Leray J. Lagrangian Analysis and Quantum Mechanics: A Mathematical Structure Related to Asymptotic Expansions and the Maslov Index

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264

IV,§2,4-IV,§2,5

1/2

Woo =

(-q--)

is a mapping 17\iR0 - C.

The calculations that establish the conditions (3.11) in III,§1 prove the

following theorem.

THEOREM 4.2.

The conditions (3.4)-(3.6), which are satisfied by the quad-

ruples of quantum numbers defining solutions of (2.3), express the existence

of a function with constant lagrangian amplitude defined mod(1/v) on

V2 but not on V.

Conclusion.

Let V be the lagrangian manifold given by equation (4.2).

Let U be a function with constant lagrangian amplitude defined mod(1/v)

on V The following is a crude interpretation of the spin:

U is lagrangian on V in the spin 0 case (Schrodinger);

U is lagrangian on V2 (and not on V) in the spin i case (Dirac).

5. The Probability of the Presence of the Electron

Recall that x is the position and p the momentum of an electron; thus

(x,p)EE3QE3=XQX".

With the conventions of remark 2, let

which is the length called the radius of the Bohr atom in the first quantum

theory. Let

, _

x'=Rl,

where Z is the atomic number of the nucleus.

THEOREM 5. The position x of the electron with quantum numbers (j =

I ± Z, 1, m, n) stays in the projection VX of V[L0, M0] onto X; VX is a

subset of X lying between two spheres and outside of a cone of revolution

around the magnetic field; the center of these spheres and the vertex of

this cone coincide with the nucleus of the atom.

Assume that on V [LO, M0] c E3 Q E3 the probability of the presence

of the electron is proportional to the invariant measure 7v [see the con-

IV,§2,5

265

clusion of the introduction to chapter III]. Then on Vx the probability of

the presence of the electron is crudely

1 + 2

d3x'/

2713 712

R,2 +

2n2R' - (1 + 1)2n2 V (1 + 4)2(x12 + x22) - m2R,2

(5.1)

A crude definition of Vx is that Vx is the subset of X where the above

radicals are all real.

Remark 5.1.

If Iml = 1 + Z, then VX is the disk where

x3 =

0, R'2

- 2n2 R' + (1 +

21)2 n2

< 0.

The probability of the presence of the electron on this disk is

1 1

d2x'

2712 712

tiI -R'2 + 2n2R' - (l + 1)2n2

Proof. On V [L 0, Mo] the probability of the presence of the electron

is, by definition,

4712

dt dt A d`P A dD, since 'i = dt A d`F A dD.

By (1.5), (1.11), and (1.12), of III,§1, the projection Vx of V[L0, Mo]

onto X is the set of points x in X for which

sin O, where cos O = o

and the second-degree equation in Q,

H[LO, Mo, Q, R] = 0, (5.3)

has real roots.

Thus Vx is indeed a subset of X lying between two spheres and outside

a cone of revolution whose centers and vertex are at 0.

Formulas (1.5), (1.6), and (1.12) of III,§l show that every point x inside

Vx is the projection of 4 points (t, T, (D) in V [Lo, Mo]. By (1.16) and (3.6)

of III,§1 and by (2.9),

d3x = - QR sin q' sin 0 dt A d`V A d (D,

dt =

RQdR

> 0,

266 IV,§2, 5-IV,Conc1usion

where I QI is defined by giving x. Then on Vx the probability of the presence

of the electron is

[

RdR]-'

d'x

r Q

54( )

By (1.12), of III,§1.

x3 = R cos'P sin 0;

therefore, from (5.2)2,

RJsin'Pf sin0 = 1 [Lo(x1 + x2)

- M'R2]12,where

Ma =

1 + 2

(5.5)

The definition of Q by equations (5.3) and (4.2), where E has the value

(4.1) and P2 = (Q2 + L2)/R2, crudely gives

Q n [-Ri2

+ 2n2R' - (1 + '-)2n2]1j2. (5.6)

It follows that

RQR _ 2n

z22

(5.7)

Formulas (5.5), (5.6), and (5.7) give the probability (5.4) its crude ex-

pression (5.1).

