On the Bethe-Lamb equations

ANNALS

OF

PHYSICS

79,

518-541 (1973)

On the Bethe-Lamb

Equations*

M. T. GRISARU AND H. N. PENDLETON Physics Department, Brandeis University, Waltham, Massachusetts 02154 AND

R. PETRASSO American Science and Engineering, Cambridge, Massachusetts 02139 Received November 9, 1972

A derivation is given of an effective Hamiltonian to be used in treating the decay of atomic states perturbed by a static applied field. This Hamiltonian is an improvement on that usually used in writing Bethe-Lamb equations. It includes the virtues of a Stark-perturbed golden rule method which has recently proved successful in treating decays of metastable states in helium, where the usual Bethe-Lamb approach fails.

1. INTRODUCTION

In recent years some controversy [ 1, 21 has arisen concerning the validity of the application of the Bethe-Lamb equations [3,4] to certain decay processes of excited atomic states. Some of these processes have been studied experimentally to a precision sufficient to reveal discrepancies between experimental results and common theoretical predictions, a situation which has led various authors to alternatives [2, 5, 61 to the Bethe-Lamb equations in the hope of removing these discrepancies. While these alternative methods do in fact lead to an understanding of the experimental data we feel that the Bethe-Lamb approach has sufficiently attractive

features

that it should

not be abandoned.

In this paper

we derive

from

first principles an effective Hamiltonian method which combines the virtues of the Bethe-Lamb approach with the success of the alternatives. We remind the reader that the Bethe-Lamb equations were devised to handle the kind of problem of which the following is typical. In the absence of external applied fields the low lying excited levels of the hydrogen atom have well-defined * Research supported in part by the National and GP-25485.

518 Copyright All rights

0 1973 by Academic Press, Inc. of reproduction in any form reserved.

Science Foundation

under Grants GP-18721

BETHE-LAMB

519

EQUATIONS

decay characteristics: the 2S,,, states, which decay predominantly via the emission of a pair of photons, have a mean life of 0.1215 set [7]; and the 2P,,, states and the 2P,,, states, which decay via the emission of a single electric dipole photon, have a mean life of 1.6 x 1O-s seconds [4]. In the presence of an applied electric (Stark) field the 2S,,, and 2P,,, states become mixed; if the Stark field is not too strong one will observe a complex decay pattern made of a long lifetime part, a part showing oscillations modulated by decay of intermediate lifetime and a short lifetime part. Tine long lifetime part will actually be a superposition of three different long-lived decays, with different lifetimes, which have not yet been experimentally distinguished. The problem is to compute the lifetimes, which depend on the Stark field, and the amplitudes of the various parts of the pattern, which in addition depend on the initial state of the 2S,,,-2P,,, mixture. Of these quantities, the ones of principal interest to experimentalists are the lifetimes of the long-lived states, since they (or an average of them) are most easily measured with precision. The elementary theory of the decay of an isolated excited atomic state [8] is begun by setting up time dependent perturbation theory, the perturbation being produced by the coupling of the atomic system to the radiation field. One makes the ansatz that the excited state, an eigenstate of the atomic Hamiltonian, decays exponentially. One then solves the equations approximately to obtain an expression for the lifetime. However, if one is dealing with several excited states coupled to each other by, for instance, a Stark field, it becomes crucial to know which linear superpositions should be assumed to decay exponentially. Since the interaction with the radiation field also produces some mixing one either has to guess the linear superposition correctly or must develop a more systematic approach. The Bethe-Lamb approach, as we believe it is usually understood, starts with the knowledge of the decay pattern in the absence of the perturbing external field. One represents the decay pattern by the solutions to a reduced Schriidinger equation in the space of the excited atomic states using an effective Hamiltonian (a matrix) which has an antihermitian part. The matrix elements of this part are proportional to the decay rates of the individual excited atomic states. One writes ffeff = Ho - iT,,2,

(].I)

where the hermitian operator HO (a matrix) has as eigenvalues the energies of the excited states while the hermitian operator r which is taken to be diagonal in the same basis as that which diagonalizes HO has the decay rates of these states as its eigenvalues. Now the presence of a perturbing Stark field is taken into account by adding the usual real hermitian Stark interaction term H, to H, , leauing T unchanged. One then solves the Bethe-Lamb equations i&,b/St = (H,, + H, - irj2)

# -= H,&,

(1.2)

520

GRISARU,

PENDLETON,

AND

PETRASSO

perhaps most succinctly by finding the eigenvectors and (complex) eigenvalues of the Stark-perturbed effective Hamiltonian. The decay rates are given by the imaginary parts of these eigenvalues; they can be obtained to a good approximation by computing the expectation values of r in the eigenstates of H, + H, (the usual Stark-perturbed atomic states). It turns out that this method, which we shall call the usual Bethe-Lamb (UBL) method, gives results in agreement with experimental data for the perturbed 2S1,, lifetime in hydrogenic atoms [9], but gives results in strong disagreement with experimental data for the perturbed 25, lifetime in helium [6]. Another approach, which we shall call the Stark-perturbed golden rule (SPGR) method, begins by ignoring the radiation field and solving the problem of the atomic system in the presence of the Stark field [5, 6, IO]. One assumes that it is the Stark-perturbed eigenstates of this problem that do the decaying. The decay rates are then calculated by doing first order time dependent perturbation theory (Fermi’s golden rule, etc.) for the coupling to the radiation field. The decay rates of perturbed metastable states so computed agree well with the experimental results in the case of helium [6]; but do not include linewidth corrections which can have observable effects in some situations in hydrogen. Apparently it was not realized for a long time that the SPGR method is quite different from the UBL method; for example the description in [4] mixes both methods. The obvious conclusion to be drawn from the preceding remarks is that neither the UBL method nor the SPGR method is generally correct. Holt and Sellin [2] have described a time-dependent perturbation theory which will yield correct results in hydrogen and helium; similar treatments had previously been given by KallCn and by Fontana and Lynch although these authors did not focus attention on the UBL-SPGR controversy [12]. As mentioned above we feel that the Bethe-Lamb method has many advantages. In Section 2 we derive a general expression for an effective Bethe-Lamb Hamiltonian, leaving some minor complicating features for discussion in later sections. In Section 3 we take a simple model situation, using it to illustrate the application of our improved Beth-Lamb (IBL) method and to examine the virtues and defects of the UBL and SPGR methods. We also indicate how to apply our IBL method to hydrogen and helium in the presence of moderately weak Stark fields. In Section 4 we discuss some general theoretical problems associated with the setting up of an effective Hamiltonian, then conclude with a discussion of some of the minor complicating features that occur in real hydrogen and helium, such as hyperfine interactions and spontaneous two-photon decays.

