Pauli spinors Φj and Χj such that :
(DS4) uj(pj) = (Φj, Χj)T
Given the expression for the matrices γ0 and γk, the DS3 relation then becomes a coupled system of two equations :
(DS5) σk pk Χj = (Ej/c - m c) Φj
σk pk Φj = (Ej/c + m c) Χj
Substituting Χj into the first equation, and taking into account the expression for the Pauli matrices σ, we obtain :
(Ej/c - m c)(Ej/c + m c) Φj = (σk pk)2 Φj = pj2 I2 Φj
Hence the expression of the observable Ej :
(DS6) Ej = ±(pj2 c2 + m2 c4)1/2
This so-called "dispersion" relationship corresponds to particles (electrons) with positive energy Ej and to antiparticles (positrons) with negative energy Ej.
It now remains to express Φj and Χj to find uj(pj)
For the electron (Ej > 0), the second equation DS5 gives :
(DS7) Χj = Φj σk pk/(Ej/c + m c)
If * denotes the complex conjugate without transposition and if wj is an arbitrary normalized Pauli spinor (such that (wj*)T.wj = 1), for example (1, 0)T for a spin oriented along the +z axis, then Φj can be chosen as follows :
Φj = ((Ej/c + m c)/(2 m c))1/2 wj to verify the normalization relation : (uj*(pj))T γ0 uj(pj) = 2 m c
For the positron (Ej < 0), the first equation DS5 gives :
(DS8) Φj = -Χj σk pk/(|Ej|/c + m c)
and Χj can be chosen as follows :
Χj = ((|Ej|/c + m c)/(2 m c))1/2 wj
Time equation :
Taking into account the constant (Ej/c) I4, relation (D4) becomes :
(DT0) i h' (1/T(ct)) dT(ct)/d(ct) I4 Ψj(x) = (Ej/c) I4 Ψj(x)
or :
(DT1) dT(ct)/d(ct) + i Qj T(ct) = 0
with :
(DT2) Qj = Ej/(c h')
which has the following solution :
(DT3) T(ct) = Cj exp(-i Qj ct) = Cj exp(-i Ej t/h')
with Cj being an arbitrary complex constant, which is equal to any phase factor of modulo 1 (Cj = ei θj with θj being a real number) in order to satisfy the normalization condition for Ψj(ct, x).
To simplify the calculations, we conventionally take Cj = 1. But this choice is arbitrary and must be reviewed if interferences or superpositions of states are studied, because the relative phases between different states play a crucial role in these situations.
Complete wave function :
Given the relationships (D2)(DS2)(DS4)(DS7)(DS8)(DT3), the complete normalized wave function for any eigenstate j is therefore :
(DC1) Ψj(t, x) = (Φj, Χj)T exp(i θj) exp(i pj.x/h') exp(-i Ej t/h')
This elementary solution Ψj(t, x) describes a free spin 1/2 fermion in the form of a relativistic plane wave, whose energy and momentum satisfy the relation Ej2 = pj2 c2 + m2 c4.
By superposition of these eigenstates (wave packets), it constitutes the basis of the relativistic quantum description, integrating the prediction of antimatter via negative-energy solutions.
Probability amplitudes :
Taking into account relation DS2, the probability density ρ is then written :
(DD1) ρ = (Ψj*(x))T Ψj(x) = (uj*(pj))T uj(pj)
D7.10. Similarities between quantum mechanics and classical mechanics :
Classical mechanics, which is deterministic and based on well-defined trajectories, contrasts with quantum mechanics which is based on probabilities, superpositions of states and other fundamental principles specific to its field.
However, despite these fundamental conceptual differences, the two approaches share several similarities :
- Description of the isolated system in terms of objects (examples : particles, solids) and internal or external interactions (examples : forces, fields, quantum interactions)
- Intrinsic data (or properties) (examples : mass, dimensions, spin, quantized energy levels of an atom)
- Observables (examples : position, velocity, energy, wave function, eigenvalue)
- Initial conditions (at t = 0) and boundary conditions of the system (example : boundary conditions for a vibrating string)
- Conservation of certain physical quantities (examples : momentum, angular momentum, energy)
- Universal physical constants (examples : gravity acceleration (g), light speed (c), Planck's constant (h))
- Resolution framework (examples : laws of dynamics, Schrödinger equation)
- Certain mathematical formalisms (examples : differential and integral calculus, linear algebra, differential equations)
- Calculation of observables (with deterministic or probabilistic results)
- Evolution of the system over time (examples : trajectories, wave function)
- Analysis and interpretation of results by comparing theoretical predictions and experimental observations, allowing models to be validated or adjusted.
