Physics

Differential Geometric Methods in Theoretical Physics: by Konrad Bleuler (auth.), Ling-Lie Chau, Werner Nahm (eds.)

By Konrad Bleuler (auth.), Ling-Lie Chau, Werner Nahm (eds.)

After numerous many years of lowered touch, the interplay among physicists and mathematicians within the front-line learn of either fields lately grew to become deep and fruit­ ful back. a number of the prime experts of either fields turned serious about this devel­ opment. This method even resulted in the invention of formerly unsuspected connections among quite a few subfields of physics and arithmetic. In arithmetic this issues specifically knots von Neumann algebras, Kac-Moody algebras, integrable non-linear partial differential equations, and differential geometry in low dimensions, such a lot im­ portantly in 3 and 4 dimensional areas. In physics it issues gravity, string conception, integrable classical and quantum box theories, solitons and the statistical me­ chanics of surfaces. New discoveries in those fields are made at a swift velocity. This convention introduced jointly lively researchers in those components, reporting their effects and discussing with different contributors to additional advance concepts in destiny new instructions. The convention was once attended by means of SO individuals from 15 countries. those complaints rfile this system and the talks on the convention. This convention was once preceded through a two-week summer season university. Ten teachers gave prolonged lectures on comparable themes. The complaints of the varsity can be released within the NATO-AS[ quantity by means of Plenum. The Editors vii ACKNOWLEDGMENTS we wish to thank the numerous those that have made the convention a hit. in addition, ·we have fun with the superb talks. The lively participation of every body current made the convention energetic and stimulating. All of this made our efforts worthy­ while.

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I Les origines de los angeles Th´eorie quantique
I. 1. Les recommendations de l. a. body classique
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(I. 1. 2) Nature ondulatoire de los angeles lumi`ere
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(II. 1. 2) Courant de probabilit´e
(II. 1. three) Valeur moyenne et ´ecart quadratique moyen
(II. 1. four) Op´erateur “impulsion” dans l’espace des coordonn´ees
II. 2. Particule dans un potentiel ind´ependant du temps
(II. 2. 1) strategies stationnaires
(II. 2. 2) Quantification de l’´energie
II. three. l. a. barri`ere de potentiel finie : l’effet tunnel
II. four. Le puits quantique
II. five. L’oscillateur harmonique
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II. 6. Appendice : Fonction g´en´eratrice des polynˆomes d’Hermite et oscillateur harmonique
(II. 6. 1) Orthonormalit´e des fonctions 'n(x) de l’oscillateur harmonique
(II. 6. 2) Valeurs moyennes et probabilit´e de transition
III Fondements de l. a. th´eorie quantique
III. 1. Equation de Schr¨odinger et ses propri´et´es
(III. 1. 1) Spectre de l’op´erateur hamiltonien et aspect de vue du calcul vectoriel
(III. 1. 2) Le vecteur d’´etat de l’espace d’Hilbert E et ses propri´et´es
(III. 1. three) Repr´esentation des coordonn´ees |ri
(III. 1. four) Repr´esentation des impulsions |pi
(III. 1. five) formula matricielle : Repr´esentation des ´etats d’´energie
(III. 1. 6) D´eg´en´erescence d’un niveau d’´energie
III. 2. constitution de l’espace de Hilbert "H et produits tensoriels d’espaces
III. three. Le processus de mesure et sa description quantique
(III. three. 1) Commutateurs et grandeurs physiques simultan´ement mesurables
(III. three. 2) Grandeurs physiques non simultan´ement mesurables : G´en´eralisation des kin d’incertitude
de Heisenberg
III. four. L’´equation d’´evolution
III. five. Les diff´erents sch´emas en m´ecanique quantique
(III. five. 1) Le sch´ema de Schr¨odinger
(III. five. 2) Le sch´ema de Heisenberg
(III. five. three) Le sch´ema d’interaction
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(III. 7. four) Sym´etrie de translation
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(III. nine. 2) M´ethode variationnelle lin´eaire
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(IV. five. 2) Exp´erience de Stern et Gerlach
(IV. five. three) Vecteur d’´etat et op´erateur de spin
(IV. five. four) Pr´ecession du spin dans un champ magn´etique
(IV. five. five) Composition de deux moments angulaires
IV. 6. Appendice : Fonctions sp´eciales associ´ees au second angulaire
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(IV. 6. 2) Les harmoniques sph´eriques
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(V. 1. 1) Potentiel `a sym´etrie sph´erique
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V. 2. L’atome hydrog´eno¨ıde
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V. three. constitution wonderful des atomes alcalins
(V. three. 1) Interactions spin-orbite
(V. three. 2) Corrections relativistes
V. four. Effet de Zeeman des atomes alcalins
(V. four. 1) Atome plac´e dans un champ magn´etique quelconque
(V. four. 2) Effet Zeeman anomal
(V. four. three) Effet Paschen-Back
V. five. Etats quantiques de los angeles mol´ecule diatomique
V. 6. Appendice : Propri´et´es des fonctions sp´eciales de l’atome hydrog´eno¨ıde
(V. 6. 1) Les polynˆomes de Laguerre associ´es
VI Transitions entre ´etats stationnaires
VI. 1. Mouvement d’une particule charg´ee soumise `a un champ ´electromagn´etique
(VI. 1. 1) Le hamiltonien du syst`eme
(VI. 1. 2) motion d’un champ magn´etique constant
(VI. 1. three) Invariance de jauge
VI. 2. Perturbations non stationnaires
(VI. 2. 1) R`egle d’or de Fermi
VI. three. Le rayonnement dipolaire
VI. four. Corrections multipolaires
VI. five. Expression quantique des coefficients d’Einstein
VI. 6. Coefficients d’absorption
VI. 7. R`egles de s´election et le spectre optique d’atome `a un ´electron
(VI. 7. 1) Les r`egles de s´election d’un oscillateur harmonique et d’un atome hydrog´eno¨ıde r´ealiste
VII advent `a los angeles th´eorie quantique non-relativiste des syst`emes
de particules identiques
VII. 1. Le formalisme g´en´eral
VII. 2. software `a l’atome d’h´elium
(VII. 2. 1) interplay d’´echange et magn´etisme
VII. three. L’approximation du champ self-consistant de Hartree et de Hartree-Fock
VIII advent `a l. a. th´eorie quantique de los angeles diffusion par un
potentiel
VIII. 1. part efficace de diffusion
(VIII. 1. 1) part efficace diff´erentielle dans le syst`eme du laboratoire
(VIII. 1. 2) Interpr´etation classique et loi de Rutherford
VIII. 2. Traitement stationnaire
(VIII. 2. 1) Equation int´egrale de l. a. diffusion et answer “approch´ee” : “Approximation de Born”
(VIII. 2. 2) Le r`egle d’Or de Fermi et l’approximation de Born
(VIII. 2. three) M´ethode des ondes partielles
Livres de r´ef´erence
– J. L. Basdevant, M´ecanique quantique, ellipses, 1986.
– J. Hladik, M´ecanique quantique, ´editions Masson, Paris, 1997.
Bibliographie
– D. Blokintsev, Principes de m´ecanique quantique, ´editions Mir, Moscou, 1981.
– J. M. L´evy-Leblond, F. Balibar, Quantique. Rudiments, Inter-Editions, Paris, 1984.
– Cl. Cohen-Tannoudji, B. Diu, F. Lalo¨e, M´ecanique quantique, tomes I & II, Hermann, 1980.
– E. Merzbacher, Quantum Mechanics, John Wiley, third ed. , 1998.
– S. Gasiorowicz, Quantum Physics, John Wiley, 1997.
– L. D. Landau, E. M. Lifshitz, Quantum Mechanics, Pergamon Press, third ed. , 1981.
– V. ok. Thankappan, Quantum Mechanics, John Wiley, 2d ed. , 1993.
– A. B. Wolbarst, Symmetry and Quantum Mechanics, Van Nostrand Reinhold Comp. , 1977.
– W. Louisell, Radiation and noise in Quantum Electronics, McGraw-Hill, 1964.
– A. Z. Capri, Nonrelativistic Quantum Mechanics, Benjamin/Cummings, 1985.
– J. J. Sakurai, smooth Quantum Mechanics, Benjamin/Cummings, 1985.
– W. Greiner, B. M¨uller, Quantum Mechanics, vol. I & II, Hermann, 1980.
– T. Fliessbach, Quantenmechanik, Spektrum Akademischer Verlag, 1995.
– R. W. Robinett, Quantum Mechanics, Oxford collage Press, 1997.

