Perturbation Theory in Quantum Mechanics. II

Perturbation Theory in Quantum Mechanics. IIAuthor(s): A. H. WilsonSource: Proceedings of the Royal Society of London. Series A, Containing Papers of aMathematical and Physical Character, Vol. 124, No. 793 (May 2, 1929), pp. 176-188Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/95140 .

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176 A. H. Wilson.

lines should be included in the sums does not arise, as their intensities are here assumed to be negligible.

Conclusions.

L. The selection rules are rigorous if there are no external fields. 2. The rule " AEk is odd " is equally rigorous, even in a uniform magnetic

field. 3. The summation rule holds if magnitudes of the order of the spin energy

are neglected.

I should like to thank Dr. P. A. M. Dirac for his criticisms of this paper.

Perturbation Theory ibn Quantum Mechanics-It. By A. H. WILSON, Emmanuel College, Cambridge.

(Communiicated by R. H. Fowler, F.R.S.-Received March 8, 1929.)

1. Introductio^n.

In a previous paper* the theory of perturbations was discussed for systems. which possess only discrete spectra. The theory is now extended to systems, with both discrete and continuous spectra, and in addition a slightly more general method of perturbation is considered. The Hamiltonian H of the perturbed system is split up into three parts Ho, H,, H2 (t), two of which can be chosen arbitrarily. The solutions of the perturbed problem can then be worked out in terms of the characteristic functions corresponding to the HIamiltonian Ho + H2 (T), where T is any arbitrary fixed time. In general, however, the series of perturbations does not converge, and different modes of splitting up the Hamiltonian will lead to different formal results. The perturbation theory does usually give the correct physical results, and it is therefore necessary to give some explanation of how this happens. In (I) the asymptotic nature of certain solutions was emphasised, but this does not extend to the more general systems treated here. It is suggested that the usual boundary conditions do not give an adequate description of any but the simplest atomic problems, and that more detailed restrictions, determined by

* ' Roy. Soc. 'Proc.,' A, vol. 122, p. 589 (1929). Referred to hereafter as (I).



Perturbation Theory in Quantumn Mechanics. 177

the experimental conditions, ought to be substituted. As this is impossible in practice, an alternative method is to regard the motion as a perturbation on that Hamiltonian Ho + 12 (T) which most nearly corresponds to the experimental conditions. (For example, we may regard an electron as free or bound according as the experiment determines the number of free or bound electrons respectively, whereas in the usual theory the initial Hamiltonian is taken as fundamental.) This criterion, together with the initial conditions, suffices to fix the division of H into the three parts H., H11112(t) which at first were chosen arbitrarily. The formal solution can be worked out on this basis, and it is then assumed that this solution is asymptotic to the one which would be obtained if the full boundary conditions were used. This suggestion is in accordance with the fact that a hydrogen atom in an electric field gives rise not only to the usual Stark effect, but also to an ionisation effect.

In ? 2 the perturbation equations are derived in a slightly more general form than has previously been given. The theories of Born,*. Diract and Oppen- heimer are particular cases of that given here, and are obtained by particular choices of Hoy H1, H2 (t). To obtain Born's theory we take H1. = 0, and for Oppenheimer's we take H2 (t) 112 (T) and T 0. In Dirac's theory we have

=_ 0, H2 (0) = 0 and T _ 0. Each of these theories is appropriate for the solution of special problems, but the more general method of this paper is required for the full discussion of the perturbation theory.

In ? 3 the necessary existence theorems are proved, and in ? 4 the validity of the perturbation theory is discussed.

2. The Perturbation Equations.?

2.1. We consider an atomic system which is subjected to an arbitrary perturbation from outside. The unperturbed system is characterised by a Schr'odinger equation

( + 2Tat (1)

where Ho and Hi are independent of the time. The differential equation (1) will involve several independent variables, but, as we are not interested in the difference between degenerate and non-degenerate systems, the theory will be the same as for only one independent variable. We suppose that equation

* ' Z. Physik,' vol. 40, p. 167 (1926). t 'Roy. Soc. Proc.,' A, vol. 112, p. 661 (1926). $ 'Phys. Rev.,' vol. 31, p. 66 (1928). ? I am indebted to Mr. L. C. Young for verifying the details of this section.

