370

BETHE, 8 ROAN, AN D STEH N

It should be a,dded that the discrepancy was calcu-
lated using the value of pz/pp measured by Bloch, Levinthal, and Packard, who found

X=pD/pal=0. 3070126&0.0000021.

However, Bitter and Siegbahn, measuring ratio, have found, respectively,
E= 0.3070210&0.0000050

the same (6)

K= 0.3070183+0.0000015.
The result (7) is larger than (5) by about two parts in one hundred thousand, and would make the dis-

crepancy (using Nafe and Nelson's determination of the
h.f.s.) 1.5X10 ', which would be hard to account for on
the basis of this calculation alone. In conclusion, then, we are not yet in a position to
estimate structural sects. In order to do so, we need,
on the one hand, a more accurate value for the deuteronproton moment ratio, and on the other a better knowl-
edge of the deuteron 5 state wave function.
ACKNOWI EDGMENT
The author wishes to thank Professor H. A. Bethe and Dr. A. Bohr, who gave generously of their time and interest, and whose help and criticism were indis-
pensable.

PH YSI CAL R EVI EW

VOLUM E 77, ~VUM BER 3

FLBRUARY l, lo50

Numerical Value of the Lamb Shift
H. A. BETHE AND L. M. BROWN
Laboratory of Nuclear Studies, Cornell University, Ithaca, New York
AND
J. R. STEHN
Knolls Atomic I'macr Laboratory, Schenectady, Nm York
(Received October 10, 1949}
The average excitation potential of the 2s state of hydrogen which occurs in the Lamb shift, is calculated
numerically and found to be 16.646&0.007 Ry. This gives a theoretical value of 1051.41&0.15 megacycles for the Lamb shift, compared with the latest experimental value of 1062&5. It is not known whether the discrepancy of 10 Mc can be explained by relativistic effects. Simple analytical approximations are discussed
which make plausible the high value of the average excitation potential and give a good approximate value for it.

N this paper, we are reporting two independent - numerical calculations of the average excitation
potential of the hydrogen atom which occurs in the
formula for the Lamb shift. ' The first calculation was done in 1947 by one of us (J.R.S.) with the help of
Miss Steward, the second in 1949 by L.M.B.

and for states with //0

— +- ( ~Z(n„

t) = 8Z4

oP
RyI

ln

Ry

j no' 3or 4 ko(no, l)

3 ci;
g 21+1 I,

(2)

where

jj= cc~~,;—= 1—/(1l/+l 1)

for for

= l+ ~~
l

(2a)

The formula for the Lamb shift of a hydrogen level np, l has been derived by many authors' and is for s states

— — „0)= ~E(

( 8Z' a' RyI l

p —l 2+5

no' 3pr 4 ko(no, 0)

6

—1q
5i ~

(1)

' W. E. Lamb and R. C. Retherford, Phys. Rev. 72, 241 (1947};
75, 1325 (1949}.
~ H. Kroll and W. E. Lamb, Phys. Rev. 75, 388 (1949}; ¹
J. B. French and V. F. Weisskopf, Phys. Rev. 75, 1240 (1949};
R. P. Feynman, Phys. Rev. 74, 1430 (1948}, and correction in
Phys. Rev. 76, 769 (1949), footnote 13; J. Schwinger, Phys. Rev.
76, 790 (1949).

In these formulas, Z is the nuclear charge, pip the

principal and l the orbital quantum number, n=e'/kc

)— the fine structure constant, Ry the Rydberg energy,
p, =mc' and kp the average excitation energy which we wish to calculate.
This average energy is de6ned by'

kp(np, ln

l)
PI (npOI

P, In) I'(E„E,,

Ry n

=XI ( ~l p. in) I'(E.-E.»n Ry

'H. A. Bethe, Phys. Rev. 72, 339 (1947}, quoted as A in the
following. The deinition is in Eq. (6}.

VALUE OF TH E LA M 8 SH I FT

TAsx.E I. Oscillator strengths for hydrogen.