BETHE-LAMB

EQUATIONS

521

2 In order to postpone the complications associated with the complexities of real atomic systems (spin, unbound states, etc.) and with the renormalization program, we first treat a schematic version of our problem which exhibits all its important features. As we go along we shall make comments concerning the steps necessary to take the complications into account: Section 4 will be devoted to a further discussion of the more perplexing ones. Our model is very similar to that of Holt and Sellin [2]; we differ from them mainly in our formalism, which is patterned after that of Messiah [13], and in our conclusion regarding the validity of Bethe-Lamb equations. We suppose that our atom has a finite number of nondegenerate states, and denote the unperturbed atomic Hamiltonian by k: the eigenstates I m) then satisfy k / m)

=

k,,,

/ m\.

(2.1)

Our full Hamiltonian is a sum of three parts: a free Hamiltonian K consisting of li and a free radiation Hamiltonian, an interaction term V coupling the radiation to the atom, and an interaction term S coupling atomic states to an externally applied field. The states describing a noninteracting photon-atom pair are denoted by j m, p) and satisfy K I m, pi = (k,,, t w,)! m, p?,

(2.3)

where p is the photon momentum. In the Hilbert space 2 describing the system of atom and photons we define two subspaces: the subspace YE??spanned by those eigenstates of K in which no photon is present, and the orthogonal subspace ,PO . The associated projection operators P and Q satisfy P+Q=l,

PQ = QP = 0.

We shall assume that V satisfies the familiar selection rules which photons be emitted and absorbed singly; in particular PVP = 0.

(2.3) require that (2.4)

We further require that V behave sufficiently well that our subsequent operator and analytic function manipulations be justified. Our aim is to extract from the time dependent Schrodinger equation for the whole system a set of equations which describe the situation in the subspace .P,, . The time evolution of the whole system is described by the operator U(t) = exp[--iHt].

522

GRISARU,

PENDLETON,

AND

PETRASSO

Defining the resolvent G(z) = (z - H)-l

(2.5)

we can write exp[-izt]

G(z) dz

(2.6)

where CO, the contour of integration in the complex z-plane, starts at + co above the real axis and returns to + co below the real axis after enclosing all the points in the spectrum of H, which lie between a lower bound b, and + co on the real axis. We define an evolution operator in the subspace Z?= by u(t) = PU(t)

P = &

f

co

exp[--izt]

g(z) dz,

where g(z) = PG(z) P.

G-8)

The operator u(t) will describe the time development of a state in HP to the extent that one only wants information about that part of the state which stays in &, (the state decays as part of it leaves &,); u(t) is sufficient to answer questions about lifetimes, but not about decay products. We define X so that

v= X+ QVQ,

X E PVQ + QVP,

(2.9)

H=X+C.

(2.10)

and in addition define B and C so that B=K+S,

C=B+QVQ,

Then, using the operator identity (z - H)-’

= (z - C)-l + (z - C)-l X(z

H)-’

(2.11)

it is easy to derive the relation g(z) = &A4 + adz) w> g(z)

(2.12)

go(z) = P(z - C)-1 P

(2.13)

where and the mass operator w(z) is given by w(z) = P VQ(z - C)-l Q VP.

(2.14)

BETHE-LAMB

We solve (2.12) in terms of an operator sional space XD , as follows:

523

EQUATIONS

inversion

taking place in the finite-dimen(2.15)

g(z) = [z - b - w(z)]--‘, where b ;= PBP = PCP = P(K + S) P = k -+ s. We define a z-dependent

reduced Hamiltonian

(2.16)

h(z) by (2.17)

h(z) k b $ w(z).

the sum of a hermitian b and of a small operator w(z) which is usually nonhermitian. If V is sufficiently small it will still be possible to find a complete set of right eigenvectors of the nonhermitian h(z) satisfying (2.18)

h(z)1 r, n, z, = h,(z)1 r, n, z),

with n running from 1 to N, the dimension of the space ZP [14]. There will also be a complete set of left eigenvectors, not the same as the right eigenvectors, satisfying (2.19) f?+(z)1 I, I?, z) == h,*(z)1 I, n. I‘ The right eigenvectors satisfy but not among themselves:

orthogonality

relations

with

(r, n, z / r, tn, z‘

the left eigenvectors,

i’ 6 ,,,,,

(2.20)

We write spectral decompositions of h(z) and g(z): 7l=N

h(z) =

C

h,(z)

I r, 4 z?
(2.21)

,?I=1

‘0)

=

n=N / r, f7, z)(l, n, z I c z - h,(z) . n-1

(2.22)

The next stage of the argument relies on analytic properties of w(z) which are consequencesof assumedanalytic properties of the matrix elements of V. (These properties can be demonstrated to hold in every order of perturbation theory.) The massoperator w(z) is an analytic matrix function in the complex z-plane cut along the real axis from the branch point b, to + co. It can be analytically continued through this cut; in particular we shall focus our attention on its continuation from above the real axis downwards, which we call w-(z). These analytic properties are shared by the vectors / r, n, ~1;; and I I, n, z J, the eigenvalues X,(z), and the

524

GRISARU,

PENDLETON,

AND

PETRASSO

matrix g(z). However, if we continue g(z) from above the cut we expect to encounter N poles just below the real axis near the eigenvalues b, of the matrix b (since w+ is small). These poles come from the vanishing of the denominators z - X,(z) in (2.22) and occur at positions z, , solutions of (2.23)

zn = hz+(z,>.

These analytic properties of g+(z) are used to distort the integration contour C,, of (2.7) to C, shown in Fig. 1. Separating the contributions to the integral of

0 :I( ‘I’ I 14 IIL---

L----,----l

1 t ----lL_----__IL---,--_---

----

I Cl FIG.

1. The deformed integration contour for u(t).

the small circular contours surrounding straight sections C, we find that

the poles from the contributions

of the (2.24)

40 = %(G + us(t), where ?l=N

u,(t) =

C exp[--iz,2] ?I.=1

I r, 4 z,>(l, n, z, I 1 -

(2.25)

~,'(4

and us(t) is the contribution from C, . The integral over the vertical sections of C, is well- known [15] to give the familiar asymptotic power dependence of decaying states associated with the low-momentum part of the photon phase space, a power dependence which should dominate the decay for very large times. Those times are usually so large that it is no longer possible to see the decaying remnant in a practical experiment. The horizontal sections of C, contribute a background term which is bounded by e-bt times a constant; if we wait long enough this background term must be dominated by the pole contributions u,(t). In the temporal region of interest in a practical experiment it is a good approximation to replace u(t) by uD(t), which we shall do from now on.