VOL. CXXIV.-A. N



178 A. H. Wilson.

(1) is separable, and that, with given boundary conditions, one of the resulting ordinary differential equations possesses a continuous set of characteristics as well as a discrete set, while the other equations only have discrete characteristics. The equations will be written in terms of the space variable which gives rise, to the continuous characteristics, the other variables not appearing explicitly.

We put (1) (/(1) (x, W) e- 2iriWt/h,

where 0(l) (x, W) is independent of the time, and W is a real parameter to be determined by the boundary conditions. Equation (1) takes the form

L {0(1)} - W(1l) = (Ho + 11, - W) b1) ? 0. (lA)

Since the boundary points are always singularities of this differential equation the condition that 1 0(1)12 should be integrable over the whole of the configuration space is a sufficient boundary condition. In practice this is nearly equivalent to the more usual condition that f(') must be finite and continuous.

If f OM 12 dx taken over the whole of the configuration space is finite, the

corresponding value of W is a discrete characteristic, and the totality of such values of W constitutes the discrete spectrum.

The theory of continuous spectra has been studied by Weyl,* and has been applied to quantum mechanics by various writers. As there seems to be some %obscurity in the applications, the theory is given briefly here for reference.

X (x, W) is a continuous function of the two variables x and W, and X (x, 0) = 0. Also,

J {X (x, W)}2dx _ (W)

exists and is a continuous function of W. Then X (x, W) is called a permissible solution of (1A) if it satisfies the equation

w L {X (x, W)}-WX (x, W) + X (x, ?) dX - 0.

W1 belongs to the continuous spectrum if and only if X (x, W1) is a permissible solution of (1A) and the function X (W) is not constant in the neighbourhood of W1.

* 'Math. Ann.,' vol. 68, p. 220 (1910).



Perturbation Theory in Quantum Mechanics. 1]79

The fundamental theorem is that any permissible solution can be expressed in the form

w x (x w_ O +f(x~ x, i(X) d X 0

where i (XA) is the indefinite integral of a function of integrable square.t In future the characteristic functions belonging to the discrete spectrum will

be denoted by Oi(1) (x), while those belonging to the continuous spectrum will be written as O(i) (x, W). The functions must be normalised as follows.

For the discrete set, J f+1? (x) bm(1) (x)* dx =a..

where the integral is taken over the whole of the configuration space, Ok(1) (x)* is the conjugate of km(') (x), and a. is the usual Kronecker a.

For the continuous set we define

AnF = j +() (x, W) dW/I, w

where the range of integration AnW is a finite interval of the continuous spectrum of W. The normalisation required in the quantum theory is then given by$ J AF . AmF* dx = A, W/h,

where A,MW is the common part of the "intervals A.W and AmW. We also have the orthogonal relation

f n) (x) AmF* dx =0.

If TO is a function of integrable square satisfying the same boundary conditions as the characteristic functions and for which L (T0) is continuous and quadratically integrable, then T0 can be expanded in the absolutely and uniformly convergent representation?

to (x) = -zC.1) (x) + O(l) (x, W) c (W) dW (2)

where

c= J' o (x) On(1) (x)* dx

and

c (W) -- - dx to (x) X ')(x, X)* d X. h dW AnI

t Weyl, loc. cit. See also Hobson, 'Functions of a real Variable,' vol. 1, 2nd ed., p. 613.

1 Oppenheimer, loc. cit. ? Weyl, loc. cit., theorem 7.

N 2



180 A. H. Wilson.

In quantum problems the spectrum falls into one of three classes. In the first class the spectrum is entirely discrete and the integral does not appear in (2). In the second class the discrete characteristics range from W = - Xo

to W = 0, and the continuous ones from W = 0 to W = oo. The integral is then taken between 0 and oo . In the third class there is no discrete spectrum, and the sum disappears from (2). The general formula includes all these particular cases, which are obtained by making f(') or 0(l) (W) identically zero when the corresponding part of the spectrum does not exist. Since the representation (2) is uniformly convergent we may multiply by To*, integrate with respect to x and invert the order of integration or integrate term by term. This gives

T Io 2dx = E IcnI2 + | jc(W)12 dW, n oo0 showing that I Ic.12 and Ic (W) 2dW are finite. Whether the Riesz-

n -oo

Fischer theorem holds in general for Sturm-Liouville functions involving continuous spectra, does not seem to be known, but it seems to be true for all cases of physical importance. For C, and c (W) to be the coefficients of a function

of integrable square, it is necessary that E IC 12 and J c (W) 12 dW should be n -oo0

finite, and we shall assume it is sufficient in all cases which occur in the quantum theory.