Transitions from to m=1
3
5 large

2$ nP
0.43486 0.10276
03..066441m93'

—0.S1S38732
0.01359 0.00305
00..010021m21'

sd

0.69576

0.12180

0.04437 3.257m

'

where (nof~ p, ~n) denotes the matrix element of an arbitrary momentum component p, corresponding to the transition from the "initial" state no, l, of energy
Eo, to the final state w, of energy E„.Note that on the
left-hand side the matrix element from the initial state
l=0 is involved, regardless of the value of / for which

ko(no, l) is to be calculated: This definition is necessary

because the sum on the left would vanish (see A) if we
used l/0 for the initial state. For s states, then, ko is

actually
E Eo

the geometric average with the weighting

of the factor

pexocoit(aEtio—nEoen)e; rgfoyr

other states, ko is defined in as close analogy to the s

states as possible. The denominator Ry in the argu-

ments of the logarithms could be omitted for s states;

for l/0, it serves to make the argument dimensionless

and appears again in Eq. (2); any other constant

energy would serve as well, but Ry is most convenient.
f It is convenient to introduce the oscillator strength

by putting

(n,l( p. n) )'=-',m(E„—E)f(nol, n)

~

~

and to use the energy change in Rydberg units,

v= (E„—Eo)/Ry.
Then (3) becomes

ln(ko(no, f)/Ry)+f(no0, n)v'=Pf(nof, n)v' lnv. (6)

The left-hand side may be evaluated by sum rules (reference 3, Eqs. (9) and (10)) and gives

As is seen from Table II, the discrete anal states n
contribute only about 2 percent of the sum in (8) for the 2s state. For the 2p state, their absolute contribution is similar while in this case the continuum gives an almost negligible contribution. In either case, no great numerical accuracy is required for the discrete states.
For transitions to the continuum, we define the
quasi-principal quantum number n by
E„=+Ry/no

Then the oscillator strength for transitions from the 2s state into an interval. dv of the continuous spectrum is

df(2s,

n)

=4v-'dv((4/3)+
)&exp( —4n

v ') arc cot(n/2))/(1

—e ' ").

(10)

The summation over n in (8) should be replaced by an integration over dv. For transitions from the 2p state
to the continuous spectrum

df(2P,

n)

=

(8)/(9e)xvp(o—dv(4ono+avrc

') cot(n/2))/(1

—e '~").

(11)

These formulas may be obtained from the expressions for the photoelectric absorption coeKcient (reference 4, Eqs. (47.19), (47.20)), using the general relation between absorption coefficient and oscillator strength,

r = Xs2or'e'(k/mc) (df/dE),

(12)

' where X is the number of atoms per cm', s the number
of electrons in the given shell, and df/dE the oscillator
strength per unit energy interval, i.e. (1/Ry)(df/dv)
It should be noted that Eq. (47.20) of reference 4, like the original formula of Stobbe, 'is too large' by a factor of 2; this mistake has been corrected in (11).
In Fig. 1 we have plotted the expression

g= ,'v'df/dv-

(13)

Qv'f(no, 0, n) =16/(3no').

For our case, no= 2, therefore

ln(ko(2, l)/Ry)=goof(2, t, n)v" lnv. - (8)

IQ

/
/
I

l

To evaluate this expression, we need the oscillator

1

strengths for hydrogen. For transitions to the discrete

f 05 &)cv /

spectrum, we have used the formula given by Bethe, 4

(Eq. (41.4)) and re-evaluated his Table 16 in which

'~Rp

'~ ~

some numerical mistakes were found, some of which

OI

IO

10

IQ&

have been previously pointed out by other authors.

I. The corrected values are given in Table
n(~6), we found it suflicient to use

For large

. g(2s, n)= oof(2s, n)v'=—0—3436n, '(1+.$n ')

(8a)

— ). g(2p, n)= sof(2p, n)v'=03149n '(1+4 24n o) (8b.

' H. A. Bethe, Etuedbuclg de Pkysik 24/1 (Verlag Julius

FIG. 1. The weight functions for the determination of the average excitation energy.

'' Note that v. in reference 4 is 4Ry.

'

M. Stobbe, Ann. This was pointed

years ago. It is most

strength df/d~ just

d. Physik 7, 661 (1930).

out to one of the authors (H.A.B.) many

easily above

verified by comparing and just below the

the oscillator
energy E =0.

