Lamb shift

Introduction

    Suppose there is  a one-electron state 0. An interation  of electrons with the electromaelegnetic field  mixes this state with a set of states N, in  which the electron occupies  some state n, and additionally there is a photon, which can be described by  its momentum and polarization. As a result, in the second order of the perturbation theory there appears an energy shift of the level |0>. This shift, as well as physical reasons producing it,  is called the Lab shift.
    A potential initiating this effect can be written in terms of the velocity of the electron and the vector potential

V = -e Overscript[v, ] · Overscript[A, ] (1)

Here the vector potential describes a photon with the given momentum and polarization

Overscript[A, ] = (4π)/(2ω)^(1/2) Overscript[ϵ, ] Exp[ (Overscript[k, ] · Overscript[r, ] - ω t)], (2)

Overscript[k, ] , Overscript[ϵ, ] are the momentum and polarization, ω = k is the photon frequency .   The normalization

factor (2π/ω)^(1/2) is justified below .      In the dipole approximation, which we will use below, the vector - potential simplifies

Overscript[A, ] ≃ (4π)/(2ω)^(1/2)   Overscript[ϵ, ]   Exp[-ω t] (3)

The matrix element, which mixes the states 0 and N reads

V_ (N, 0) = -e (4π)/(2ω)^(1/2) Overscript[v, ] _ (n, 0) ·   Overscript[ϵ, ] (4)

For the energy shift one finds

δE_0 = Underscript[∑, N] (| V_ (N, 0) |^2)/(E_0 - E_N) (5)
= 4π e^2Underscript[∑, k, ϵ] Underscript[∑, n] 1/(2ω) (| Overscr ... ript[v, ] _ (n, 0) ·   Overscript[ϵ, ] |^2)/(E_0 - E_n - ω) (5)

Here it is taken into account that the summation over the photon intemediate states results in the integral over the photon momentum and sumation over its polarixation.  

Normalization of the vector potential

Preliminary remarks

Suppose one needs to fulfill summation over a set of a continuum states of some particle. One can  take some large volume, call it V, and assume that it is greater than the volume in which physical events considered take place. Then instead of summation over the continuum states one fulfilles summation over quasi-continuum set of states, for which periodical boundary conditions in the volume V are valid. After that one rewrites the summation as the integral using the following formulae  

Underscript[∑, n] ψ_Overscript[p, ] _n (Overscript[r, ]) ψ_O ... (V) (Overscript[r, ]) ψ_Overscript[p, ]^((V) *) (Overscript[r, ] ') (6)

Here the superscript (V) indicates that the wave functions on the right-hand side are normalized in the volume V. Instead of this normalization it is convenient to introduce normalization for unit volume, in which the following obvious identity holds

ψ_Overscript[p, ]^(V) (Overscript[r, ]) = 1/V^(1/2) ψ_Overscript[p, ]^(V = 1) (Overscript[r, ]) (7)

Then the necessary summation reads

Underscript[∑, n] ψ_Overscript[p, ] _n (Overscript[r, ]) ψ_O ... ] (Overscript[r, ]) ψ_Overscript[p, ]^* (Overscript[r, ] ') (8)

where the superscript (V=1) is suppressed.
    Thus, to fulfil summation over the continuum set of states one can integrate  over the momentum ^3p/(2π)^3,  normalizing wave functions as "one particle per unit volume".

Photons

〈0 | Overscript[A, ] _Overscript[k, ]^* Overscript[A, ] _Overs ... 2754;] | 0〉 = | 〈1 | Overscript[A, ] _Overscript[k, ] | 0〉 |^2 (9)

A photon represents a harmonic oscillator. Therefore the avaraged kinetic and potential energies equal half of the total energy. In the vacuum this total energy is half of the frequency. In this case

Overscript[T, -] = Overscript[U, -] = 1/2E = 1/4ω (10)

The kinetic energy of the oscillator can be written of from the energy of the electromagnetic field in a unit volume, which reads

dε/dV = (| Overscript[E, ] _Overscript[k, ] |^2 +| Overscript[H, ] _Overscript[k, ] |^2)/(8π) (11)

Here the energy is calculated "per unit volume", precisely as is necessary for the purposes of integration over the continuum states of the photon, see above. Since

Overscript[E, ] _Overscript[k, ] = -∂/∂tOverscript[A, ] _Overscript[k, ] (12)

the first term in the energy represents the kinetic energy

T = | Overscript[E, ] _Overscript[k, ] |^2 = ω^2Overscript[A, ] _Overscript[k, ]^* Overscript[A, ] _Overscript[k, ] (13)
Overscript[T, -] = 〈0 | T | 0〉 = ω^2/(8π) 〈0 | Overscript[A, ] _Overscript[k, ]^* Overscript[A, ] _Overscript[k, ] | 0〉 (13)

Combining this with the previous disscussion one finds

Overscript[T, -] = ω^2/(8π) | 〈1 | Overscript[A, ] _Overscript[k, ] | 0〉 |^2 = 1/4ω (14)