BETHE-LAMB

525

EQUATIONS

It is understood that in (2.25) all the z-dependent terms have been obtained by analytic continuation from above the real axis before setting z = z, ; if we wanted the negative t behavior of u(t) we would do all the continuations from below the real axis, going onto a different sheet of g(z), and hitting poles in the upper halfplane instead of the lower (the theory possesses time reversal symmetry). In order to identify the decay rates we assume that the atomic system is described at t = 0 by a density matrix p which acts in the %‘, subspace. The experimentally measured decay curve N(t) is given by N(t) = Trace[u(t) put(t)] = c eeYntp, +

n

C e-i”““te-(l’2)(~“+~m)tpnn ?zfr?l

, (2,26J

where yn = -21m 2,)

(2.27)

%l?7 = Re(z, - z,*),

(2.28)

p71= (r, n. z, I r, n, z,><4 n, z, I p I 1, 4 -G/U

(2.29)

- ~,‘(z,)1”,

and prim =

(r3m, z, I r, n, z,Xl, n, z, I p I 1,m, z,>/[l - ~,‘kJl[l

- A,‘(z,>l.

(2.30)

Clearly the term in (2.26) which survives longest is the term from the first sum associated with the smallest constant y1 ; obviously we identify r;l as the longest lifetime present in the complex decay curve. The interpretation of the other yn as decay rates seems equally straightforward, although some doubt is cast on the precision of all these identifications by the following considerations. Since each eigenvector I r, n, z,) is computed using a different Hamiltonian h(z,), we no longer even have orthonormality between right and Ieft eigenvectors, that is

(1, n, z, I r, m, z,> f a,, .

(2.31)

Because of this fact, and of the presence of the [I - h,‘(z,] factors in the denominators of (2.29, the u(t) operators do not possess the semi-group multiplication property we have come to expect of evolution operators, that is.

41)&J i 4t1+ tz>.

(2.32)

One consequence of (2.32) is that it is impossible to define states which exhibit one single simple exponential decay curve; strictly speaking there are no decaying eigenstates. Another consequence is the impossibility of writing u(t) = exp[--i%rt] 595/79/z-16

(2.33)

526

GRISARU,

PENDLETON,

AND

PETRASSO

as an exact expression; it is therefore, impossible to write a set of Bethe-Lamb equations whose solutions reproduce exactly the decay pattern displayed in (2.26). However, since w(z) is small compared to b, we expect that the dependence on z of the eigenvectors of b + w(z) is slight, so that the right eigenvectors are approximately orthogonal to the left eigenvectors. (Such an expectation can be shown to be valid if the norm of w(z) is small compared to the smallest difference of eigenvalues of b.) We then define the Bethe-Lamb approximation by temporarily ignoring the variation with z of the eigenvectors and eigenvalues of h(z). That is, we assume that we may replace I r, n, z,} in (2.25) by j r, n, 5) for some fixed 5, with a similar replacement for I 1,n, 2,). We note that z, = X,(z,J, and assume that we may replace the z, on the right-hand side of this relation by 5, obtaining z, w h,(c). Finally we suppose that we may ignore X,‘(z) compared to 1, so that we have that (2.34)

where h(C) is our candidate for Bethe-Lamb effective Hamiltonian. Let us now reconsider the variation with z of the eigenvalues and eigenvectors of h(z). The pole locations z, will be close to the eigenvalues b, of b when ~(0 is small, and so will the approximate pole locations h,(c), no matter what we choose for 5. But if we choose 5 close to a particular z, we can expect that particular h,(c) to be a better approximation to z, than the other &&‘) will be to z,,, ; furthermore, we expect the error, which will always be small, to increase as 1 5 - z,,, 1increases. A similar consideration applies to the eigenvectors. In the applications of the Bethe-Lamb method, one is interested in the decay of a single state or of a group of states of nearly equal energies. The contribution to N(t) from n’s associated with states very different in energy plays a small role; therefore, one makes the best choice of Bethe-Lamb Hamiltonian by picking 5 to be near the energy of the group of decaying states of interest. If it turns out in a particular case that the results of the decay pattern calculations depend significantly on the choice of 5, one must conclude that in that case the Bethe-Lamb approximation is bad, and that the decay pattern cannot be adequately described by Bethe-Lamb equations. In practice we choose 5 to be a real number; keeping in mind that we are analytically continuing to 5 from the upper half plane, we find that w(C) = lim PVQ[S + ie - B - QVQ]-l QVP = L(n - iT(5)/2.

(2.35)

c-+0+

In order to obtain more information from (2.35) we first put the denominator in a more convenient form. We note that the best energy to use for the atomic state I b, m) is not the eigenvalue b, of b in that state, because L(c), the hermitian part of w(c), will contribute

BETHE-LAMB

EQUATIONS

527

a ievei shift to the state / b, m) which can profitably be included in the energy we associate with the state. We introduce an operator D which differs from the operator B only in that it assigns to the Stark perturbed atomic states the best energy values d, that atomic physics can provide (such as those of Bethe-Salpeter [4] or Erickson 1161). The operator D-B will be small (second order in V) embodying the radiative energy corrections to the atomic states. Using the identity (2.1 l), we have that JC(~;)= lim PVQ[< + in - D]-l QVP + lim EVQ[< + ic - 01-l r-0’ \: [(B - D) + QVD][<

Inserting the appropriate intermediate sions for I,(<) and I’(<):

+ ie -

r-4 L B - QVQ]-’

QVP.

(2.36)

states into (2.36) gives the following expres-

(2.37) T(5) = 2n c p(i - d,) O(< - d,) j d2Qn PV 1b, nt

n7,p;:,(b,

n7,p

/

VP + . . . . (2.38)

where the higher order terms can be computed by iterating the second term of (2.36). The function p appearing in (2.37) and (2.38) is the density of states function for the photons; the theta function in (2.38) ensures that the intermediate atomic states have an energy smaller than <. Notice that sandwiching the r of (2.38) between vectors representing a Stark-perturbed initial state and setting 5 to be the energy of that state yields the Stark:perturbed golden rule expression for the decay rate 161. The observed decay rates are given by the imaginary parts of the eigenvalues of the Bethe-Lamb Hamiltonian matrix Heff = b + L(L) - ir(5)/2

where H,, is the atomic which would presumably energy levels. When s = in the &-representation,

= f& + .S-t R(i) - ir(0/2-

(2.39)

Hamiltonian matrix (with the Stark field switched off) be used in the UBL method: it has Lamb-shifted atomic 0 the hermitian matrix R(5) has a vanishing diagonal part and has very small off-diagonal parts: R(5) = US) - (4 - k).

(2.40)

Since R is small, the main difference between the UBL and the IBL methods lies in the form of F.

