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I wonder if there is a detailed derivation of the quasi-electron lifetime:

1τk=2π1V2k,qσ|Vq|2fk(1fkq)(1fk+q)δ(ϵkqϵk+ϵk+qϵk)

from Fermi golden rule. Although the result is stated in many literature and textbooks, I did not find an explicit derivation from Fermi golden rule anywhere so far.

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The following argument follow basically this proof.


The golden rule is a straightforward consequence of the Schrödinger equation, solved to lowest order in the perturbation H of the Hamiltonian,

(H0+Hit)nan(t)|neitEn/=0,

where En and |n are the stationary eigenvalues and eigenfunctions of H0 .

Rewrite this equation as that of the time evolution of the coefficients a an(t),

idak(t)dt=nk|H|nan(t)eit(EkEn)/.

This equation is exact but normally cannot be solved in practice.

For a weak constant perturbation H which turns on at t=0, we can use perturbation theory. Namely, if H=0 it is evident that a an(t)=δn,i, which simply says that the system stays in the initial state i.

For states ki, ak(t) becomes non-zero due to H0 and these are assumed to be small due to the weak perturbation. Hence, one can plug in the zeroth order form a an(t)=δn,i into the above equation to get the first correction for the amplitudes ak(t),

idak(t)dt=k|H|ieit(EkEi)/,

which integrates to iak(t)=2k|H|ieiωt/2sinωt/2ω

for ω(EkEi)/, for a state with ai(0)=1,ak(0)=0, transitioning to a state with ak(t) (again, ki).

The transition rate is then

Γik=ddt|ak(t)|2=2|k|H|i|22sinωtω,

a sinc function peaking sharply for small ω. At ω=0, sin(ωt)/ω=t, so the transition rate varies linearly with t for an isolated state |k !

By dramatic contrast, for states of energy E embedded in a continuum, they must be all accounted for collectively. For a density of states per unit energy interval ρ(E), they must be integrated over their energies, and whence the corresponding ωs,

Γif=2dωρ(ω)|f|H|i|2sinωtω.

For large t, the sinc function is sharply peaked at ω0, and negligible outside [2π/t,2π/t]; the density and transition element can be taken out of the integral, so that the rate Γif=2ρ|f|H|i|2dωsinωtω

is now merely proportional to a constant Dirichlet integral, π.

The time dependence has vanished, and the constant decay rate of the golden rule follows.

As a constant, it underlies the exponential particle decay laws of radioactivity. (For excessively long times, however, the secular growth of the ak(t)s invalidates lowest-order perturbation theory, which requires ak<<ai.)


If it helps, try to have a look to this article http://onlinelibrary.wiley.com/doi/10.1002/9783527665709.app6/pdf

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