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

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254

1V,§2,1-IV,§2,2

2CT di sinh 2 p = i(A - C )a[curl B] - i< p + B, curl B>T tanh p

A+Ca[(p+B)n Aj

by (1.10); (1.15) follows by (1.21).

Proof of (1.16) and (1.17). Let y E E3;

e±°Ta[Y]e+vt = a[Y] cosh2 p

± Z{zo[y] - 7[y]T}sinh2p - Tu[y]rsinh2p. (1.23)

Now by the definition (1.10) oft and the commutation formula (1.20),

(Ta[y] - a[y]t} sinh 2p = ic[(p + B) A y].

By (1.10) and (1.19),

TQ[Y]

Csinh2p(P + B,y> -

a[Y]T,

whence

TQ [Y]T =

C sinh 2p

(p + B, Y>T -

of y].

By (1.24) and (1.25), (1.23) may be written

(1.24)

(1.25)

Ce±pt j[y]e+PT= AQ[Y]

- (p + B,y>Ttanhp ± ii[(p + B) A y],

whence the two formulas

{e°Ta[Y]e-nt + e-°'Q[Y]e°T}

= Au[y] - (p

+ B, y>ttanhp,

{eP'u[Y]e °T - e °tcr[Y]e°T} = ia[(p + B) A Y],

which prove (1.16) and (1.17), respectively.

2. The Reduced Dirac Equation for a One-Electron Atom in a Constant

Magnetic Field

Let us choose A, B, and C such that the function H defined by (1.4),

is the hamiltonian of such an electron: H has to be the function defined

IV,§2,2

255

by the formulas (4.15) and (4.18) of III,§1, up to the factor p. Thus,

A is a function of R, the function R i-- RA(R) is affine,

B = '(-bx2, bx1, 0), where b e R.

In the expression (1.4)1 for H, neglect B2, as is done in (4.20) of III,§1.

In the expression (1.14) for J, similarly neglect the term

i A

a[B A

Ax] in comparison with 2

[

ax A

B],

because, on V, xAx/(A + C) will be negligible in comparison with 1. Then

the expressions (1.4) and (1.14) for H and J reduce to

H(x, p) =

-[P2 + bM + C2 - A2(R)],

(2.1)

where b and C E R and R i4 RA(R) is affine,

The matrix of lagrangian operators associated to the matrix

H + 1 J,

where H and J are defined by (2.1) and (2.2), will be denoted by a, and

called the reduced Dirac operator.

Remark 2.

Thus H, defined by (2.1), becomes the function defined by

formula (4.20) of III,§1, up to the factor p. It follows that

Ac E+ERZ

E-Yf

C=µc,

where

c is the speed of light,

p and E are the mass and charge of the electron,

Z is the atomic number of the nucleus,

E is the energy level of the atom, which is close to µc2,

.' is the intensity of the magnetic field.

We replace the study of the Dirac equation by the study of the system

256

IV,§2,2

a,U=(aLl -Lo-

v'Lol')U=(aM-Mo- im'+Zva3)U=0,

(2 .3)

where the vector U has two components, which are lagrangian functions

defined on a compact lagrangian manifold V in E3 O E3, and where I'

and m' are real and have negligible squares.

Theorem 2 of §1, can be applied to system (2.3); its statement becomes

more precise by means of the following two lemmas.

LEMMA 2.1.

An approximate value for the function a defined by (2.4)3 of

§1 is

e[L0,MO]=

(1+NL)2+2Mo(1+NL)NM+N,K,

(2.4)

where

NL = NL[Lo, MO],

NM = NM[LO, Mo]

Proof.

Let us explicitly give the values of fo and go, defined by (2.4)1

and (2.4)2 in §1. By (2.1),

H[L, M, Q, R] = L22+ Q2

+ 2M +

ZCZ

zA2(R),

(2.5)

whence

HL=R, HM=2,

now by (2.3) of III,§2,

A, = - HL,

p, = - HM ;

by (1.5) of III,§2, we have, writing F for F [L,, Mo],

A, dt = 0,.1 = 27rNL,

Jr.

p,dt=A,p=27rNM;

hence,

Jr,

Jr.

2dt = -27rNM;

dt = -21rNL,

IV,§2,2

257

the last formula and formula (2.4)1 of §1, where f = b/2 by the choice

(2.2) of J, give

.fo = - iNm.

(2.7)

By the choice (2.2) of J and the definition §1, (2.4)2 of go,

_ _Lo

g0 2

rR(A+C)dt;

(2.8)

now by (2.14) of III,§1 and (2.5), we have, on F,

dt =

dt 0 H th 0

(2 9)

, us > ;

Q =

RHQ

hence relat

ions (2.8) and (2.6)1 may be written

= -

10)

2 J

n L

r Q A

Jr. Q R

Let us calculate an approximate value forgo, namely, its value for b = 0.