It can also be verified by checking the sum rule analogous, to (7),

Springer, Berlin, 1933), p, 273.

Zf(2p, n)v'=0, see Table II.

372

BETHE, 8 ROYVN, AND STEHN

as. logv, for 2s and 2p. The integral under the 2s curve, after addition of the discrete spectrum, gives unity
(Eq. (7)).The smallness of g for the 2p state is apparent. For the numerical integration, we have used the
integration variable n instead of v, inserting
(14)
It is easily seen that the integrand in (7) then behaves
as de for small n, and that in (8) as de loge. Because
of the singularity of logn at n=0, numerical integration
is not feasible. However, it is easy to expand the
TABLE II. Results of numerical integration.

Contribution to 5: Discrete Continuous
Tot, al

9s State

Karlier

Later

Integration

0.02645 0.9736
1,00005

0.02645 0.97340 0.99985

2p State Later
Integration
—0.081366 +0.081336 —0.000030

high by 0.00006, an error considerably smaller than the
rough estimate of 1 part in 10,000 from the fourth diGerences. As a check, the sum in (8) wilkout the
logarithmic factor was also calculated (denoted by S);
according to (7), this should give unity for 2s and zero
for 2p. The deviation of 1.5 parts in 10,000 (Table II)
for the s state indicates the accuracy attained; for the
p state, the error is one-fifth of this. One can be doubtful whether it is more accurate to use the result (8) as it
stands, or to divide it by the numerically calculated 5;
both results are given in Table II, and their mean is
used as the final result. The probable error is taken as
the whole difference between the two results; the main
source of error is presumably in evaluation of the
integrand (or of the discrete contribution) since the
error from Simpson's rule should be less than 0.0001.
The probable error for the 2p state was estimated from the diGerence of the earlier and the later calculation
which was 0.0002.

Contribution Discrete

to (8)

—0.0465

Continuous

2.8593

—0.04649
2.85840

Total
Total/5
Adopted ln(kp/Ry) kp/Ry

2.8128 2.81191 2.8127 2.81233
2.8121+0.0004
16.646 a0.007

—0.030669 +0.000685 —0.02998
—0.0300+0.0002
0.9704+0.0002

integrand (apart from the log) into a power series in n, which permits analytical integration. This procedure
was used for n up to 0.05, while from there to 0.1 it
was used as a check on the numerical work: The results
agreed within 0.00005 which is beyond the accuracy at tempted (0.0001).
The numerical integration was done by Simpson's rule, using in the final calculation for the s state intervals' as follows:
from n =0.05 to 0.1, interval 0.0125 from n=0. 1 to 0.2, interval 0.025 from n=0. 2 to 1, interval 0.05.
From n= 1 to ~, we used s= 1/e as a variable, choosing
intervals of 0.05 for s. The fourth differences of the
integrand were of the order of 1 percent of the integrand itself (usually less) so that the error in the result may be expected to be about 1 part in 10,000. The earlier calculation was done with twice the intervals; the
results of the two calculations agree within 0.0009 unit, after correction of a numerical mistake of 0.0031 unit in the earlier integration, and of 0.0007 unit in the
discrete contribution, both of which were subsequently discovered. The error in Simpson's rule is roughly proportional to the fourth power of the interval used, so that the result of the second calculation should be
' The integrand was calculated at the intervals h stated. The
integral from x to x+2h is then )hfdf(x)+4f(x+h)+f(x+2h) j.

These results can now be inserted into the Lamb
effect formulas (1), (2). In the numerical factor in
front, we should insert the Rydberg constant for
hydrogen, rather than for infinite mass. This can be
seen from the derivation the velocity matrix elements
in A Eq. (6) are the same whether the reduced mass
or the full mass is used for the electron whereas the
energy diGerences E„—E are proportional to the
electron mass and thus to Ry~. This can also be traced
through A Eqs. (8) to (11). There is some doubt
whether to use the reduced mass of the electron also in
p= mc' in (1); we shall do so which may cause an error of 1/1840= 0.0005 in the value of the parenthesis in (1).
A larger uncertainty exists because of the neglect of
relativistic corrections in (1) (see below). For 0, we take the value deduced from hyperfine
structure'

1/n= 137.041~0.005.