D7.11. Sources relative to quantum physics :
[ASP] Alain Aspect, Si Einstein avait su, Odile Jacob, 2025.
[BOU] Alain Bouquet, Noyaux et particules.
[COH] Cohen-Tannoudji, Mécanique quantique, Tome I, CNRS Editions
[CHA] ChatGPT, le moteur d'Intelligence Artificielle développé par OpenAI.
[PER] Perplexity, le moteur d'Intelligence Artificielle développé par Perplexity AI.
See detail.
E4.1. Introduction :
Genetics is the science that studies the laws of heredity, that is, the transmission of traits and genetic information from parents to their offspring.
It includes six main categories :
1. Cell and Genome Organization
Structure and organization of genetic material (genes, chromosomes, etc.) within the cell.
2. Genetic Regulation
Control of gene expression without modification of the DNA sequence. The main factors are as follows :
- Non-coding DNA sequences (not translated into proteins), which are the anchoring platforms for proteins, providing spatial and temporal control of the gene expression.
- Regulatory proteins, which are of two main types : histones which compact the DNA around them, and transcription factors which bind to non-coding sequences to activate or repress gene expression.
- Long non-coding RNAs, which act as indirect regulators, for example, guiding proteins to specific regions of DNA.
- Epigenetic modifications, which are reversible additions of chemical groups to DNA molecules and histones, thus altering DNA accessibility.
3. Genetic Variation
Production of genetic differences. The main mechanisms are as follows :
- Mutation : Permanent modification of the DNA sequence at the gene, chromosome, or entire genome level.
- Recombination (crossing-over) : Exchange of DNA segments between homologous chromosomes during meiosis, creating new genetic combinations.
- Polymorphism : Coexistence of several normal genetic forms in a population, as demonstrated by the ABO blood group system in humans, which produces a diversity of groups (A, B, AB, O, etc.).
- Transposition : Change in the position of a DNA segment within the genome.
- Genetic drift : Random change in the frequencies of different forms of a gene in a small population, often due to chance events (catastrophes, isolation) that reduce the number of individuals.
- Migration (or gene flow) : movement of individuals from one population to another, resulting in the arrival or departure of certain gene forms in the host population.
- Random mating : reproduction where partners are chosen at random, promoting genetic mixing within the population.
4. Genetic Repair
Correction of errors or damage in DNA. The main mechanisms, classified by increasing severity of the damage, are as follows :
- Direct repair : For a slightly damaged base
- Base excision repair (BER) : For an absent or severely damaged base
- Copy error repair (MMR) : For an erroneous sequence following DNA replication
- Nucleotide excision repair (NER) : For a damaged sequence
- Single-strand break repair (SSB repair) : For a broken strand, with the intact second strand serving as a template
- Double-strand break repair (DSB repair) : For a broken double strand
5. Genetic Transmission
Transmission of hereditary information at the individual level (between cells of the same organism) or transgenerationally (from one generation to the next). The main mechanisms are as follows :
- Replication : Faithful copying of DNA before each cell division.
- Mitosis : Cell division of somatic (body) cells producing two identical daughter cells with the same number of chromosomes as the parent cell.
- Meiosis : Cell division of germ (reproductive) cells producing two gametes (egg and sperm), each containing half the number of chromosomes. This division does not fragment the genetic information but intelligently redistributes it between the gametes through shuffling and control mechanisms, allowing fertilization to reconstitute a complete and unique genetic heritage.
- Fertilization : Fusion of two gametes (egg and sperm) to form an egg cell (zygote) with a full chromosome number.
- Mendelian inheritance : Transmission of hereditary traits via genes located on nuclear chromosomes (therefore inherited from both parents), according to Mendel's laws.
- Non-Mendelian inheritance : Transmission of hereditary traits that do not follow Mendel's laws, either due to the location of genes outside the nuclear chromosomes (such as mitochondrial DNA inherited only from the mother, or chloroplast DNA), or due to specific mechanisms affecting nuclear genes (such as parental imprinting, incomplete dominance, dynamic mutation, sex-linkage).
