Lamb shift calculated by simple noncovariant method

9 Pages·0.15 MB·English

by  Guang-jiong Ni

Lamb shift calculated by simple noncovariant method
Guang-jiong Ni
∗
Department of Physics, Fudan University, Shanghai, 200433, P. R. China
Jun Yan
†
Department of Physics, New York University, 4 Washington Place, New York, NY, 10003
The Lamb Shift (LS) of Hydrogenlike atom is evaluated by a simple method of quantum elec-
trodynamics in noncovariant form, based on the relativistic stationary Schr¨odinger equation. An
→−
induced term proportional to p4 in theeffective Hamiltonian is emphasized. Perturbative calcula-
tion ofsecond orderleads totheLSof 1S state andthat of 2S −2P statesin H atom with
1/2 1/2 1/2
the high accuracy within 0.1%
9
9 I. INTRODUCTION
9
1
The experimental discovery of Lamb Shift (LS) in 1947 and its theoretical explanations that followed are of great
n
importance for the establishment of Quantum Electrodynamics (QED). (Ref. [1]- [3]). The experimental value of the
a
energy difference between 2S and 2P states for Hydrogen atom reads (in unit of microwavefrequency)
J 1/2 1/2

8
E(2S ) E(2P )=1057.845MHz (1)
1/2 − 1/2
1
while the absolute LS for 1S state is [4]
v 1/2
6
1 ∆EH(1S1/2)=8172.86MHz (2)
0
1
0 ∆E (1S )=8184.00MHz (3)
D 1/2
9
9 TheLSonlyaccountsfortheorderof10 6 orO(α3)ofthatofthebindingenergyforelectron,i.e.,thatofRydberg
−
/
h energy
p
t- Ry =RH = 1α2µ=3.28805128 109MHz (4)
n 2 ×
a
u where µ is the reduced mass of electron and α= e2 =1/137.0359895. (h¯ =c=1).
4π
q The theoretical investigation over 50 years reveals that:
:
v (a) The main contribution of LS comes from the difference of radiative correction (i.e., the perturbative energy
i stemming from emitting a virtual photon and then absorbing it) in different states of electron.
X
(b) The difference between the wave functions of S states and P states is important. The electron in S states has
r moreprobabilityto moveinto the vicinityofnucleus. Inotherwords,ithas morehighmomentumcomponentsinthe
a
momentum representation of wave function for S states.
In some literature, in the integration of momentum k of virtual photon, the range of k was often divided into
two regions. For low k from k = 0 to, say, k = αm = αm, the binding effect of electron is taken into account
e
in noncovariant theory, whereas for high k up from k = αm the covariant theory of QED is applied. This kind of
treatment seems to us is difficult to avoid the double counting in virtual electron states conceptually. Moreover, the
so-called long wave (i.e., E1) approximation was used in the noncovariant theory at low k region as eikr 1. But
∼
exp(ikr) exp(iαma) exp(i) 1 (a=1/αµ being the Bohr radius) is also doubtful to be a good approximation.
∼ ∼ ∼
We wish to restudy the problem in noncovariant scheme. First of all, the mystery of LS is not only related to the
small scale of energy shift shown at Eqs.(1)-(3) but also to the high accuracy of the noncovariant calculation based
on the Stationary Schr¨odinger Equation (SSE):
H ψ =εψ (5)
0
∗E-mail: [email protected]
†E-mail: [email protected]
1
with
p2 Zα
H = (6)
0
2µ − r
and
Z2α2µ
ε= (7)
− 2n2
According to the theory of special relativity (SR), the energy of free electron reads
p2 p4
E = µ2+p2 =µ+ + (8)
2µ − 8µ3 ···
p
The magnitude ratio of the third term to the second one is approximately h4pµ22i ∼ h2pµ2iµ1 ∼ |µε| ∼ O(α2) in a
Hydrogenlike atom. So at first sight, the relativistic modification to H would be expected to account for an energy
0
decreaseof ε to orderof 105MHz.However,in fact, the LS showsan energy increaseof S states to orderof 103MHz
only. This is a mystery we should consider first before the LS could be understood in the noncovariant formalism of
QED.
II. RELATIVISTIC SSE
In Refs. [5]- [6], it was argued that the SSE, Eq.(5), is essentially relativistic as long as the eigenvalue ε is related
to the binding energy B as follows:
2ε
B =Mc2[1 (1+ )1/2] (9)
− Mc2
where M = m+m is the total mass of Hydrogenlike atom with m being the mass of nucleus. Obviously, when
N N
ε Mc2, B ε as expected.