(15)

" Then the constant factor in (1), the "Lamb constant,

is, when expressed in frequency units:

(o.'/3a )Ry = 135.549&0.015 mega, cycles. (16)

The uncertainty arises entirely from that of a. The
first term in the parenthesis in (1) is
1n(p/ko(2s)) = 10.5336—2.8121= 7.7215+0.0006. (17)

The error includes 0.0004 from Table II, and 0.0005 from the questionable use of the reduced mass in p, .
The value quoted by French and %eisskopf, 7.6906, referred to the difference
ln(p/ko(2s)) —ln(Ry/ko(2P)) = 7.7215 —0.0295 = 7.6920,

according to our present calculation, the small diBerence between this and 7.6906 being due to numerical mis-

' H. A. Bethe and C. Longmire, Phys. Rev. 75, 306 (1949),

VALUE OF THE LAMB SHIFT

takes in our earlier calculation. The results are given in
Table III. The error of the shift of 2s includes 0.11 Mc
from the uncertainty of the Lamb constant (16), i.e. mostly of n, and 0.08 Mc from that of ln(p/ko(2s)), of which only 0.05& arises from the numerical calculation. For the p levels, the uncertainty is all due to the
calculation of ko(2p). The measured Lamb shift is the difference between
the shifts of 2s and 2p~ state, vis.
I (2s) —L(2p~) = 1051.41+0.15 megacycles (18)

in agreement with the calculated values given by French and%eisskopf, and Lamb

previously and Kroll.

'

The latest published experimental value" is

1.(experimental) = 1062&5 Mc.

(19)

The discrepancy between (18) and (19) is twice the

experimental error, which probably has been generously
estimated. It is possible that this discrepancy is due to

the neglect of relativistic corrections in the derivation

of formula (1). These may not be negligible because

the wave function of the hydrogen ground state has

large Fourier components of high wave number. A rough

estimate'shows that these corrections are of relative

order 0., but without the logarithmic factor which gives
the main contribution to (1). Thus the rough estimate

would give a correction of order 1 Mc, too small to

explain the discrepancy. Moreover, superficial reasoning

would lead one to expect a decrease of the theoretical

value. However, an explicit calculation of the relativistic

corrections is now very urgent and may give a diferent

result.

It has often been pointed out that a higher theoretical

result would be obtained if the vacuum polarization

eGect were left out of (1), and that this effect can be

separated from the main part of the Lamb eGect in a

, relativistically
is represented

invariant by the

way. term

The
——'

vacuum polarization
in (1) and therefore

contributes an amount

—27.110 Mc

(20)

to the Lamb eGect. If it were excluded, the result would be 16 Mc higher than the experimental value so that
the agreement is not improved.

In this section, we shall try to give a qualitative

understanding of the very large result, 16.646 Ry, which

we found for the average excitation potential of the

l2ns(Esta—te.EoA) swiwthas

pointed out, lnko the weight factor

is v'f.

the average
If we take

of the

average of E„—E0 itself, rather than of its logarithm,

with the same weight factor v f, we get infinity. This

"R. C. Retherford and %. E. Lamb, Phys. Rev. 75, 1325
I', 1949).

TABLE III. Lamb shifts.

First term in parenthesis Total parenthesis Level shift in megacycles

2S
7.7215&0.0006 7.6707+0.0006 1038.53&0.14

2@1/2
+0.0300&0.0002 —0.0950~0.0002 —12.88&0.03

2@3/2
+0.0300&0.0002 +0.0925~0.0002 +12.54&0.03

is bees.use we have (neglecting constant factors)

2(E 2(E 2 Eo)&f

Eo) I po I

(i'll /~+)o

I

I

pe'
=((Sl /&&) )oo= o((&&) )oo= o ' po'4' dr. (21) +4

This integral diverges at r=0 as dr/ro if Po refers to an

s state. If the angular momentum of the state |ko is not

Ezero—, Etoheis

integral infinite

is finite. Now since the average of (for an s state), it is understandable

that the average of ln(E„—Eo) is large

Further information can be obtained from the

asymptotic behavior of the oscillator strength f, given

by Eq. (10), for large frequencies; this is, considering

(14):

v'df = (8/3or) v '*dv = (16/3m. )dn.