Thus considering creation (or annialation) of one photn one should have

〈1 | Overscript[A, ] _Overscript[k, ] | 0〉 = (4π)/(2ω)^(1/2) (15)

Energy sfift

δE_0 = 4π e^2Underscript[∑, ϵ = 1, 2] ∫^3k/(2π)^3Un ... | Overscript[v, ] _ (n, 0) ·   Overscript[ϵ, ] |^2Ω (16)

Here dΩ represents integration over the spherical angle of the photon momentum

∫Underscript[∑, i = 1, 2] | Overscript[v, ] _ (n, 0) ·   Overs ... rscript[v, ] _ (n, 0) |^2, Overscript[n, ] = Overscript[k, ]/k . (17)

Thus, summation over the polarization and integration over the angles for a dipole transition gives a factor 2/3

δE_0 = (4π e^2)/(2π)^31/22/34πUnderscript[  ∑, n] | Over ... n, 0) |^2Underoverscript[∫, 0, arg3] ω ω^21/ω1/(E_0 - E_n - ω) (18)
= 2/(3π) e^2Underscript[ ∑, n] | Overscript[v, ] _ (n, 0) |^2Underoverscript[∫, 0, arg3] ω ω/(E_0 - E_n - ω) (18)

Renormalization

Repeating the argument for the vacuum one can write

(δE_0) _vac = (2e^2)/(3 π) Underscript[ ∑, n] | Overscript[v, ] _ (n, 0) |^2Underoverscript[∫, 0, arg3] ω ω/-ω (19)

since in the vacuum the operator of velocity does not have nondiagonal matrix elements. For the energy difference one finds

(δE_0) _ren = δE_0 - (δE_0) _vac =  = (2e^2)/(3π) Underscript[ & ... ;] _ (n, 0) |^2Underoverscript[∫, 0, arg3] ω (E_0 - E_n)/(E_0 - E_n - ω) (20)

In the following discussion the subscript ren (for renormalized) is suppressed.
    The  physical picture considered can be only correct in the nonrelativistic approximation. Therefore the integration over ω should run only up to m (not to infinity, where different formulas need to be worked out).

δE_0 (2e^2)/(3π) Underscript[ ∑, n] | Overscript[v, ] _ (n, ... ;) Underscript[ ∑, n] (E_0 - E_n) | Overscript[v, ] _ (n, 0) |^2Log[m/(E_0 - E_n)] (21)
Overscript[v, ] _ (n, 0) = 1/mOverscript[p, ] _ (n, 0) ≡1/m〈n | ... 9001;n | Overscript[r, ] | 0〉≡ (E_0 - E_n)/mOverscript[r, ] _ (n, 0) (22)
δE_0 = -(2e^2)/(3π) Underscript[ ∑, n] (E_0 - E_n)^3 | Overscript[r,  ... Underscript[ ∑, n] (E_n - E_0)^3 | Overscript[r, ] _ (n, 0) |^2Log[m/(E_0 - E_n)] (23)

This expression has a real and imaginary parts. The latter will be discussed when radiative widths are considered. For the real part, which is represents a proper energy, one finds

Re[δE_0] = (2e^2)/(3π) Underscript[ ∑, n] (E_n - E_0)^3 | Overscript[r, ] _ (n, 0) |^2Log[m/(| E_n - E_0 |)] (24)

In absolute units

Re[δE_0] = 2/(3π c^3) Underscript[ ∑, n] ω_ (n, 0)^3 | Overscript[d, ] _ (n, 0) |^2Log[(m c^2)/(| E_n - E_0 |)] (25)

where

ω_ (n, 0) = (E_n - E_0)/ (26)

Large Log approximation

For atoms

E_n - E_0∼Z^2Ry (27)
Log[(m c^2)/(| E_n - E_0 |)] ≃Log[1/(Zα)^2] ≫ 1 (28)

assuming that (Zα)≪1.

Re[δE_0] ≃ (2e^2)/(3π ) Log[1/(Zα)^2] Underscript[ ∑, n] (E_n - E_0)^3 | Overscript[r, ] _ (n, 0) |^2 (29)