528

GRISARU,

PENDLETON,

AND

PETRASSO

At this point one should proceed to solve the secular equation det (&r

- A) = 0

(2.41)

by suitable methods of numerical analysis, obtaining the N roots h,(c) and the Bethe-Lamb decay rates y,, = -21m A,({). In the next section we shall exhibit an approximate solution to (2.41) using a scheme designed to illuminate the BetheLamb method. The effective Hamiltonian of (2.39) is essentially the same as that of Breit and Teller [7], which unfortunately has not received the attention it deserves. 3

This section is devoted to a comparison of our improved Bethe-Lamb method, the usual Bethe-Lamb method, and the Stark-perturbed golden rule method. The main difference between our IBL method and the UBL method lies in the assumption made in the UBL method that F is field independent and is diagonal in the representation in which K is diagonal; but in fact F is found to have significant off-diagonal matrix elements and some dependence on the Stark interaction S. To display these effects in their simplest form we make further simplifications in the model used in Section 2. We allow one state 1a) to represent the ground-state complex (lS,,, , F = 0, 1 in H, 15, in He); another state 1b) to represent the metastable excited complex G% , F = 0, 1 in H, 25, in He); a third state [ c) to represent the excited state complex most strongly admixed to the metastable complex in the presence of an external field (2P,,, , F = 0, 1 in H, 2lPr in He); and a fourth state 1d) to represent all the other atomic states significantly coupled to the metastable complex by the external field (2P,,, , F = 1,2; nP,(F),n 2 3 for H; nlP1, n 3 3 for He). In applications these four single states will be replaced by sets of states; see Section 4. Each of the operators Ha , R, s, and r will then be a 4 x 4 matrix. The operator H,, is of course diagonal in the representation just defined; we assume that its eigenvalues are ordered so that E, -=cEb < EC < Ed . In real life EC < Eb for hydrogen, but for simplicity we adopt the order mentioned: it prevents the decay of the analog of the 2S,,, state to the 2P,,, state (in real hydrogen that decay is negligible because of the smallness of the associated phase space for the decay photon). We shall require that EC - Eb < Ed - EC < E, - E, . The matrix s is taken to obey a parity selection rule. We say that the states 1a) and I b) are even, that the states I c) and 1d) are odd, and insist that s connect an even state only to an odd state, and vice versa. We ask that the matrix elements (m j Y I n, p) obey a similar selection rule in m and n. Since we are interested in the decay of the perturbed j b) state we choose 5 = do ,

BETHE-LAMB

EQUATIONS

529

the Stark-perturbed energy of that state. Then the only intermediate atomic state allowed by the theta-function in (2.38) is the perturbed ground state, which is predominantly even with a small odd admixture for weak Stark fields. As a consequence when we compute the matrix elements of r in the unperturbed basis we find that they are small (first order in s) if they connect states of differing parities, even smaller (second order in s) if they connect two even states, and largest if they connect two odd states. Henceforth we shall abbreviate the phrases “first order in the Stark field” and “first order in s” by “O(,Sl)“, and the phrases “second order in the Stark field’ and “second order in s” by “O(S2)“. For similar reasons the matrix R also has O(N) matrix elements connecting states of differing parities, and by construction its diagonal elements are O(S2). We now find that eigenvalue h of the effective Hamiltonian (2.39) associated with the state j b); we employ a method designed to focus attention on the different roles played by the various matrix elements of Herr for the case of weak Stark fields. For this purpose we present Herr in the block matrix form

II f (f M’1

(3.1)

where M is that 3 x 3 submatrix of Herr involving the states 1c), / d), and / a), f is the 3-vector (I&,, , Hbd , Hba), and p is Hbb . The eigenvector equation (Heff - X)l r, b, 5) = 0

(52)

can be rewritten as the pair p-AXff’(l)=O,

(3.3)

f + (M - A) 9 = 0,

(3.4)

where we have chosen the normalization of the 4-vector I r, b, 5) so that its components in the unperturbed basis I b), / c>, / a>, j a) are (1, & , #d , 4,). From (3.3) and (3.4) we find that X = p - f * (M - X)-l f.

(3.5)

In inverting M - h we make use of the fact that four of the matrix elements of M are O(S1); denoting them by Tij and the remaining ones by Nij instead of Hij enables us to specify a convenient splitting of M: M =

M=N+T,

(3.6)

530

GRISARU,

PENDLETON,

AND

PETRASSO

Using (2.11) we find (M - ii-1 = (N - A)-1 - (N - A)-lT(N

- X)-l + ... .

(3.7)

The matrix element Hba = R&S) - ir,,(&‘)/2 = fa represents the attempt by radiative correction terms to mix the even parity states. In the case of hydrogen and in the case of helium, a careful evaluation of (3.5) using both terms (3.7) shows that such a term contributes about one part per million to the decay rate, since the effect is proportional to the ratio of the Lamb shift to the Lyman-a frequency. Therefore we shall setf, = 0 henceforth, which allows us to assert that the vector f is O(S2) (sincef, and fd really are). We shall compute X to O(S2) since the imaginary part has no zero or first order parts. Then using the condition that f is O(Sl), we have that A=p-f.(N-A)-‘f+...)

(3.8)

where we need keep only the zero order part of N - h. We split N by writing N=(:

;

i),

N=X+Y.

(3.9)

The matrix element Y,, = YdC has a contribution from R and one from r. These two terms are a priori of the same rough order of magnitude (barring selection rules) in hydrogen and helium, namely of the order of lo* to lo9 Hz. The smallest element of X - X is XC, - h, which is of the order of EC - Eb . In the case of hydrogen this number is also of the order of 10sHz, so a perturbation treatment of Y does not seem indicated. However, as will be discussed in more detail in Section 4, a rotational selection rule operates in this case, forcing the part of Y independent of the Stark field to vanish. As a consequence Y,, < EC - Eb , and a perturbation treatment is justified. In the case of helium the smallest element of X - h is still EC - Eb , but this number is now of the order of 1014Hz, so Ycd < EC - Ea and again a perturbation treatment is justified. Using (2.11) again we have that (N - X)-l = (X - X)-l - (X - X)-l Y(X - A)-’ + --a . Substituting (3.10) into (3.8) and returning to the original notation elements off, X, Y, and p, we obtain h = Hbb -

(3.10)

for the matrix

BETHE-LAMB

531

EQUATIONS

Because we are interested in the decay rate yb we take the imaginary part of each of the five terms in (3.1 l), multiplied by minus two. We write Yb

=

Yl

+

Yz

+

Y3

+

Y4

+

Ys

-t

“.

(3.12)

.