- Epigenetic transmission : Transmission, at the individual or transgenerational level, of regulatory marks of gene expression, without modification of the DNA sequence.
6. Evolution and Natural Selection
Given the hereditary variations between individuals, natural selection favors certain traits within this diversity : individuals carrying genetic characteristics that are advantageous for their survival and reproduction in a given environment produce more offspring.
This concept of natural selection evolved in four stages :
- Lamarck (1809) : The giraffe lengthens its neck to reach high leaves and passes this acquired trait on to its offspring.
- Darwin (1859) : In giraffes, those born with longer necks through random individual variation survive better and reproduce more, thus producing more offspring.
- Neo-Darwinism (1930-1940) : In giraffes, those born with longer necks due to random genetic mutations survive better and reproduce more, thus producing more offspring.
- Epigenetics (1942) : In giraffes, in addition to random genetic mutations, difficulty reaching high leaves could alter the expression of genes involved in neck growth, without changing the DNA sequence. These epigenetic modifications could be temporarily passed on to their offspring.
E4.2. Organization of the cell and the genome :
A hierarchical description of the eukaryotic cell (with nucleus) is given as follows [PER][CHA] :
Note : The number shown is relative to the adult human being.
Eukaryotic cell = Plasma membrane + Cytoplasm + Nucleus. Number = 37 trillion
|
| Plasma membrane = Envelope separating the interior and exterior of the cell.
|
| Cytoplasm = Cytosol + Organelles
| |
| | Cytosol = Aqueous medium containing ions, nutrients, and enzymes.
| |
| | Organelles = Free ribosomes + Endoplasmic reticulum + Mitochondria + Golgi apparatus + Lysosomes + Cytoskeleton
| | | Free ribosomes = synthesize proteins from messenger RNA. Number = 10 million
| | | Endoplasmic reticulum (ER) = Synthesizes lipids (smooth ER) and other proteins (rough ER, with attached Ribosomes).
| | | Mitochondria = Produce cellular energy (ATP) through cellular respiration. Contain their own DNA (mitochondrial genome). Number = 100 to 10,000 (according to cell type).
| | | Golgi apparatus = Sorts, packages and transports molecules (mainly proteins and lipids) to their final destination (internal or external to the cell).
| | | Lysosomes = Digest cellular waste. Number = 300 to 500
| | | Cytoskeleton = Maintains cell shape and participates in cell movement.
|
| Nucleus = Nuclear membrane + Nucleoplate + Chromatin + Nucleoli
| |
| | Nuclear membrane = Double envelope separating the interior and exterior of the nucleus.
| |
| | Nucleoplate = Gelatinous medium containing enzymes and ions.
| |
| | Chromatin = Nuclear genome + Regulatory proteins + Associated RNAs + Epigenetic modifications
| | |
| | | Nuclear genome = Set of Chromosomes. Number = 23 pairs of chromosomes.
| | | | Chromosome = DNA molecule (double helix consisting of two antiparallel strands of Nucleotides) = Genes + Non-coding sequences
| | | | | Gene = DNA sequence containing the information necessary for the synthesis of either a protein (coding gene) or a functional RNA (non-coding gene). Number = 20,000 coding genes per nuclear genome.
| | | | | Non-coding sequences = DNA sequences not translated into proteins, which are the anchoring platforms for Regulatory proteins, providing spatial and temporal control of the gene expression. Proportion = 98 % of the nuclear genome
.
| | | | | Nucleotide = Phosphate group + Deoxyribose sugar + Nitrogenous base. Number = several million per chromosome
| | | | | | Nitrogenous base = Adenine (A), Thymine (T), Cytosine (C), or Guanine (G). Number = 3 billion pairs of nitrogenous bases per nuclear genome.
| | |
| | | Regulatory proteins = two main types : histones which compact the DNA around them, and transcription factors which bind to non-coding sequences to activate or repress gene expression.
| | |
| | | Associated RNAs = Main mediators of gene expression
| | | | Messenger RNA (mRNA) = Copies a coding Gene (the blueprint for protein production) and transports it to the Ribosomes.
| | | | Transfer RNA (tRNA) = Transports amino acids to the Ribosomes and positions them according to the provided blueprint.