≪ ≃−
Eqs.(5)-(7) together with Eq.(9) was derived from the time-dependent Schr¨odinger equation of two-body (say
electron-proton) system in combination with a basic symmetry
θ( r , r , t)=χ(r ,r ,t) (10)
−→e −→p −→e −→p
− − −
The explanation is as follows. The electron and proton are all not pure. They not only have a particle state
θ(r ,r ,t), but also have a hiding antiparticle state χ(r ,r ,t), θ and χ are coupled together according to the sym-
−→e →−p −→e −→p
metry (10). For one body system, this symmetry leads to Klein-Gordon equation (without spin) or Dirac equation
(with spin) (Refs. [7] [8]). The symmetry that “the space-time inversion is equivalent to particle-antiparticle trans-
formation” shown as (10) is the essence of SR. The various strange effects of SR are nothing but the reflections of
antimatter which is in a subordinate status (χ < θ ) and is just displaying its presence tenaciously [9].
| | | |
Therefore, at the level of quantum mechanics, the Hamiltonian H in SSE Eq.(5) is enough in the form of Eq.(6).
0
No p4 term like that in Eq.(8) is needed for relativistic correction. The latter is brought into consideration implicitly
at the final stage ε B as shown by Eq.(9).
→−
III. SEMIEMPIRICAL CALCULATION OF LS
Based on SSE, we consider that in an effective Hamiltonian a term b p4 will be induced at QED level by radiative
2
correction. It will lead to an energy shift of Hydrogenlike atomic level as:
8n b Z4 1
∆ERad = Znl b p4 Znl =[ 3] 2 (a= ) (11)
Z,nl h 2 i (2l+1) − n4a4 αµ
(cid:12) (cid:12)
(cid:12) (cid:12)
The coefficient b can be fixed from experiment by the LS between 2S 2P states in Hydrogen:
2 1/2 1/2
−
b 3
2
= (1057.845 0.087)MHz=1586.637MHz (12)
a4 2 −
H
2
where 0.087MHz is the small correction stemming from the finite nucleus radius.
To calculate the absolute LS of 1S state, Eq.(2), we should add three extra contributions:
(a) The vacuum polarization in QED induces a decrease in charge of electron: [10], [11]
α4Z2
∆α= (13)
−3πn2
which leads to an increase of energy
2Z4α3
∆EVp = R (14)
Z,nl 3πn4 y
For Hydrogen 1S state, it reads
∆EVp =271.140MHz (15)
H,1S
(b) The relativistic correction from Eq.(9) reads
ε2 1 µ R
∆ERel = = α2 y (16)
−2M −4 M n4
which yields a decrease in energy:
∆ERel = 23.814MHz (17)
H,1S −
(c) The correction from finite nucleus radius (r ) reads
N
4 1 r
∆ENu = ( N)2R δ (18)
nl 5n3 a y l0
which contributes a small increase of energy
∆ENu =0.697MHz (19)
H,1S
Altogether, we obtain theoretically
∆ETheory =8181.208MHz (20)
H,1S
The deviation between (20) and the experimental value, Eq.(2) is only 0.1%.
IV. CALCULATION OF B FROM THE FIRST PRINCIPLE
2
We are now in a position to derive the value of coefficient b from the first principle of QED in noncovariantform.
2
Consider an electron with charge e is moving with (three dimensional) momentum p in the center of mass system
−→
−
of Hydrogenlike atom and is coupled to the electromagnetic field via two kinds of interactions ( [1], [12])
H(1) = e −→Aˆ −→ˆp (21)
µc ·
e¯h ˆ
H(2) = −→σ −→ −→A (22)
2µc ·∇ ×
p2
According to the perturbation theory in quantum mechanics, the energy shift due to (21) from original ε = −→
p 2µ
will be
i H(1) p 2
∆E(1) = −→ (23)
p (cid:12)(cid:10) (cid:12)ε (cid:12)ε (cid:11)(cid:12)
Xi (cid:12) (cid:12) p−(cid:12) i (cid:12)
3
where the stationary state p = 1 ei−→p −→r is normalized in a volume V. The intermediate (virtual) state i is
|−→i √V · | i
composed of a plane wave eigenstate of SSE (Eq.(5)) −→q and a virtual photon with continuous momentum −→k, see
q2 ˆ
Fig. 1. So the εi in the denominator of (23) reads εi = −→2µ +ωk, (ωk = −→k = k). The quantized potential −→A of
(cid:12) (cid:12)
electromagnetic field reads as usual: (cid:12) (cid:12)
(cid:12) (cid:12)
−→Aˆ(−→r,t)=Z (2dπ−→)k3/2√21ωk λX=1,2−→ǫ −→k,λ(cid:18)aˆ−→k (t)ei−→k·−→r +aˆ†−→k(t)e−i−→k·−→r (cid:19) (24)
AftertheintegrationofmatrixelementofH(1) withrespecttospace,wesubstitute oneδ functionδ −→p −→q −→k
(cid:16) − − (cid:17)
by V and then perform the integration with respect to q, yielding
(2π)3 −→
αp2 1
∆E(1) = dη(1 η2)I (25)
p − πµ Z −
1
−
∞ dk
I = (26)
Z k+ξ
0
where η = −→p·−→k and ξ =2(µ pη).