(22)

— This shows that already the average of v~ (E Eo)&,

with the weight factor v df, is infinite. The small

contribution of the discrete transitions (Table II) is

further evidence.

The most concrete picture is obtained from Fig. 1 in

which ooo'df/dv -is plotted against 1ogv. The maximum of the curve is seen to come at v= 7 which is reasonably

high, and the curve falls ofI' much more slowly towards

higher than towards lower v. The required average

logk0 is simply the center of gravity of the curve of

Fig. 1, and a value of 16.6 for ko looks entirely reason-

able.

It is easy to obtain simple analytic estimates of lnk0.

This can be done by approximating the exact oscillator

strength (10) by a simpler expression which can be

easily integrated. Since we know already that the high

frequencies are the most important, the simpler expres-
f sion must be a close approximation to at the high

frequency end but need not be so for low frequencies.

The simplest approximation is to set

— dG= ovodf = (8/7r) dne

which agrees with (22) for small n, and to choose a in such a way that the integral of (23) over all n is unity as it must be according to (7). This gives

(24)
Then the average of lnv becomes
lnn— ln(ko/Ry) = (1nv) = —2(inn) = 2a]tdne "
=—2(lna+C) = 2(ln(8/or)+C), (25)

BETHE, B ROUN, AND STEHN

where C is Euler's constant. Inserting numerical values,

ln(kp/Ry) = 3.026, kp-—20.6 Ry,

(26)

which is slightly larger than the correct values

ln(kp/Ry) = 2.812, kp= 16.65 Ry.

(27)

This simple calculation therefore gives the correct order of magnitude for kp.
To get a closer approximation, we consider the
f asymptotic behavior of in more detail. Expanding
(10) in a power series in n, but keeping the power series in the denominator, one gets

The former condition gives

(k+1)a= v.

(32)

The solution is
k = 8/(pr' —8) = 4.279, = a (pr —8)/pr = 0.5951. (33) — The average of inn is then

—(inn) = lna+

lna, n
(1+an) "+'

= lna+ +(k —1)—O(0)

(34)

, — dG=

'

v'df

=—
1+pm+

(8/~)dn
'— (-', pr' 2-, )n'+

(8/pr)dn (28)
1+3.14n+4.33n'+

The exponential in (23) gives instead

(8/ )dn dG= spv'df=—
1+(8/pr) n+ 32m. 'n'+

(8/pr)dn
(29)
1+2.55n+3. 24n'+

Obviously, this expression falls o6 too slowly with

increasing n. This error is compensated by a too-rapid

n~~, decrease for very large n: Actually, g should behave as

dv n 'dn for

whereas (23) gives an exponential

decrease. It is therefore understandable that (23) gives

too high a value for the average lnkp.

To improve on the behavior for small n, we choose

dc= (8/pr)dn
(1+an)k+1

and determine the two constants u and k so that the

linear term in n in the denominator of (28) is correctly

given, as well as the integral. The latter condition

requires

ka = 8/pr.

where 4'(n) is the logarithmic derivs, tive of the factorial
function, i.e. of F(n+1) (Jahnke-Emde's definition).
With our numerical values

ln(kp/Ry) = (lnv) = —2(inn) = 2.7818 kp = 16.148.

(35)

This is very close to the correct values (27), and somewhat smaller. Expansion of (30) for small n gives

(8/pr)dn
1+(k+1)an+p(k+1)ka" n"+--

(8/pr)dn

1+xn+4n'+ )

(36)

which is rather close to the correct expression (28), even

in the quadratic term. (36) is still above (28) for small

n, so that we might still have expected a too large

result for lnkp, it seems that the curve of (30) crosses

the correct curve (10) three times.

These calculations make the large value of the

average excitation potential appear plausible. More-

over, they make it likely that kp is nearly independent

of the principal quantum number: The asymptotic

expression for the oscillator strength for small n,

einxccleupdtingfortearmfsacotof rrenlaotiv' ewhoircdher

n, is also

independent occurs in the

of no, sum

rule (7). Therefore, all the information we used in

determining the constants a and k in (30) is independent

of np, and the estimate (36) applies to all values of np