| | | | Ribosomal RNA (rRNA) = Forms the backbone of the Ribosomes and assembles amino acids into protein chains according to the provided blueprint.
| | | | microRNA = blocks specific messenger RNA, preventing their translation and thus reducing the production of target proteins.
| | | | Long non-coding RNA (IncRNA) = Indirect regulator, for example, guiding proteins to specific regions of DNA.
| | |
| | | Epigenetic modifications = Reversible additions of chemical groups to DNA molecules and histones, thus altering DNA accessibility.
| |
| | Nucleoli = Synthesize ribosomal RNAs. Number = 1 to 5
E4.3. Epigenetics :
Epigenetics is a fundamental biological mechanism that continuously regulates gene expression based on context (physiological or environmental variations), without altering the DNA sequence, unlike genetic mutations.
This regulation is achieved through reversible chemical modifications, such as :
- DNA methylation, by adding methyl groups to the DNA.
- Modification of regulatory proteins (histones) around which DNA is wrapped.
These modifications are triggered by various internal or external stimuli, including :
- Climate variations (temperature, humidity)
- Dietary changes (deficiency, excess, nutritional quality)
- Habitat changes (urbanization, pollution, migration)
- Physiological stress (illness, intense exercise)
- Traumatic stress (emotional shock, violence)
These modifications dynamically control access to genes and their expression levels, allowing their activation or repression without altering the underlying genetic content.
Two levels of plasticity are distinguished :
1. Individual plasticity : When stimuli are moderate or transient, the changes remain confined to somatic (body) cells and are not transmitted to subsequent generations.
For example :
- The Arctic fox changes its coat (white in winter and brown in summer). Fox cubs are born with a coat adapted to their season of birth, but their ability to change color then depends on environmental stimuli (study by Zimova M., Mills L.S., Nowak J.J., 2016).
- In humans, skin tanning in response to the sun is a temporary adaptation specific to each individual (study by Slominski A., 2004).
2. Transgenerational plasticity : When stimuli are intense or prolonged, the changes also affect germ cells (sperm and oocytes). These marks are then transmitted to offspring and influence their phenotypes (behavioral, physiological, and morphological traits) for one to three generations before fading.
For example :
- In water fleas (Daphnia), parental exposure to predator signals induces the development of morphological defenses (helmets or spikes) in the offspring, even if the young have never encountered a predator themselves (study by Tollrian R., 1995).
- Male mice exposed to prolonged cold give birth to offspring better adapted to low temperatures (study by Chan J.C., 2020).
- Male mice conditioned to fear an odor transmit a specific hypersensitivity to that odor to their offspring (study by Dias & Ressler, 2013).
- In humans, malnutrition before pregnancy in the mother or father can have a lasting effect on the child's health, even if the child subsequently grows up in a normal food environment (study by Gete DG., Waller M., Mishra GD., 2020).
Conversely, activities such as music or cognitive learning, even when practiced intensively, have not yet demonstrated any hereditary impact.
E4.4. ARN :
RNA (ribonucleic acid) is a molecule essential for genetic organization and regulation.
It copies genetic information from DNA (present in the cell nucleus) and transmits it to ribosomes (in the cell cytoplasm). The latter then "read" this messenger RNA as a blueprint for producing proteins (coding RNA).
Once produced, proteins are transported to their sites of action (cells, organs, tissues). Some, like hormones, travel through the blood to reach distant organs.
These proteins ensure the body's functioning. They are involved in genetic regulation (histones and transcription factors), hormonal regulation (hormones), digestion (enzymes), transport of oxygen in the blood (hemoglobin), immune defense (antibodies), and tissue repair (muscles, skin, nails).
Other types of RNA (non-coding) do not contribute to protein production. They are functional and regulate gene expression :
- by blocking specific messenger RNAs to prevent their translation (e.g., microRNAs), reducing the production of target proteins.
- by controlling access to DNA through chemical modifications without altering its sequence (e.g., long non-coding RNAs).
E4.5. Sources relative to genetics :
[CHA] ChatGPT, le moteur d'Intelligence Artificielle développé par OpenAI.
[PER] Perplexity, le moteur d'Intelligence Artificielle développé par Perplexity AI.
See detail.
Last page update : 19 April, 2025.