pk −
·
V. THE RENORMALIZATION IS A PROCEDURE TO RECONFIRM THE MASS.
As in the calculation of QED in covariant form, we also encounter the divergent integral I, Eq.(26), here in the
integration of three dimensional momentum k of virtual photon. To treat the divergence, we will follow the spirit of
a simple but effective method used in covariant quantum field theory, which evolved from the so-called differential
renormalization in the literature [13]- [16], then was proposed by Ji-feng Yang [17] and applied extensively in Refs.
[11], [18], [19] (see also the discussion in [20]). Here the trick is as follows.
Take the derivative of integral with respect to the parameter ξ having a mass dimension:
∂I ∞ dk 1
= = (27)
∂ξ −Z0 (k+ξ)2 −ξ
which is convergentnow. Then we reintegrate Eq.(27) with respect to ξ for returning back to I
I = lnξ+C (28)
1
−
where an arbitrary constant C appears. Substituting (28 ) into (25), one obtains
1
αµ 2 p p µ p 2 p p µ p
∆E(1) = [( ( )2+ )ln(1+ )+( ( )2 + )ln( 1 )
p π 3 µ µ − 3p µ 3 µ − µ 3p (cid:12) − µ(cid:12)
(cid:12) (cid:12)
16 p 2 4 4 4 p (cid:12) (cid:12)
( )2+ +( ln2+ lnµ C )( )2] (cid:12) (cid:12)
1
− 9 µ 3 3 3 − 3 µ
=b(1)p2+b(1)p4+ (29)
1 2 ···
α 4 4 4
b(1) = ln2+ lnµ C (30)
1 πµ(cid:18)3 3 − 3 1(cid:19)
α 2
b(1) = (31)
2 πµ3 (cid:18)−15(cid:19)
Note that, however, the term b(1)p2 will be combined into the kinetic energy term in original H . They are
1 0
indistinguishable. The appearance of arbitrary constant C precisely reflects the fact that we can not calculate the
1
4
reducedmassofanelectronviatheperturbationcalculationof∆E(1). SothevalueofC ischosensuchthatb(1) =0,
p 1 1
implying that the value of reduced mass µ (yet not µ , see below) is reconfirmed as an observed mass which can
obs
only be fixed by the experiment (not by theory).
Next turn to H(2), which induces the spin flip process between p, 1 and q, 1 states, yielding
−→ ±2 −→ ±2
(cid:12) (cid:11) (cid:12) (cid:11)
1 i H(2(cid:12)) p,s 2 (cid:12)
∆E(2) = −→ z
p 2 (cid:12)(cid:10) (cid:12) ε (cid:12) ε (cid:11)(cid:12)
i,sXz=±12 (cid:12) (cid:12) p−(cid:12) i (cid:12)
α 1
= dηJ (32)
−2πµZ
1
−
∞ k2dk
J = (33)
Z k+ξ
0
For this divergent integral, derivative of third order is needed to render it convergent:
∂3J 2
= (34)
∂ξ3 −ξ
So after reintegrating with respect to ξ, we have:
J = ξ2lnξ+C ξ2+C ξ+C (35)
2 3 4
−
αµ 2µ p p p p 22 p 4
∆E(2) = [(1+ )3ln(1+ ) (1 )3ln( 1 )] ( )2
p π {3p µ µ − − µ (cid:12) − µ(cid:12) − 9 µ − 3
(cid:12) (cid:12)
2C C 4 (cid:12) (cid:12) 4 4 p
+4(ln2+lnµ) 4C 3 4 + l(cid:12)n2+2(cid:12)+ lnµ C ( )2
− 2− µ − µ2 (cid:18)3 3 − 3 2(cid:19) µ }
=b(2)+b(2)p2+b(2)p4+ (36)
0 1 2 ···
αµ 2C C
b(2) = 4(ln2+lnµ) 4C 3 4 (37)
0 π (cid:20) − 2− µ − µ2(cid:21)
α 4 4 4
b(2) = ln2+2+ lnµ C (38)
1 πµ(cid:18)3 3 − 3 2(cid:19)
α 1
b(2) = (39)
2 πµ3 (cid:18)−15(cid:19)
We shall fix three arbitrary constants C , C and C carefully. First look at the b(2)p2 term which should be
2 3 4 1
combinedwiththeterm p2 withµalreadyfixed. Anymodificationonµmustbefiniteandfixed. Sotheonlypossible
2µ
choice of C is to cancel the ambiguous term 4lnµ in b(2), leaving
2 3 1
β 2α 4
b(2) = β = ln2+2 (40)
1 2µ π (cid:18)3 (cid:19)
TheconstantsC andC mustbechosensuchthatb(2) =0,whichmeansthatwereconfirmtheSSEasourstarting
3 4 0
point. There is always no rest energy term in SSE.