2.2. If To is the wave function of the system at time t = 0, and if the system is isolated and subject to no external perturbations, then the wave function at any subsequent time is

F = zc"tp (1) ? fJ- VI1) (W) c (W) d W. (3) n -oo0

Now suppose that at a given time, say t 0, a perturbation H2 (t) is added, giving a Schr6dinger equation

-1 2 h a 0. ~~~~~~~(4) (HO + H, + 2( ) 7r at)

Also suppose that we can solve the equation

ITo + ] 2 (T) 2ih a (2) =? (5) 27t at) 5

where T is any fixed time. The characteristic functions of (5) will, in general, consist of a discrete set and a continuous set, which will satisfy the same



Perturbation Theory in Quantum Mechanics. 181

orthogonal conditions as 4,1). We denote the continuous characteristics by ,u, and take the range of values to be (- oo, oi ).

We assume a solution of equation (4) in the form

I4 = 2cA4,,l) + 4,(1)(W) c (W) dW co

+ a., (t) 4,(2) +J4(2) (L) y (L,) t) dL. (6)

The first line is merely the wave function T of the undisturbed system, the coefficients On and c(W) being determined by the initial conditions as in equation (2). am (t) is a function of the time only, and y (V, t) is a function of V. and t. These coefficients have to be determined so that (6) gives a solution of equation

(4) satisfying the condition that i 41,2 dx is finite. After substituting (6) in

the differential equation (4), we expand all the non-vanishing terms of the 00

resulting expression in the form E oc. n 4,(2) + J co(2)

( tL) 3 (p.) d [. Assurming

that the unknown coefficients can subsequently be determined so as to allow the necessary inversion of the order of integration, we obtain, by equating to zero the coefficients of 4 (2) and of 4(2) (i[), the following set of differential equations for the quantities a" (t) and y (t, t)

ih/2tin * i(t)- tgH, a . (m t) + J [n ( )* Y ( [t t) d . (7)

ih/27u. jQt, t) -g( , t) + I FHm([) am(t) +J H (, A) y (X, t) d?. (8)

The coefficients in (7) and (8) are given by the formal expansions

H2 (t) T =E gm (t)(2) +J (2) Fg ( [ t) d [

{H. + H2(t) H2 (T)} 44,(2) , Z Hnm in(2) + J4(2) (V.) Hm (U) d t and

H( ., ?X) =-- { Hl{H + 1-H2(t) -H2(T)}dx |(2)(X)dxi 4,(2) ( )*dt. A2 d~tdX i

In the next section we shall examine under what conditions it is possible to obtain a solution of the perturbation equations (7) and (8) satisfying

I an (t) 2 finite and J y ( , t) 2 d, finite, which is a necessary condition n

for fl 4 12 dx to be finite. Whether this is a sufficient condition does not, as it



182 A. H. Wilson.

happens, arise, since the restrictions imposed are already too heavy for most quantum problems, and we conclude that in general the perturbation theory does not give a solution satisfying the boundary conditions.

3. The Existence Theorems. 3.1. Simplifying the notation, we have the following equations,

dw = as(z) + uuij wi + |u (V) v(t, z) du (i = 1, 2,...) (9) dz dv (t,z) = b (tt, z) + E; u( ()j ?X (+ , ?u ) v (?, z) d?X. (10) dz i-0

In (I) the solution of the corresponding equations was obtained as a matrix. This is not possible here, since in addition to the matrix of the coefficients u4j there is also the nucleus u (,u, X) and the two sets of functions ui (v), u (I)p which we may call " mixed nuclei." We have therefore to introduce the idea of limited nuclei, corresponding to limited matrices.