By (2.5), in that case the equation of I, is

F:Q2 =

A'(R)A"(R) - Lo,

where

A'(R) = RA(R) + CR,

A"(R) = RA(R) - CR;

(2.11)

A' and A" are affine functions; by remark 2, A' is increasing and A'(R) > 0

for R > 0; it follows by an easy calculation using residues that

(' L° AR

dR = 2n;

(2.12)

Q A'

now by (2.11)1,

AR1+ AR

A' R A+C'

then the relations (2.10) and (2.12) prove that

go = -R(1 + NL).

(2.13)

Formulas (2.7) and (2.13) and formula (2.4)3 of §1 prove the lemma.

258

IV,§2,2-IV,§2,3

The assumptions made in §1,2, including those of §1, example 2, are

satisfied, as it is proved by the following lemma.

LEMMA 2.2.

The values of the physical quantities defining A, B, and C

(remark 2) are such that for the energy levels E used in section 3,

fo = -7rN,H,

go = -7r(1 + NL),

NL',

NLM, NM2

are small in comparison with 1.

Proof.

N[-,,] is expressed by (4.8) of III,§1, where Ao, Bo, and Co

are defined by identifying formulas (4.6) and (4.20) in III,§1:

M = aZrUC2(1c2

E2fi

Mo/h)

1-in

[Lo,

[(L)2

- a2Z2]'

(2.14)

where Z, c, p, e, E, and ,Y are the physical quantities defined by remark 2,

is the dimensionless fine-structure constant

(2.15)

13he

( means of the order of magnitude of), and

Z hh

is the Bohr magneton.

(2.16)

In section 3, we choose

0 < (µc2 /E) - 1 ^_, a2Z2 /(2n2), where n is an integer,

the magnetic energy /3,Y to be very small in comparison with µc2, and

2Mo/h to have integer values which are not large,

Lo/h > 2.

The lemma follows.

3. The Energy Levels

Notation. Formulas (4.23) and (4.25) of III,§1 have already defined and

used the function F given by

F(n, k) (3.1)

)

IV,§2,3

259

Note the sign of its derivatives:

F,, > 0,

Fk > 0.

Using F, it is possible to give the relation

hk + N [hk, hm] = hn,

(3.2)

where N is expressed by (2.14), the form

E2 = Uc2[pc2 + 2/3.,Ym]F2(n, k).

In other words, to the degree of approximation used here, and assuming

E > 0,

E = Uc2F(n, k) + f3.3f°m.

(3.3)

In these formulas, a is the fine-structure constant (2.15) and /3 is the Bohr

magneton (2.16).

THEOREM 3.

The energy levels E for which system (2.3) (where l' and m'

are real and have negligible squares) has admissible lagrangian solutions

(definition 2 of §1) on a compact lagrangian manifold are defined by the

quadruples of quantum numbers

j±Z, m,

n (3.4)

such that

m -

?, and n are integers

(3.5)

and

lml <j,

0<l<n.

(3.6)

Up to some negligible quantities, E is expressed as a function of these

quantum numbers as follows:

E=Uc2F\n,I+2

fl2

2 Fk + 214+ 1

+ /3.dm.

(3.7)

U C2 2 4

The + sign is the same in (3.4) and (3.7).

Remark 3.1.

For ,Y = 0, since Fk > 0, (3.7) reduces to

260

IV,§2,3

E = µc2F(n, k),

where

k=j +ZEZ.

Proof.

By theorem 2 of §1, the energy levels E such that the system

(2.3) has solutions defined on compact lagrangian manifolds are those

that satisfy condition (2.5) of § 1. From the approximate value of e [Lo

, M0]

given by lemma 2.1, these values of E are approximately those satisfying

the condition

h(I + Z) + N[h(I + Z), hm]

=din

(3.10)

ml I + Z, 1 < n;

1, m - Z, n integers;

1 + NL and NM small.

Then by the equivalence of the relations (3.2) and (3.3), a crude ap-

proximation for E is

µc2F(n, 1 + z)

µcz [1

2nzz

(3.11)

which is sufficient to justify the use of lemma 2.2.

From example 2 of §1, admissible solutions of the system (2.3) cor-

respond to a choice of 1, m, n, and E satisfying (3.10) if and only if

ImI <l±Z,

which amounts to (3.6) and (3.4), where the ± sign is the same in (3.4) and

(3.10).

Let us now express E as a function of (1, m, n).

Since (3.2) is equivalent to (3.3), on the one hand (3.10) may be written

E=µczF(n±Z (1+NL)z+214m1(1+NL)NM+Ny,I+ZI

+ f3.tm ;

//(3.12)

on the other hand, the following two relations are equivalent (for all

dk, dm, dn):

(1 + NL)dk + NMdm = dn,

Fkdk) + /3.$" dm = 0.