Now, VX, being the subset of X where (5.5) and Q are real, is crudely

defined by the condition that the radicals entering into (5.1) be real.

Remark 5.2.

The probability of the presence of the electron defined by

theorem 5 is extremely different from that defined by wave mechanics,

but closely connected to the first quantum mechanics: the electron stays

in a compact subset VX of X that is the union of the trajectories used by

the first quantum mechanics; the probability of its presence results as

simply as possible from the equations characterizing V.

Conclusion

This survey has elucidated Maslov's quantization. On the one hand,

theorem 3 of IV,§2 proves that it is related to classical wave mechanics

IV,Conclusion

267

and preserves its essential results; on the other hand, remark 5.2 of IV,

§2 proves that it is related to the first quantum theory and preserves the

simplicity of its calculations. Chapters I and II have shown how Maslov's

quantification replaces the Bohr-Sommerfeld quantization by a quantiza-

tion that mathematical motivations justify.

We considered only one-electron atoms (or ions). Let us review very

briefly how quantum mechanics has studied several-electrons atoms. It

has been by using the Schrodinger equation.

i. There are rich theoretical results (by T. Kato, B. Simon, and many other

mathematicians); but they do not lead to numerical results.

ii. There are some numerical results (by E. A. Hylleraas, C. L. Pekeris,

and many other physicists) that strikingly agree with experimental results;

but their mathematical accuracy is not completely established.

[In its present state Maslov's quantification can reobtain neither (i)

nor (ii).]

iii. There is also a very crude method that foresees the behavior of the

plasmas; it pays attention to only one of the electrons, to which is applied

the theory of one-electron atoms. V P. Maslov asserts that such a proce-

dure becomes more efficient when his quantification is used.

That last fact and the fact that (i), (ii), and (iii) together do not at all

constitute a complete theory show how also physics called for a coherent

elucidation of Maslov's quantification and for the required tools.

Bibliography

1. Cited Publications

[1] Arnold, V. I.

On a characteristic class intervening in quantum conditions, Funct. Anal. App!. 1 :1-14,

1967 (in Russian and English translation).

[2] Bethe, H., and Salpeter, E.

Quantum Mechanics of One- and Two-Electron Atoms, New York: Springer-Verlag, 1957.

[3] Buslaev, V. C.

Quantization and the W.K.B. method, Trudy Mat. Inst. Steklov 110: 5-28, 1970 (in Russian).

[4] Cartan, E.

La theorie des groupes finis et continus et !a geometrie differentielle, traitees par la methode

du repere mobile, Paris: Gauthier-Villars, 1937 (chapter Xl).

[5] Cartan, E.

Lefons sur les invariants integraux, Paris: Hermann, 1922.

[6] Choquet-Bruhat, Y.

Geometrie diJjerentielle et systemes exterieurs, Paris: Dunod, 1968.

[7] De Paris, J. C.

Probleme de Cauchy a donnees singulieres pour un opbrateur dfferentiel bien decomposable,

J. math. pures app!. 51: 465-488, 1972.

[8] Carding, L., Kotake, T., and Leray, J.

Uniformisation et dbveloppement asymptotique de la solution du probleme de Cauchy

lineaire a donnees holomorphes; analogie avec la theorie des ondes asymptotiques et

approchees, Bull. soc. math. France 92:263-361, 1964 (introduction and chapter 7).

[9] Lefschetz, S.

Algebraic topology, Amer. Math. Soc. Colloq. Publications 27, 1942 (chapter 111, §5).

[10] Maslov, V. P.

Perturbation Theory and Asymptotic Methods, Moscow: M.G.U., 1965 (in Russian and

English translation).

[11] Maslov, V. P.

Theorie des perturbations et methodes asymptotiques, suivie de deux notes complementaires

de V. I. Arnold, et V. C. Buslaev, Paris: Dunod, 1972 (translated by J. Lascoux and R. Senor).

[12] Morse, M., and Cairns, S.

Critical Point Theory in Global Analysis and Differential Geometry, New York: Academic

Press, 1969 (§4, reduction theorem).

[13] Schwartz, L.