3.2. A nucleus G (A, ,u) is said to be limited with bound M provided

I J xJ G ( X, ,) X (X) y (V.) d X d | < M for all X, x(X), y(p,), where x(X), y(R) are complex functions of the real variables X, ji, satisfying

00 rX I X(X) 12dx 1 y (p) 12d[L < Similarly a mixed nucleus G (X) is limited with bound M provided

|z __G (X))xiy(X) dS X< M i= 1 -x

for all N, X, xi, y (k), subject to OD 0OD

E lX,l6 12 ty(X) 12 (X < Most of the properties of limited matrices extend at once to limited nuclei. Those that are used in the sequel are given below without proof, the proofs being trivial extensions of those given by Hellinger and Toeplitz* for matrices.

We define the product of two nuclei G1 (?, p.), G2 (I, ,u) to be the nucleus

J G1 (?, v) G2 (v, p.) dv. There are also obvious definitions of the product

of a nucleus with a mixed nucleus, of two mixed nuclei, and of a mixed nucleus with a matrix.

* ' Math. Ann.,' vol. 69, p. 289 (1910).



Perturbation Theory in Quantum Milechanics. 183

1. The product of two limited nuclei with bounds M. M' is limited with bound MM'.

2. If G (A, tL) is limited with bound M, then

- J G (X, ) y ( t) dp. dx K M and

X i i 0 G(X,t) x(X)dX d0t<M

where

- X $(X) 1 2dX< 1, - 0 y (,u) 12 d,tL <I and conversely.

3. With the same hypothesis,

- 00i G (Xi p) 12dX<M, i 0 G (X, p) 11 d, < M.

3.3. We now proceed to solve the differential equations. We assume that the point zo is not a singularity of any of the equations, and that it is possible to surround zo by a finite domain D free from singularities. We have then to find solutions of (9) and (10) reducing to given values wo and v ([, zo) at zo,

and satisfying I I Wi 12 finite and I v (,, Z)I2d,u finite. We also assume that 00~~~~~~~~0

I as(z) 12 and I b ( ,, z 12 d,u are bounded, and that ui1, ui (,),'u (.,.)1, u ([,, A)

are all limited throughout the whole of D. The coefficients are functions of z, but it is convenient not to show the dependence on z explicitly. The method adopted is to solve the equations (9) formally, leaving v (,L, z) arbitrary. The value of w, is then substituted in (10) giving rise to an integro-differential equation for v (R, z) only, which can be solved for v (,u, z).

We therefore first solve the non-homogeneous system dwd - E ujw) + c. dz Ij

.

The corresponding homogeneous system was treated in (I), and it was shown that the necessary and sufficient condition for the existence of a solution of the required type is that a certain matrix Q (u) is limited. It is sufficient that the matrix (ui;) is limited, and the solution is given by w -Q (u) wo. We now make the further assumption that the inverse matrix Q-'- exists, such that Q (u) ?-1 (u) = 1. To solve the non-homogeneous system we make the substitution

w ?Q (u) W,



184 A. H. Wilson.

where W denotes the set (W1, W2, ...). We then have

dw dW dz ~~~dz

- UW + C,

where c is the set (ci, C2, ...). Denoting as usual the operator | dz by Q, we

obtain on integration

w =Q(M6) qO + Q (u) Q Q-1 (it) c. (1 Also

S i12 E QiWi? + I QirQ Qrjiiej|2

<A{Z IIQ911 -1C 1 < A {E I E jWj? 12 + E .,I zQi f Qrj~ } $ 3 i ri

where A is a constant. Since both 52 and QQ Q-: are limited, this last expression is finite provided I I 0j 12 is bounded in D. It was proved in (I) that Q (u)

is majorised by exp (sM), where M is the bound of the matrix (uij), and s is the length of the curve from zo to z. The series (11) for w, is therefore majorised by an exponential series, and the differentiation is justified.