We may replace the h in the denominators of yZ through y5 by Eb since the numerators are O(S2) and the zero order value of h is just Eb The contribution y1 is rbb , which we evaluate using (2.38). The rate r,,, is O(S2); it arises from the presence in the Stark-perturbed ground state of small pieces of the odd parity states I c) and 1d‘). The result for y1 is then 71

=

‘&,ScaI(Ec

-

&YY

Ybceb

+

(&d~do/(&

-

-‘%)‘)

Ybddh

f

0@3),

(3.13)

where ybCCbis given by = 2’dEb

Ybrrh

-

&) 1 d2G, (b / v 1C, P>(c,

(3.14)

p j k’ i bi

and a similar expression holds for ybddb . Expressions such as these, which have a subscript sequence in which the parity order is (even) (odd) (odd) (even) are all roughly of the order of magnitude of the decay rates of the unperturbed / c) or , d; states ( lo8 to 10QHz in hydrogen and helium); the differences arise entirely from the dipole matrix elements inside (3.14). We shall also encounter other “allowed” sequences, such as (even) (odd) (even) (odd); the sequence is allowed if the first pair contains both parities and the second pair also contains both parities. For yZ we have that y.J

= 2

IrncHb$b)

Re(EH,;2

cc

-

Eb)

+

2

Im(/f$)

b

y(?;Fb)

= yz1 + yz2

ee

b

Eb)/[&

-

(3.15)

Then Y21

=

+bcrcb

+

rb&b%%

-

Eh2

-i-

irk]

(3.16)

Ed))

+

ow).

(3.17)

&rzc].

(3.18)

where r,b is O(Sl), and to that order is given by rL,b

=

(sacYcad&

-

a)

+

6-%dYeadb/(Ea

-

Then y2r1 , the first of the four terms making up yZ1, is given by YZll

=

Sbc&-Ycacb&

The largest contribution Y22

=

rcc[(sbc

+

-

Eb)I&

-

&)[6%

-

-6)’

+

to yb comes from yzz : Rbc)&b

+

f&b)

-

irbcrcbwc

-

a2

+

g%).

(3.19)

532

GRISARU,

PENDLETON,

AND

PETRASSO

Each factor inside the brackets is O(Sl), but the largest by far are the factors S,, and SC, . We have just seen that r,, is at best of the order of a typical “allowed” Stark energy (say S,,) times a dimensionless factor y/(E, - Ea). That dimensionless factor is smaller than 1O-5 for both hydrogen and helium, so we make a negligible error in dropping such terms from (3.19). Since the matrix element Rh, is roughly of the same size as I’,, (see 2.37), we have the approximate expression (3.20) where we have evaluated r,, to lowest Stark order. In the UBL method, the factor ycaac is replaced by the actual decay rate of the unperturbed 1c) state. This replacement is equivalent to replacing, in (3.14), Eb - E, by E, - E, and it leads to the fractional error 3(E, - Eb)/(Eb - EJ. This error is insignificant in hydrogen ((2 x 10-6) but serious in helium (-l/10). On the other hand the SPGR method uses the correct ycaacwhich is computed directly from the oscillator strengths. It omits the term (2) rz, in the denominator of (3.20), an omission which is insignificant in helium, as we have mentioned earlier, but produces a fractional error of about l/400 in hydrogen which is not much smaller than the present experimental fractional uncertainties of 2 x 1O-2 [ll]. We can at this point check the validity of writing our expressions to the lowest significant order in the Stark field. In order to obtain a decay rate ya which is measurable with some precision the external field must be adjusted to make yb approximately the inverse of the transit time of the quenching apparatus, that is, of the order of 103Hz. Since ycaaois not smaller than lO*Hz, the ratio S,,/(E, - Eb), which is the largest one that occurs in any of the contributions to yb , must be less than 10-2. In beam foil experiments ratios larger than this may be encountered, so that one might wish to find the solution to (2.41) without resorting to our pedagogically motivated perturbation expansions. Now that we have identified the dominant contribution to the decay rate we reconsider the contribution y1 . The ratio of y1 to y22 is of the order of 6% - Ed2/(&

- EJ2,

which is less than 3 x lo-l3 in hydrogen, and less than l/900 in helium. Since the present experimental fractional uncertainty in yb for helium is approximately 2 x 1O-2 [6] the presence or absence of the term y1 is at present experimentally unchecked. In the UBL method no such term appears; in the SPGR method it does. Let us now consider the relative sizes of yzl and yz2 . The ratio y21/y22 is of the order of 2(Ec - Eb)/(Eo - E,) + 2(E, - Eb)/(Ed - E,), assuming all allowed yii to be of roughly the same size. This estimate of order is not significantly in error even in hydrogen where some y’s, such as yeaad and y&,bd, are zero. Then for

BETHE-LAMB

533

EQUATIONS

hydrogen y21/y22 is less than 3 x 10-6, whereas for helium it is of the order of l/10. Contributions of the yZI type are absent from the UBL expressions (because the ground state is not taken to be Stark-perturbed) but are present in some SPGR expressions [6]. The expressions for yX1 and ys2 are so similar to those for yZl and yZZ that we need not discuss them in detail. The y3 contribution is smaller than the yZ contribution by a ratio which is of the order of magnitude of (E, - EJ2/(Ed - Eb)2, which is 1O-2 in hydrogen and 6 x 1O-2 in helium. Therefore y3 contributions are significant in both hydrogen and helium; both the UBL and the SPGR method provide estimates of them, though the SPGR method provides better ones through its use of oscillator strengths rather than the UBL’s unperturbed decay rates. In helium similar contributions from higher P-states must also be included; in hydrogen it is not necessary to do that because for such states the ratio (EC - Eh)/(Ed - E,,) is less than 10-6. The contributions y4 and ys are absent from UBL expressions because in the UBL effective Hamiltonian Hcd and Hdc are taken to be zero. The real part of H cd 3 Red, is especially small in real atoms because of the smallness of Lamb shifts in P-states (although this would not be true in our oversimplified four state model) so we shall omit it. Then we have that

- Im(&&d

ImKHz - E&&d - &)I /H,,-Eb12jHdd-

EJ2

red

*

(3.21)

The second term yd2 will involve a numerator which must contain a product of three r’s, whereas the first term, yal , need have only one r in the numerator. The ratio y42/y41 is of the order of y2/(Eb - E,)(E, - E,) or smaller, and so is smaller than IO-l2 for both hydrogen and helium. We shall neglect the contributions of the imaginary parts of Hbc and Hdb to Re(HbeHdb) for reasons already mentioned in discussing yz2 in the paragraph following (3.19). Then (3.22) An order of magnitude estimate of y4/y2 is given by the ratio (EC - Eb)/(Ed - E,,) which is $ in helium. The contribution from y5 is the same as that from y4, so we see that their joint contribution is the second most important one in yb, being larger than that of y3 by a factor of the order of 8 in helium. The contributions from higher P-states are even more important for y4 and ys than they are for y3 , so a summation over many states of the d type must actually be performed for helium.