Effective potential ∼ (e^2/m) ΔU

Re[δE_0] = (2e^2)/(3π ) Log[1/(Zα)^2] Underscript[ ∑, n] (E_n - E_0)^3 ... E_0) 〈0 | Overscript[p, ] | n〉〈n | Overscript[p, ] | 0〉 (30)
(2e^2)/(3π ) Log[1/(Zα)^2] 1/2Underscript[∑, n] (〈0 | ... Overscript[p, ] | 0〉) = Re[δE_0] = e^2/(3π m^2 ) Log[1/(Zα)^2] S_0 (30)
S_0 = Underscript[∑, n] (〈0 | Overscript[p, ] | n〉〈n |[H, Ov ... 〈0 |[H, Overscript[p, ]] | n〉〈n | Overscript[p, ] | 0〉) (31)
Underscript[∑, n] | n ><n | = 1S_0 = 〈0 | Overscript[p, &# ... 62754;] | 0〉 = 〈0 |[Overscript[p, ],[H, Overscript[p, ]]] | 0〉 (32)
H = Overscript[p, ]^2/(2m) + U (r) (33)
[H, Overscript[p, ]] =  Overscript[∇, ] U (34)
[Overscript[p, ],[H, Overscript[p, ]]] = [Overscript[p, ], O ... #8711;, ] U] = Overscript[∇, ] · Overscript[∇, ] U = ΔU (34)
S_0 = ΔU (35)
Re[δE_0] = e^2/(3π m^2 ) Log[1/(Zα)^2] 〈0 | ΔU | 0〉 (36)
δU_eff (r) = 1/(3π  ) Log[1/(Zα)^2] e^2/m^2 ΔU (r) (37)
U = -Ze^2/r (38)
Δ1/r = -4π δ (Overscript[r, ]) (39)
ΔU = 4π Z e^2 δ (Overscript[r, ]) (40)
δE_0 = (4Z e^4)/(3 m^2 ) Log[1/(Zα)^2] φ_0^2 (0) (41)

In the log approximation only the s-states exibit the Lamb shift. Other states are also influenced by the Lamb correction, but is does not have the large log factor.

φ_ns_ (1/2) (0) = Z^3/(π a_0^3)^(1/2) = (Z^3α^3 m^3)/π ^(1/2) (42)
δE_ns_ (1/2) ≃ (4Z e^4)/(3 m^2 ) (Z^3α^3 m^3)/π Log[1/(Zα)^2] = (4Z^4 α^5 m)/(3 π) Log[1/(Zα)^2] (43)
δE_np_ (1/2) ≃δE_np_ (3/2) ≃0 (43)

Accurate relativistic calculations give

δE_ns_ (1/2) = (4Z^4 α^5 m c^2)/(3 π) (Log[1/(Zα)^2] + 19/30 + L_n) (44)
{{n,   1,   2,   3,   ∞}, {L_n, -2.984, -2.812, -2.768, -2.721}} (44)
   E_0∼Z^2α^2mδE_0/E_0∼Z^2α^3Log[1/(Zα)^2] (45)
In Hydrogen (46)
δE_ (2s_ (1/2)) - E_ (2p_ (1/2)) ≃1050 MHz (46)

Radiative decay of excited levels

For a quasistationary state 0 the energy has its real and imaginary parts

E_0 = Re[E_0] -  Γ/2 (47)

Γ is called the width

ψ∼ Exp[- E t] = Exp[- Re[E_0] t - Γ/2t], Probability ∼ | ψ |^2∼Exp[- Γ t] (48)

This means that probability of the decay W (of the excited state) per second equals the width

W = Γ≡Γ/ (49)

From the expression for the Lamb shift (see the Lamb shift)

δE_0 = (2e^2)/(3π) Underscript[ ∑, n] (E_n - E_0)^3 | Overscript[r, ] _ (n, 0) |^2Log[m/(E_0 - E_n)] (50)

one finds

Im[δE_0] = (2e^2)/(3π) Underscript[ ∑, n] (E_n - E_0)^3 | Overscript[r, > ... ^2 = -2/3Underscript[ ∑, E_n<E_0] ω_ (0n)^3 | Overscript[d, ] _ (n, 0) |^2 (51)

This means that the radiative width of the state 0 equals

Γ/2 = 2/3Underscript[ ∑, E_n<E_0] ω_ (0n)^3 | Overscript[d, ] _ (n, 0) |^2 (52)

Correspondingly, the probability of the radiative decay of the state 0 is

W = Underscript[ ∑, E_n<E_0] W_ (0n) = 4/3Underscript[ ∑, E_n<E_0] ω_ (0n)^3 | Overscript[d, ] _ (n, 0) |^2 (53)

Here

W_ (0n) = 4/3ω_ (0n)^3 | Overscript[d, ] _ (n, 0) |^2≡ (4ω_ (0n)^3)/(3 c^3) | Overscript[d, ] _ (n, 0) |^2 (54)

is the probability of the decay into some given state n, while W is the total probability of the radiative decay,

Overscript[d, ] _ (n, 0) = e Overscript[r, ] _ (n, 0) = e∫ψ_n^* ( ... t[r, ]) ^3r≡e〈ψ_n^* | Overscript[r, ] | ψ_0〉 (55)

is the dipole matrix element and

ω_ (0n) = E_0 - E_n≡ (E_0 - E_n)/ (56)

is the frequency of the transition. It is instructive to compare W_ (0n) with the classical expression. In the classical approximation the energy rate radiated by a dipole Overscript[d, ], which oscillates with the frequency ω is

P = (4ω^4)/(3c^3) | Overscript[d, ] |^2 (57)

It follows from here that the probability of radiation of one quantum is

W = P/( ω) = (4ω^3)/(3 c^3) | Overscript[d, ] |^2 (58)

Compare this calssical expression with the quantum one

W_ (0n) = (4ω_ (0n)^3)/(3 c^3) | Overscript[d, ] _ (n, 0) |^2 (59)

They mathch nicely.

Created by Mathematica  (October 15, 2006) Valid XHTML 1.1!