Hence, the nonzero contribution of b(2)p2 does bring a finite and fixed modification on µ so that
1
µ
µ = (41)
obs
1+β
5
whereµ isthereducedmasseventuallyobservedinexperiment. However,asdiscussedinEq.(8),thereisnotermlike
obs
p4 in SSE. The relativistic correction is left to the modification of ε to B at the final stage. So an unobservable
−8µ3
obs
term is also modified, the difference 1 1 1 p4 should be treated also as an unobservable background of
−8(cid:16)µ3obs − µ3(cid:17)
modification in p4 term, which should be subtracted off from the observable one. Hence while b =b(1)+b(2) =b(2),
1 1 1 1
but instead of b =b(1)+b(2) = α 1 , we should have a renormalizedbR as
2 2 2 πµ3 −5 2
(cid:0) (cid:1)
1 α
bR =b + 3β+3β2+β3 = (1.942816878) (42)
2 2 8µ3 πµ3
(cid:0) (cid:1) obs
VI. COMPARISON WITH THE EXPERIMENTAL VALUES
Let us use Eq.(42) to evaluate the LS of H atom:
5bR
∆ERad = bRp4 = 2 =7901.629MHz (43)
H,1S 2 1S a4
(cid:10) (cid:11) H
Adding the contributions from Eqs.(15), (17) and (19), we get
∆ETheory =8149.653MHz (44)
H,1S
which is smaller than the experimental value Eq.(2) up to 0.28%.
For the LS between 2S and 2P states,
1/2 1/2
2 bR
∆ERad = 2 =1053.551MHz (45)
H,2S1/2−2P1/2 3a4H
After adding a small contribution due to the finite nucleus radius, we have
∆ETheory =1053.638MHz (46)
H,2S1/2−2P1/2
which is also smaller than the experimental value Eq.(1) up to 0.40%.
For further improvement, we keep all pn (n 4) terms. So we manage to evaluate the renormalized radiation
≥
correction as follows
p2 p2
∆ERad(p)=∆E(1)+∆E(2) b(2)p2 ( p2+µ2 µ )+( p2+µ2 µ ) (47)
−→ p p − 1 − q obs− obs− 2µobs p − − 2µ
with µ µ=dµ= βµ and p2+µ2 p2+µ2 = µobsdµ as we wish to keep the explicit dependence on
obs− − obs obs− √p2+µ2
p p obs
α throughout Eq. (47) being of order O(α). Then we calculate the expectation value of ∆ERad(p) for a fixed state
−→
numerically, yielding
∆ERad =7920.533MHz (48)
H,1S
∆ERad =1057.550MHz (49)
H,2S1/2−2P1/2
∆ERad =7922.688MHz (50)
D,1S
∆ERad =1057.838MHz (51)
D,2S1/2−2P1/2
After adding the other three corrections Eqs.(14)-(18), we are pleased to see the results
∆ETheory =8168.557MHz (52)
H,1S
6
∆ETheory =1057.637MHz (53)
H,2S1/2−2P1/2
∆ETheory =8186.181MHz (54)
D,1S
∆ETheory =1058.363MHz (55)
D,2S1/2−2P1/2
coinciding with the experimental data to a high accuracy (<0.1%).
∼
VII. SUMMARY AND DISCUSSION
(a)WeproposeasimplebuteffectivemethodforcalculatingtheLSinanoncovariantform. Twocrucialobservations
are as follows:
(i) The SSE is essentially relativistic as long as its eigenvalue ε is related to the binding energy B by Eq.(9).