534

GRISARU,

PENDLETON,

AND

PETRASSO

Since the UBL method omits y4 and y5 entirely it is quite unsatisfactory for the analysis of helium decays. The SPGR provides an approximate version of (3.22) in which the r’s in the denominators are absent; as we have seen that approximation is eminently satisfactory for helium. The UBL method works for hydrogen because the terms of the y4 type computed using (3.22) for d-type states with IZ > 3 are of the order of 10” of yZ , and so can be neglected; and the prototype d state, the 2P,,, state, yields a vanishing F,, because of the rotational invariance selection rule noted by Holt and Sellin [2] and discussed in the next section. Collecting all the contributions that may be of significance in hydrogen or helium, and letting the variables m and II range over the pair c, d, we obtain

bm

+ m;,, @,,f+

c m.n

nb

maan

“,;

[~bw&a%nanb 6%

+ -

6%

-

+ ii~;,,,lKE, QW,

Eb)(En

-

Eb)

- Ebj2 + tY;,,,l

&nSmaYbmml(‘% E,)2 + ir:,,,]

-

Eb)

(3.23) -

The first term in (3.23) is the UBL expression [ll] for weak Stark correct oscillator strength rates ymaan replacing the more common rates. The second term represents the effects caused by the coherent mixing of states in the initial decaying state ] b), and except for denominator, is given by the SPGR expression [5, 61. The third term effects caused by the coherent Stark-induced mixing of states in 1a), and except for the y’s in the denominators, is given by the SPGR

fields with the experimental Stark-induced the y’s in the represents the the final state expression [6].

4 In this section we discuss the modifications necessary in order to apply the method of the previous sections to a realistic situation. To illustrate these modifications we return here to the decay of the Stark-perturbed 2S1,, state of hydrogen and the Stark perturbed 2’S’, state of helium. We write the full Hamiltonian for the atom interacting with the quantized radiation field as H = Hat,,

+ %,a + Vc + V + Vs ,

(4.1)

where Hat,, includes the Coulomb interaction, Hrad is the Hamiltonian of the free radiation field, and V, is the external (Stark) potential. The expression V, + V denotes a particular way of splitting up the J * A interaction between the atom and the radiation field that we now describe: V is that part of the interaction which

BETHE-LAMB

EQUATIONS

535

allows for emission and absorption of photons of relatively low energy (“visible photons”) and which therefore is mainly responsible for the processes we are interested in, namely, decays of excited atomic states. V, allows for emission and absorption of high energy photons (“gammas”) and is mainly responsible for excitation of the atom to relatively high energy continuum states. The precise place where the division between low and high energy photons is made is not important; all that matters is that the separation be made at energies well above those of the excited states of interest. The interactions V, and V are also responsible for the radiative corrections to the atomic energy levels and states. We shall treat the problem of renormalization rather cavalierly by assuming that a renormalization program [f 71 can be carried out without interfering with the above splitting, although the renormalized forms for the operators may be quite complex. We shall also ignore infrared divergence problems since devices are known for supressing them. We thus take it for granted that all the difficulties of quantum electrodynamics either can be overcome or have no bearing on the problem at hand. In principle we can proceed as follows: the system described by the Hamiltonian (4.2) consists of physical photons and an atom whose levels have been shifted by radiative corrections produced by V, (the V corrections are missing so that the Lamb shift produced by the above Hamiltonian is not correct) but whose states are stable against decay since the V coupling is absent (at least the low-lying ones are stable; a high one may decay if its energy is large enough to allow emission of what we call gammas). Our full Hamiltonian is then H=K+V+V,=BAV.

(4.3)

We assume that we know the eigenstates and eigenvalues of K. We then split the Hilbert space in which K operates into 3?! and Xc , XV being spanned by eigenstates of K in which no real photons are present; thus the atomic states in $, must be of sufficiently low energy that they cannot emit real gammas. We assume that the matrix element (m / V I n,p> of the renormalized interaction Hamiltonian V is well approximated by the traditional dipole approximation [18] expression e(nz I r j n) . (0 / E ] p) for the atomic states and photons of interest in our computations. We define the operator b = P(K + V,) P = k + s, whose eigenstates are Stark perturbed

atomic states, and

W(Z) = PVQ[z - B - Q VQ]-l Q VP.

(4.5)

536

GRISARU,

PENDLETON,

AND

PETRASSO

We obtain then an effective Hamiltonian whose eigenvalues will yield the decay rates. In practice we are faced with two problems. The first one is that the eigenstates and eigenvalues of b are not exactly those of the Stark perturbed low lying atomic states because the operator s = PV,P does not couple in the higher states whereas Vs does. The difference, however, will be slight provided we make .%$ large enough so that all the states which are appreciably coupled by the Stark field to the states of interest (e.g., 2S,,,) are included. It turns out that in order to achieve accuracy sufficient to make the theoretical uncertainty smaller than the experimental ones, one should include states with energies up to +25eV in the case of helium [6], and up to -3eV in the case of hydrogen (that is, one need include only the 2&,, ,2P,,, , and 2P,,, states). Therefore in the case of helium the subspace X9 in which the effective Hamiltonian b + w(c) operates is quite large and includes continuum states. The treatment of&r we have given in Section 3 can obviously be generalized to the case in which the operator M acts in a large space, so long as the connecting vector f is O(S1) and so long as the off-diagonal elements of N may be treated as a perturbation during the process of inverting N - A. We shall then obtain our formula (3.23), where the sums over m and n run over all states in .&, except b states; some part of those summations may be replaced by integrations over continuum states representing ionized atoms. Clearly the choice of ~97~also dictates the frequency at which the division between visible and high energy photons should be made since the states in 7ZDhave to be stable against decay to high energy photons. Therefore, F’, should include interactions of the atom with photons of frequency greater than approximately 3 x 1016Hz for hydrogen, and approximately 2 x 10leHz for helium. This brings us to our second problem, that an appreciable part of the radiative correction is produced by V. In principle, the method we have outlined in Sections 2 and 3 will handle this problem. It treats the difference between the eigenenergies of K + S and those of the real part of the radiatively corrected Hamiltonian D as a perturbation, which is handled using the usual technique of perturbation theory. That procedure may cause some concern in connection with the 2S1,, - 2Pll, subspace of the hydrogen problem since the perturbation looks large compared to the splitting. There is probably no cause for alarm, however, since we have chosen states which are those indicated by degenerate perturbation theory, that is, those that diagonalize that piece of the perturbation operator that lives in the 2Sl12 - 2P,,, subspace. The greatest problem posed by this method is that we won’t actually know what the perturbation operator really is until a renormalization program that respects our visible-gamma splitting is implemented, so at the moment we must content ourselves with the estimate that the operator probably has a norm which is of the order of the largest Lamb shift computed by standard quantum electrodynamics, which then allows us to neglect it for most purposes.