Extra resources for Differential Geometric Methods in Theoretical Physics: Physics and Geometry

Example text

Thus we have a subalgebra A C C((z)) (8) satisfying the condition AnC[[zJ]=c. (9) A commutative algebra B satisfying (B-1) and (B-2) produces a pair (A, L), where L is a monic pseudo-differential operator of order 1 obtained by taking the n-th root of P, and A is a subalgebra of C((z)) with (9) obtained by replacing L -1 in (6) by z. Now the question: how can we go back from an abstract algebra A satisfying (8) and (9) to the algebra B of ordinary differential operators? An easy answer is this: simply replace z by L -1.

Therefore, we obtain a compact Riemann surface by attaching a point at infinity to the plane curve and a globally defined vector bundle on it. (ordP ,0rdQ) = rankB. In our example of (2), Cis an elliptic curve, p is the point at infinity and F is a line bundle. Let us denote by M 1 the moduli space of all data (C, p, L, v) consisting of an algebraic curve of arbitrary genus g, a smooth point p E C, a line bundle Lon C of degree g-l which has no non-trivial global holomorphic sections, and a non-zero tangent vector v E TpC.

By continuing this process infinitely many times, he obtained 00 (6) Therefore, (7) 20 where C ( ( z )) denotes the set of all formal Laurent series in z with finite poles at z = O. But since C((L- 1 )) is commutative, so is Bp! Let us consider the abstract version of Schur's argument. Since Q is a differential operator, m of (6) is always positive. Thus we have a subalgebra A C C((z)) (8) satisfying the condition AnC[[zJ]=c. (9) A commutative algebra B satisfying (B-1) and (B-2) produces a pair (A, L), where L is a monic pseudo-differential operator of order 1 obtained by taking the n-th root of P, and A is a subalgebra of C((z)) with (9) obtained by replacing L -1 in (6) by z.

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