IV,§2,3

This equivalence means that

µC2Fk fl

µc

N '

L M

whence since F. > 0,

F,,

(1 + NL)2 + 214+ 1(1 + NL) NM + NM

261

° fj2O2

+ 2l

aAC2

+ µ2C4

Thus (3.12) is equivalent to (3.7).

Remark 3.2.

The energy levels obtained by theorem 3 are exactly those

obtained by the classical theory of the Dirac atom: see, for example,

Bethe and Salpter [2]. For ,° = 0, their formula (14.29), p. 68, can be

identified with our formula (3.8) and, for ,Y A 0, their formulas (46.12),

(46.13), and (46.15), p. 211, can be identified with our formula (3.7): see

table for the correspondence between their notation and ours.

Table

Bethe-Salpeter [2]

Eo = mc2

E+

(E+ + E_)

AE=E+-E_

Leray (Remark 2 and Section 3)

µc2

µc2F(n, 1 + 1)

PC2F(n, 1)

µc2F(n, ( + )

µC2 Fk

#jrl(µc2Fk)

c2 2

J(°

fl2 *,2

f3t°m ± Z Fk +

21 + 1

Fkµc2 + u24

Remark 3.3.

When Q.)° is small in comparison with uc2Fk, formula

(3.7) evidently may be written

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IV,§2,3-IV,§2,4

E = µc2F(n, k) + R.Jfgm,

where g = (j +

z)(l +

z);

g is Lande's factor [see Bethe-Salpeter [2], (46.4) and (46.6) p. 209].

4. Crude Interpretation of the Spin in Lagrangian Analysis

The crude approximation consists of neglecting [ii' (pct) in comparison

with az; cz

itself, and hence F,,, in comparison with 1.

Remark 4.1. Wave mechanics suceeds in evaluating the energy levels of

the helium atom at the cost of this crude approximation: see a summary

of these very laborious calculations in Bethe-Salpeter [2], chapter II.

In the equation H = 0, this approximation identifies the hamiltonian

H of the relativistic electron [(4.20) of III,§1] with the nonrelativistic

hamiltonian [(4.19) of III,§1], where .J° = 0. In this second hamiltonian,

E denotes the nonrelativistic energy.

The crude statement of theorem 3 is the following.

THEOREM 4.1.

Let (j = I ± ?, 1, m, n) be a quadruple of quantum numbers

defining a system (2.3) that has admissible lagrangian solutions on a compact

lagrangian manifold V; it is a quadruple satisfying (3.5)-(3.6). The crude

expressions for the nonrelativistic energy E and for Lo and Mo as functions

of this quadruple are

E -µc2

z 27n,

Lo = h(1 + z),

MO = hm.

(4.1)

The crude equations for V are

z z

V:H=0,where H=2µ

-E

L=h(l+z),

M=hm.

(4.2)

Remark 4.2.

In the case

mI =j=1+i,

we have, by (4.2) and the definition of L and M (III,§1,1), that, on V,

M I = L; thus x3 = p3 = 0;

whence

dim V = 2, contrary to the definition that dim V = 3.

IV,§2,4

263

In order to give a meaning to this result, other than by reintroducing

nonzero 1' and m', one perhaps has to use the notion of a lagrangian dis-

tribution and to refer to some of the papers of Voros [25], [26].

Notation.

Let SO (3) be the group of rotations p : E3 -+ E3 leaving the

origin invariant. Let it act on E3 ® E3 as follows:

Let SO(2) be the subgroup of rotations leaving 13 (III,§1,1) that is, the

direction of the magnetic field, invariant. SO(2) evidently leaves V in-

variant. More precisely, by sections 1 and 2 of I,§1,

V = SO(2) x T2,

where T2 is a 2-dimensional torus, every point of which is left invariant

by SO(2). Recall (remark 1.1 of §1) that the universal covering group of

SO(3) is SU(2), which is a covering group of order 2. Let SU(1) be the

covering group of SO(2) of order 2. Then we have the commutative

diagram

SU(1) c-, SU(2)

SO(2) c-. SO(3)

where the vertical arrows represent natural projections of covering groups;

the horizontal arrows are inclusions. Let

V2 = SU(1) x T2.

V2 is a covering space of V of order 2; SU(1) acts on V2, leaving each

point of T2 invariant.

Definition 4. Let V be a lagrangian manifold, c°R0 its phase in a 2-frame

R0, and ji a function f,' - C. By theorem 2.2 of II,§2,

f7 1!2

WA,

OR. = dix

/3e

is the expression in Ro of a lagrangian function U defined mod(1/v) on

V. The definition of a lagrangian function on V (definition 3.2 of II,§2)

may be generalized as follows: Let V be any one of the covering spaces

of V; 0 is said to be lagrangian on V when the restriction of OR. to v0.