BETHE-LAMB

EQUATIONS

537

Let us now consider in detail the decay of the Stark-perturbed 2S,,, state in hydrogen. We first note that there are actually four of these states and four lS,,, states in the ground state complex corresponding to the four distinct ways the spins of the electron and proton may be assigned. The formalism of Section 3 is easily extended to take care of this case, or any case in which the a and b spaces have a finite dimension greater than one. The vector f will then be a rectangular matrix, the scalar p will be a finite-dimensional square matrix, and in the computations of objects such as ymaan one must sum over a complete set of “ground states” a. Thus with the Einstein summation convention applying to the letter a, and with the understanding that b states are chosen to diagonalize TVand are not summed over, the formula (3.23) and the derivation leading up to it will apply to our generalized case. Note that we can enlarge the b space even further if that proves convenient; if, for example, the Stark field were strong enough to produce almost as much 2P,,, state as 2S,,, state in the perturbed 2S,,, state, one would include the 2P,,, state in the b space and would diagonalize the p submatrix of Heff exactly [IO], while still treating the Stark coupling to other states as a perturbation. The hyperfine interaction in hydrogen places the F = 0 state lower in energy than the three F = 1 states, which are split by the Stark field into an mF = f 1 doublet and an mF = 0 singlet. The two mF = 0 states are also mixed by the Stark field, but the correct Stark perturbed states are easily found. The three different groups of states so singled out have different lifetimes, which may be calculated using (3.23) if the Stark field is large enough to allow us to ignore the spontaneous decay of the unperturbed 2S,,, states. In using (3.23) for this purpose the letter b will range over the four Stark perturbed states, and is to be chosen to be the same at all places in the formula. The resulting variations in the lifetimes turn out to be of the order of 20 %, as one might expect: the appropriate ratio is that of the hyperfine splitting to the Lamb shift splitting, which is approximately l/5. Next let us consider the selection rule [2] affecting the factors ylllaan appearing in (3.23). Since our ground state complex also consists of four states these numbers are a sum of four terms of the form of (3.14) with appropriate replacements of letters. In principle one should use hyperfine-corrected eigenstates (in the absence of the Stark interaction) for the intermediate a states, but the fractional error involved in not doing so is of the order of the ratio of the hyperfine splitting to the Lyman-a: energy, and so is negligible. The following argument concerning the selection rule applying to ymaan is unaffected by the choice of basis states / a: , as will be evident. We let m stand for any 2P,!, state, and n stand for any 2P,,, state. Then

538

GRISARU,

PENDLETON,

AND

PETRASSO

where i labels the four states in the ground state complex. We observe that the operator V is rotationally invariant, and that the projection operator within the braces is also rotationally invariant. Therefore the matrix element in question is of a product of three rotationally invariant operators, and so must conserve J as well as m, . Hence the element indicated must be zero, as claimed in Section 3. Notice that if we did not sum over i the operator in the braces would not be rotationally invariant, and the matrix element wouldn’t vanish. In the polarization experiments of Ott, Kauppila, and Fite [19] the projection operator occurring in the position of the braces incorporates information about the direction in which the photons have been emitted, thus breaking rotational invariance and allowing the 2Pll, - 2P,,, matrix element occurring in the expectation value of the polarization to have a nonzero value. A more transparent treatment of the rotational invariance selection rule is obtained by considering the zero Stark field limit of the expression for w(z) (we remind the reader that ymaan is computed in that limit): w(z) = PVQ(z - K - QVQ)-l QVP;

every operator occurring in (4.7) is manifestly rotationally invariant, so w(z) is also. Finally, let us consider the spontaneous decays of the metastable unperturbed 2S1,, states of hydrogen and of the metastable unperturbed 2?S’,,state of helium. That decay in helium proceeds via two photon emission; in hydrogen a similar two photon decay [7, 8,9] dominates the single magnetic photon decay. So far our formalism has not displayed such two-photon effects: in order to obtain them we must compute the decay rate to at least fourth order in the interaction strength V. We could attempt to do this by first computing the Bethe-Lamb effective Hamiltonian h(c) to fourth order in V, and then follow that by a similarly accurate solution of the secular equation associated with h(c). This method has the disadvantage that we can no longer be confident that it makes little difference what point 5 we choose for the evaluation of h(z). The reason for our hesitancy is that the actual location z, at which we should compute h(z) differs from the unperturbed value (which is what we use for 5) by terms quadratic in V (see below). Thus

and since w’ and z, - 5 both have significant parts second order in V, the fourth order contribution to w might have an important part which would be lost by the procedure outlined above. We therefore return to the exact expression (2.23). We must then compute the eigenvalue X,(z) of the operator b + w(z); we solve the eigenvalue equation using second order Rayleigh-Schrodinger perturbation theory treating w(z) as the

BETHE-LAMB

539

EQUATIONS

perturbation. We then solve (2.23) by iteration to sufficient accuracy to incorporate all fourth order contributions from V. The result is Z n-

- b, + (bn / ufb,) / bn,\[l + (bn i d(b,) -+- zn

(bn / w(b,)

/ bn>]

/ bi)(bi j w(b,) I btzj 6, - bi

Inserting here w(b,,) = PVQ[b,L + i.z - B - QVQ]-l Q VP and keeping terms through fourth order in V gives us an explicit form for Z, from which the fourth order contribution to the decay rates may be computed: z, = b, + (bn j PVQ(b,

+ ic - B)-l QVP 1b/j:,

- (bn 1PVQ(b,

+ ic - B)-l QVP 1 bn?

/‘. (bn I PVQ(b,

+ ic -

B)-2 Q VP / bn)

f (bn I PVQ(bn + ic --- B)-’ x QVQ(ba

+ ie -

QVQ(b,

+- if - B)-l

/3)-l QVP / b/T;>

+cif,,____ bn-l bi(bti

1PVQ(b,

x (bi I PVQ(blE f

ic - B)-1 QVP ) bn,>.