Theorie des distributions, Paris: Hermann, 1951.

[14] Segal, 1. E.

Foundations of the theory of dynamical systems of infinitely many degrees of freedom (1),

Mat. Fys. Medd. Danske Vid. Selsk. 31 (12):1-39, 1959.

[15] Shale, D.

Linear symmetries of free boson fields, Trans. Amer. Math. Soc. 103:149-167, 1962.

[16] Souriau, J. M.

Construction explicite de 1'indice de Maslov. Applications. 4" International Colloquium on

Group Theoretical Methods in Physics, University of Nijmegen, 1975.

[17] Steenrod, N.

The Topology of Fiber Bundles, Princeton: Princeton University Press, 1951 (§2, §15).

270

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[18] Weil, A.

Sur certains groupes d'operateurs unitaires, Acta math. 111:143-211, 1964.

[19] Whittaker, E. T., and Watson, G. N.

Modern Analysis, Cambridge: Cambridge University Press, 1927.

2. Related Publications

[20] Crumeyrolle, A. -

Revetements spinoriels du groupe symplectique et indices de Maslov, C. R. Acad. Sci.

Paris 280: 1753-1756, 1975; Algebre de Clifford symplectique et revetement spinoriel du

groupe symplectique, C. R. Acad. Sci. Paris 280:1689-1692, 1975.

[21] Crumeyrolle, A.

Algebre de Clifford symplectique, revctements du groupe symplectique, indices de Maslov

et spineurs symplectiques, J. math. pures appl. 56:205-230, 1977.

[22] Dazord, P.

La classe de Maslov-Arnold; L'operateur canonique de Maslov, Seminaire de Geometrie;

University Claude Bernard [Lyon 1], 1975/76; Une interpretation geometrique de la classe

de Maslov-Arnold, J. math. pores appl. 56:231-250, 1977.

[23] Guillemin, V., and Sternberg, S.

Geometric Asymptotics, Providence; American Mathematical Society, 1977.

[24] Malgrange, B.

Integrales asymptotiques et monodromie, Ann. Sci. Ecole Norm. Sup. 7 (4):405-430, 1974.

[25] Voros, A.

The W.K.B: Maslov method for non-separable systems, Colloques internationaux du

C.N.R.S., 237: Geometrie symplectique et physique mathematique. Aix-en-Provence, 1974.

[26] Voros, A.

Asymptotic h-expansions of stationary quantum states, Ann. Inst. Henri Poincare 26A:

343-370, 1977.

3. Preliminary Publications

[27] Leray,J.

Solutions asymptotiques des equations aux dbrivees partielles (unc adaptation du traite de

V. P. Maslov), Convegno internazionale: metodi valutativa nella flsica matematica; Accad.

Naz. dei Lincei, Roma, 1972.

[28] Lcray,J.

Complement a la thborie d'Arnold de I'indice de Maslov, Convegno di geometrica simplettica

et fisica matematica, 1stituto di Alta Matematica, Roma, 1973.

[29] Leray, J.

Solutions asymptotiques et groupe symplectique, Colloque sur les operateurs de Fourier

integraux et les equations aux derivees partielles, Nice; Lecture Notes, Springer-Verlag, 1974.

[30] Leray, J.

Solutions asymptotiques et physique mathbmatique, Colloques internationaux du C.N.R.S.,

237: Geometrie symplectique et physique mathematique; Aix-en-Provence, 1974.

[31] Leray, J.

Solutions asymptotiques de 1'equation de Dirac, Trends in Applications of Pure Mathematics

to Mechanics, Conference at the University of Lecce, London: Pitman, 1975

[32] Leray, J.

Analyst lagrangienne et mecanique quantique, Seminaire du College de France, 1976-1977,

and R.C.P. 25, Strasbourg, 1978.

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271

[33] Leray,J.

The meaning of Maslov's asymptotic method: the need of Planck's constant in mathematics,

Bulletin of the American Mathematical Society, to appear, 1981. (Symposium on the Mathe-

matical Heritage of Henri Poincare, April 1980).

The present publication is a revised version of [32], which encompasses [27]-[31], and

[33] constitutes a comment of the present publication.