Applying this to equation (9), we can obtain a solution in the form (11) pro-

vided E a, + (ui ) v (t, z) d[ is bounded in D. This will certainly be 1' ~~~~~2

satisfied if both I I a, 12 and I u ( ji) v ( t z) d bounded in D, which is

true since ui (,u) is limited throughout D. Thus, provided we can subsequently determine v ([, z) such that J [v ( [t z) 12 dp is finite, the above solution for wi satisfies all the conditions.

3.4. We have now to obtain a solution of equation (10). On substituting the expression for wi in (10) we obtain an integro-differential equation of 1 the following form

(d f (ti, z) + ( Ku , z) v (X, z) dX. (12)

Provided If ( L, z) 121 t is bounded and the nucleus K (t, A; z) is limited

with bound N throughout D, we can solve this equation by successive approxi- mations.




Defining

_00

it is easily seen that K& (,u, ? ; z) is limited with bound s"-' N"/(n - 1) ! The nucleus

F ( ,u, X ; z) = K, (tL, x ; z) + K2 ( G4 Z) +**

therefore exists and is limited with bound

N(1 +sN+ 21s2N2+ ...) Nexp (sN)

and is uniformly limited in D. Now put v ( ,, zo) + Qf (u, z) vo (t,, z) and consider the expression

v(,u,z) -vo(uL,z)+Q F(00 F( .; z)vo(X?,z)dX. (13)

It satisfies equation (12) formally. Also

Iv (t, z) - vo (1 p,z) < Q 0 I | Fl (I, ; Z) 12 dx J I vo (),) 12dX}

< exp (sN) { J IVo (X) 12 dX}

where vo (X) is the upper bound of vo (X, z) in D. Since j vO (X, z) 12dX is - OD)

00

bounded in D, and therefore J I vo (?) 12 d? is finite, the series (13) has an

exponential series as its majorant, and the differentiation is justified. Further

IV [LZ)1 O re 2

dz)t2dp = v0(, Z) + QJ F(p, ?; z)vo(?, z) dX d2 coe 00 - 00

<AAt Ivo (tL Z) 12 d[ + 0

|Q j F (,u, X ; z) vo (X z) d;k d} _~~~~~~~~~~~~~~0

'IV VtL Z) 12

where A is a constant, and since F Q(, ?; z) is limited J_ v d(, z) 2d,> is 00

finite. We have therefore to show thatf(,u, z) and K (t, X; z) satisfy the conditions

specified above.



186 A. H. Wilson.

Now f ( , z) = b ( e, z) + ;u (L), {.Q1jwj? + Z Q, Q n -I a,}

ij r rZ 00~~~~~~~~~~~~~~~~~~~

and J If( u, z) 12 dR will be finite provided both lb (p, z) 12 dp and _ 00 _ 00 r0 J lzu (L) { QJWJ + >Qir Q E2-'r. aj} 12 dCl

are finite. The first condition is true by hypothesis; and since ua (,) is limited the second is satisfied if

E EQijWj + , Q-k Q Q-1 a,b 12 S j t

is finite. This is true as Q and Q Q-1 are limited and z i aj 12 is bounded in D.

We next have

K (tL, )<; z) =u ( ,, A) + E u ( 4)i Qi,.Q Q1 r j 43( e) 1~~~~~73 ~ ~ r irj

and is limited by ? 3.2 theorem 1. This completes the proof of the existence of solutions of equation (10).

4. Perturbations in Atomic Problems.

We now revert to the notation of ? 2, and consider the perturbation theory- in the light of the existence theorems just proved. In the first place, consider- the ordinary theory when only discrete characteristics occur. Equation (7) then becomes

ih/2n . am (t)- E Hmn an(t)

which was fully treated in (I). The necessary and sufficient condition for the existence of a solution was shown to be that the matrix Q (H) is limited. In general this means that the matrix (Hm.) must be limited, as only a very singular type of unlimited matrix (Hmn) could give rise to a limited Q (H). The perturbation theory was originally given for the case when a small parameter occurs in the perturbing term, and it is usually stated that the series of perturbations is convergent for sufficiently small values of the parameter. This is now seen to be irrelevant, as the convergence is that of an exponential series if (Hm,) is limited, and otherwise the series probably does not converge at all. The smallness of the parameter is of importance in that, as was shown in (I), it sometimes allows an asymptotic solution to exist when no exact solution is possible.