+ ie -

B)-l Q VP j bij (4. IO)

We want to evaluate this expression when j bn) is the Stark perturbed metastable state. The second and third terms in (4.10) are O(S2). The second term gives us the one photon contribution to the decay rate of Section 3. The third term represents the fourth order contribution mentioned in connection with (4.8). Since it is O(S2) it will be negligible (for weak Stark fields) compared to U(S0) contributions from the fourth and fifth terms. This implies that an effective Hamiltonian h(c) computed to fourth order in V will give the correct O(S0) results for the decay rates. We therefore look for O(S0) contributions from the fourth and fifth terms and so use Stark unperturbed states everywhere and replace B by K. We insert complete sets of intermediate eigenstates of K at every place where the factors (b, + ic - K)-l appear. Because of selection rules these eigenstates describe an unperturbed atom accompanied by zero, one or two real photons. The factors become replaced by energy denominators (6, + ie - E)-l which we rewrite as Pr Val(b, - E)-l - 7&3(bn - E) where E is the energy of the eigenstate of K. To obtain the imaginary part of Z, we keep terms involving one or three 6functions and examine the restriction imposed by the resulting energy conservation conditions. These conditions together with selection rules make the fifth term unimportant (in the case of helium there is no intermediate atomic state of lower

540

GRISARU, PENDLETON,

AND PETRASSO

energy consistent with the selection rule; in the case of hydrogen the allowed intermediate 2P,,, state must be accompanied by a photon of such low energy that the term is negligible). In the fourth term replacing the first or last denominator by a a-function leads to negligible contributions for the same reasons. The only appreciable contribution comes from replacing the middle denominator by a a-function: Yzphotons

=

(2s I v I nP, q)
rr

En

+ c

n

w

+

wq -

q, P>

&

I v I np, PXP, HP i V / lS, q, p> 2 i En + wg - 42,

x 6(E2, - El, - cup - w,) d3p d3q,

(4.11)

the usual two photon spontaneous decay rate [7].

CONCLUSIONS

In this paper we derived a system of Bethe-Lamb equations, starting from the usual theory of matter interacting with radiation and studying its implications in a subspace describing atoms which have not emitted photons. We emphasize that these equations are approximate, and that it is not possible to obtain an exact reduced Hamiltonian. As a way of describing the evolution of decaying states we expect the reduced Hamiltonian method to be convenient and of satisfactory accuracy under the conditions that usually hold in atomic physics. It should also provide a convenient framework for considering the decay patterns of atoms in time-varying external fields, although the theoretical justification of such an extension has not been provided in this paper. We remind the reader that the method of solution of the Bethe-Lamb equations which we used in Section 3 was chosen primarily for pedagogical purposes, even though it does provide a practical computational method in the weak field case. The diagonalization of the N x N reduced Hamiltonian matrix using standard techniques of numerical analysis in conjunction with modern digital computing facilities will usually be the preferred method of obtaining the detailed numerical predictions of the Bethe-Lamb description of atomic decays. The perceptive reader will realize that we are bothered by the renormalization problem. We have become accustomed to paying lip service to the methods invented in the late 40’s, and have no doubt that those methods are adequate for treating atomic physics problems of a simple nature. However, we do not yet have a logically consistent scheme for handling the renormalization problems of the kind encountered in this paper. It is conceivable that experimental technique will eventually

BETHE-LAMB

541

EQUATIONS

reach a precision so fine that details of the structure of the renormalized interaction operator V will be subject to experimental investigation; by that time we hope to have a logically consistent renormalization scheme that will satisfactorily account for such details.

ACKNOWLEDGMENTS We wish to thank S. Berko, H. Gould, H. Holt, V. Hughes, A. Mills, W. Ott, A. Ramsey, and I. Sellin for stimulating of this paper.

C. Johnson, conversations

W. Lamb, E. Lipworth. concerning the topic

REFERENCES 1. C. E. JOHNSON, Bull. Amer. Phys. Sot. 17 (1972), 454. 2. H. K. HOLT AND I. A. SELLIN, Phys. Rev. A 6 (1972), 508. 3. W. E. LAMB, JR. AND R. C. RETHERFORD, Phys. Rev. 79 (1950), 549; W. E. LAMB, JR., Phys. Rev. 85 (1952), 259. 4. H. A. BETHE AND E. E. SALPETER, “Q uan t urn Mechanics of One- and Two-Electron Atoms.” Springer-Verlag, Berlin, 1957. 5. H. K. HOLT AND R. KROTKOV, Phys. Rev. 144 (1966), 82. 6. R. PETRA~~O AND A. T. RAMSEY, Phys. Rev. A 5 (1972), 79. 7. G. BREIT AND E. TELLER, Astrophys. J. 91 (1940), 215; J. SHAPIRO AND G. BREIT, Phys. Rev. 113 (1959), 179; S. KLARSFELD, Phys. Lett. A 30 (1969), 382. 8. V. WEI~~KOPF AND E. WIGNER, 2. Phys. 63 (1930), 54. 9. C. Y. FAN, M. GARCIA-MUNOZ, AND I. A. SELLIN, Phys. Rev. 161 (1967), 6; H. W. KUGEL, M. LEVENTHAL, AND D. E. MURNICK, Phys. Rev. A 6 (1972), 1306; M. LEVENTHAL, D. E. MURNICK, AND H. W. KUGEL, Phys. Rev. Left. 28 (1972), 1609; G. P. LAWRENCE, C. Y. FAN, AND S. BASHKIN, Phys. Rev. Lett. 28 (1972), 1612. 10. G. L~DERS, 2. Naturforsch 5a (1950), 608; G. DRAKE, Canud. J. Phys. 50 (1972), 1896. 1 I. C. Y. FAN, M. GARCIA-MUNOZ, AND I. A. SELLIN, Phys. Rev. 161 (1967), 6. 12. A. 0. G. KXLLBN, “Handbuch der Physik,” Vol. 5, Springer-Verlag, Berlin, 1958: P. R. FONTANA AND D. J. LYNCH, Phys. Rev. A 2 (1970), 347. 13. A. MESSIAH, “Quantum Mechanics,” Vol. 2, Chap. 21, Sect. 13, Wiley, New York, 1962: see also L. P. HORWITZ AND J. P. MARCHAND, Rocky Mounrain J. Math. 1 (1971), 225. 14. M. M. STERNHEIM AND J. F. WALKER, Phys. Rev. C 6 (1972), 114. 15. W. R. FRAZER AND S. PEYTON, Ann. Phys. (N. Y.) 40 (1966), 46. 16. G. W. ERICKSON, Phys. Rev. Lett. 27 (1971), 780. 17. J. GLIMM, Advances ifz Math. 3 (1969), 101; J. GLIMM AND A. JAFFE, in “1970 Les Houches Lectures” (C. Dewitt and R. Stora, eds.) Gordon and Breach, New York, 1972. 18. M. GOEPPERT-MAYER, Ann. Phyx 9 (1931), 273. 19. W. R. OTT, W. E. KAUPPILA, AND W. L. FITE, Phys. Rev. A 1 (1970), 1089.

595/79/2-V