(ii)ThemaincontributiontoLSiscomingfromthedifferentmeanvaluesofoperator p4 inS andP statesasshown
−→
in the semiempiricalcalculation. The analysisconvincedourselvesthat the use of three dimensional momentum p is
−→
much suitable than that of four dimensional one. Actually, a simple one-loop calculation in covariant form of QED
gaveus the value of LS for H atom with accuracyonly 5%for 2S 2P states and evenworse ( 22%)for 1S
1/2 1/2 1/2
− ∼
state [11].
(b) The subtlety ofSSE canbe seenfurther by the following comparison. At free moving condition, the relativistic
effect is contained in the definition ε = p2 with rest (reduced) mass being a constant containing all the radiative
2µ
correction. On the other hand, when the electronis bound in a Hydrogenlike atom, its Hamilton H contains no rest
0
massandno explicit(negative) p4 termeither. The relativistic effectis containedinthe definition ε= E2 M2 (with
−→ 2−M
E and M being the total energy and mass of the system) and Eq.(9) implicitly. However, the radiative correction
does induce a small (positive) term of p4 at the level of QED.
−→
(c) The definition of ε and Eq.(9) also indicate that there is no any negative energy eigenstate in SSE. The hiding
antiparticlestate(whichistheessenceofspecialrelativity)isalreadytakenintoaccountinderivingSSEwithEq.(9).
In other words, no further explicit virtual positron state should be considered in our calculation. Actually, we had
struggled for years before eventually realizing that only the simple formulas (23) and (32) with Fig.1 are needed in
the perturbative calculation of second order.
(d) The effective Hamiltonian of Hydrogenlike atom for evaluating the LS can be summarized as
p2 Zα α4Z3
H = +∆ERad(p)+ (56)
eff 2µ − r −→ 3πn2r
while the third term ( bRp4) is the QED modificationto the firstterm, the fourthterm could be seenas that to the
≃ 2
second term (see Eqs.(13), (14)).
Eq.(56) is applicable to j = 1 states. The P state is pushed up by extra spin-orbit coupling to form the fine
3/2
structure. ForHydrogen,E(2P ) E(2P )=1.09691 104MHz about10timesofLS.Furthermore,thehyperfine
3/2 1/2
− ×
structure (hfs) in Hydrogen, stemming from the interaction between the magnetic moments of electron and proton,
is smaller than the LS. The energy splitting of 1S states is well known as ∆E (hfs) = 1420.406MHz. As we
1/2 1S1/2
don’t take the magnetic moment of nuclei into account, the hfs is not considered in this paper.
In deriving the H , the Pauli interaction term H(2) between electron spin and the external magnetic field has to
eff
be addedtothe RSSE sothatthe spin-orbitalcoupling(hyperfinestructure)andLScanbe calculatedquantitatively.
Thisisthe pricewemustpayfortheuseofRSSE.Ontheotherhand,thoughtheDiracequationinexternalfieldcan
predict the electron spin with g =2, it fails to take the difference of reduced mass of electron into account because it
is a one-body equation. Furthermore, it predicts a too low ground state (1S) and a too large splitting of 2P and
3/2
2P states. The weakness of Dirac equation seems to us is due to its overestimation of the antiparticle ingredient
1/2
in the electron as discussed in Ref. [8]. Actually, only after long time study on Dirac equation with its difficulty
in calculating LS, especially the absolute LS of 1S states, could we believe in the advantage of using RSSE as the
starting point.
(e)The reasonableresultin this paperagainshowsthe correctnessandeffectivenessofthe renormalizationmethod
usedinRefs.[17]-[20]. Fortreatingthedivergence,whichwarnsusofthelackofourknowledgeabouttheparameters
(mass, charge, etc.), the renormalization is a procedure to reconfirm the parameters step by step rigorously. Here
it is interesting to see an example of finite and fixed renormalization. Note that, however, we must consider the
7
contribution of H(1) first to define the parameter µ before to consider that of H(2) for bringing µ to µ . This is the
obs
onlyreasonablelogic. The inverselogic,i.e.,to considerH(2) firstandthenH(1) next,wouldleadto inconsistentand
wrong result.
ACKNOWLEDGMENTS
We thank Prof. D. Zwanziger, Mr. Hailong Li and Mr. Haibin Wang for discussions. We also thank Mr. Sangtian
Liu in NYU who gaveus a lot of help in LATEXand figure of this paper. This work was supported in part by the NSF
of China.
(cid:0)!p
(cid:0)!q (cid:0)!k
(cid:0)!p
1
FIG. 1. The only one loop diagram calculated for radiation correction in the noncovariant formalism of QED. At the two
vertices, either H(1) or H(2) is used (no interference between them would occur due to no polarization in the plane wave p
state).
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