When we come to deal with the more general case of systems with continuous as well as discrete spectra the interpretation is not quite so simple. In the




first place the perturbations do not usually lead to limited matrices, and secondly there seem to be no solutions corresponding to the asymptotic ones of the simpler case. The question as to when the conditions laid down in ? 3 will be satisfied is a difficult one, and no general rule can be given. It would seem probable, however, that they will be satisfied if and only if the perturbing term in the differential equation for + is not the leading term at any of the singularities of the equation. For in this case the introduction of the new term will not affect the nature of the solution in the neighbourhood of the boundary points. When this is not so, we shall obtain entirely different results corresponding to the different divisions of the Hamiltonian H into the parts Ho, H1, H2, two of which can be chosen arbitrarily. Of course these formal solutions will not be convergent, but some of them will have a physical mean- ing. As there is no a priori reason for preferring one divergent series to another, the question arises as to which solutions we are to accept and which we are to reject.

Oppenheimer bases his theory on what he calls " nearly orthogonal functions." The property that he demands of these functions has nothing to do with the convergence of the series of perturbations, and can only affect the rate of convergence. It is therefore on the same footing as the parameter which occurs in the other perturbation theories. An alternative suggestion is that the boundary conditions are inadequate to deal with problems in which there is any ambiguity in the interpretation of the results. In the simple case of an isolated hydrogen atom the attraction of the nucleus is the only force acting on the electron, and there can be no ambiguity. When we come to consider the effect of a constant electric force on a hydrogen atom the conditions are entirely altered. At a large distance from the nucleus the constant field is greater than that due to the nucleus, and it is impossible to tell which part of the field the electron is in. If initially the electron is near the nucleus in a state of given energy, there is a probability that it will escape into a state of equal energy in that part of the field where the external force predominates, and the atom will be ionised. To observe this ionisation of atoms by an electric field we would not be concerned with the number of atoms possessing electrons in a given stationary state but with the number of free electrons moving under the action of the field, and we should have to arrange our experimental conditions accordingly. This would introduce boundary conditions of which we cannot take account in the mathematical solution of the problem. On the other hand, if we were interested in the effect of the electric field on the spectrum of the atom we should have to deal with the number of electrons



188 0. W. Richardson and F. S. Robertson.

in given stationary states of the atoms, or alternatively in the stationary states as modified by the external field. This would require an entirely different set of experimental conditions which would introduce entirely new boundary conditions into the complete mathematical solution. When we neglect these boundary conditions and substitute idealised ones we can scarcely expect our mathematical treatment to be complete. One way. of turning the difficulty without introducing the explicit boundary conditions is to work out a formal solution in that system of orthogonal functions which most nearly satisfies the experimental conditions, and to assume that this is asymptotic to the real solution. Thus for the hydrogen atom in an electric field we may work out the solution in terms of the functions corresponding to a free electron in an electric field. This will give the dissociation phenomena, and is just Oppen- heimer's result. If, however, we use the functions corresponding to the stationary states of the unperturbed atom we obtain the ordinary Stark effect. The suggestion put forward above does therefore offer a not unreasonable explanation of the apparent inconsistencies of the perturbation theory.

The Emission of Soft X-Rays by Different Elements at Higher Voltages.

By 0. W. RICHARDSON, F.R.S., Yarrow Research Professor of the Royal Society, and F. S. ROBERTSON, M.I.E.E., Senior Lecturer in Electrical Engineering, King's College, London.

(Received March 18, 1929.)

In a former paper* we investigated, by the photoelectric method, the efficiency of 14 elements as emitters of soft X-rays under various exciting voltages up to 500. We found that the efficiency had only an extreme variation by a factor of about 2 among all the elements tried which had a range of atomic number from 6 (carbon) to 79 (gold). It was also found to be a periodic function of the atomic number having for the elements tested a maximum about the middle of the periods falling away to a minimum value at the end. This is in contrast to the behaviour of ordinary X-rays for which the efficiency is proportional to the atomic number. It is to be remembered that the efficiencies

* ' Roy. Soc. Proc.,' A, vol. 115, p. 280 (1927).



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