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Fermi’s Golden Rule

Emmanuel N. Koukaras

A.M.: 198

Abstract

We present a proof of Fermi’s Golden rule from an educational perspective without compromising formalism.

1. Introduction

Fermi’s Golden Rule (also referred to as, the Golden Rule of time-dependent perturbation theory) is an equation for calculating transition rates. The result is obtained by applying the time-dependent perturbation theory to a system that undergoes a transition from an initial state |i〉 to a final state |f 〉 that is part of a continuum of states. The following sections provide the calculations and notions involved in deriving the final equation, as well as some concluding remarks (synopsis).

2. Time dependent perturbation theory

For a great deal of problems the time-independent perturbation theory suf- fices. Nevertheless, there are cases in which we want to study how systems respond to imposed perturbations and then settle into stationary states after an interval, in other words, study the transitions induced by a perturbation between stationary states of the unperturbed system. In cases like these, we use time-dependent perturbation theory to calculate, amongst other things, transition probabilities. We assume that the hamiltonian H of the system can be written in the form: H = H 0 + W (t) (1)

where ,H 0 is the hamiltonian of the unperturbed system ,W(t) is a perturbation applied to the system.

Since H 0 is the unperturbed hamiltonian, the time independent Schr¨odinger equation is satisfied:

H 0 |ϕn〉 = En|ϕn〉 (2)

The wavefunctions |ϕn〉 are related to the time-dependent unperturbed wavefunctions by: |ψn(t)〉 = |ϕn〉e−iEnt/¯h^ (3)

The time dependent Schr¨odinger equation for the system is:

H|ψ(t)〉 = [H 0 + W (t)]|ψ(t)〉 = i¯h ∂|ψ(t)〉 ∂t

with the state of the system |ψ〉, at a time t, expressed as a linear combina- tion of the {|ϕn〉} basis functions:

|ψ(t)〉 =

n

cn(t)|ψn(t)〉 =

n

cn(t)|ϕn〉e−iEnt/¯h^ (5)

We insert this relation in (4) and project the result on |ϕn〉 :

H 0

k

ck(t)|ϕk〉e−iEk^ t/¯h^ + W (t)

k

ck(t)|ϕk〉e−iEk^ t/¯h

= i¯h

∂t

k

ck(t)|ϕk〉e−iEk^ t/¯h^ ⇒

k

ck(t)Ek|ϕk〉e−iEk^ t/¯h^ +

k

ck(t)W (t)|ϕk〉e−iEk^ t/¯h

= i¯h

k

∂ck(t) ∂t

|ϕk〉e−iEk^ t/¯h^ + i¯h

k

ck(t)|ϕk〉

iEk ¯h

e−iEk^ t/¯h^ ⇒

Encn(t)e−iEnt/¯h^ +

k

ck(t)Wnk(t)e−iEk^ t/¯h^ = i¯h ∂cn(t) ∂t

e−iEnt/¯h^ + Encn(t)e−iEnt/¯h^ ⇒

k

ck(t)Wnk(t)e−iEk^ t/¯h^ = i¯h

∂cn(t) ∂t e−iEnt/¯h^ ⇒

∂cn(t) ∂t

i¯h

k

ck(t)Wnk(t)eiωnk^ t^ (6)

where ,Wnk = 〈ϕn|W (t)|ϕk〉 , the perturbation matrix element ,ωnk = (En − Ek)/¯h

Up to this point we have made no approximation. The difficulty in solving (6) arises from the fact that the coefficients are expressed in terms of themselves. In order to evaluate the coefficients from (6) we make two assumptions:

1. The system is initially in state |i〉, thus, all of the coefficients at t= are equal to zero, except for ci : cj (t = 0) = δij
2. The perturbation is very weak and applied for a short period of time, such that all of the coefficients remain nearly unchanged.

With these in mind, (6) gives us:

∂cn(t) ∂t

i¯h

ci(t)Wni(t)eiωnit^ (7)

so for any final state the coefficient will be (cf (t) ≈ cf (0) = 0):

the unperturbed hamiltonian H 0. The probability of finding the system in the eigenstate |ϕf 〉 is:

Pif (t) = |〈ϕf |ψ(t)〉|^2 (9)

Using (8) we have:

Pif (t) = (^) ¯h^12

∫ (^) t

0

eiωf it ′ Wf i(t′)dt′

2 (10)

3. High frequency harmonic perturbation

The case of an oscillating (i.e. harmonic) perturbation is a most important one. Once the effects of an oscillating perturbation are known then the gen- eral case can be evaluated since an arbitrary perturbation can be expressed as a superposition of harmonic functions. An example of an oscillating per- turbation is an electromagnetic wave such as a laser pulse. We define the oscillating perturbation having the form:

W (t) = 2W cos(ωt) = W

eiωt^ + e−iωt

Inserting this in (8) we obtain:

cf (t) = Wf i i¯h

∫ (^) t

0

eiωt ′

• e−iωt

eiωf it ′ dt′

Wf i i¯h

ei(ωf i+ω)t^ − 1 i(ωf i + ω)

ei(ωf i−ω)t^ − 1 i(ωf i − ω)

Thus, equation (10) becomes:

Pif (t) =

W (^) f i^2 ¯h^2

ei(ωf i+ω)t^ − 1 i(ωf i + ω)

ei(ωf i−ω)t^ − 1 i(ωf i − ω)

2 (13)

At this point we make an approximation assuming that the oscillating angular frequency of the perturbation has a value near the Bohr angular frequency of |ϕi〉 and |ϕf 〉, ωf i:

ω ' ωf i which can also be written:

|ω − ωf i| ø |ωf i|

With this approximation, the first term in equation (13) becomes negli- gible compared to the second one. This is made obvious from the fact that

the exponential factor (eix^ = cosx + isinx) cannot become greater than 1. Thus, the prominent term is defined by the denominator. When ω ' ωf i (usualy of high frequency ωf i ≈ 1015 sec−^1 ) the denominator of the second term goes to zero (the situation is reversed when ω ' −ωf i, so we need not examine this case separately). The second term (also called the ”resonant term”) can be written:

A− =

ei(ωf i−ω)t^ − 1 i(ωf i − ω)

= ei(ωf i−ω)t/^2

ei(ωf i−ω)t/^2 − e−i(ωf i−ω)t/^2 i(ωf i − ω)

= ei(ωf i−ω)t/^2

sin[(ωf i − ω)t/2] (ωf i − ω)/ 2

Thus, the probability becomes:

Pif (t) =

W (^) f i^2 ¯h^2

sin[(ωf i − ω)t/2] (ωf i − ω)/ 2

2 (15)

and, by introducing the function F (t, ω − ωf i):

F (t, ω − ωf i) =

sin[(ωf i − ω)t/2] (ωf i − ω)/ 2

we obtain:

Pif (t) =

W (^) f i^2 ¯h^2

F (t, ω − ωf i) (17)

remarks As we can see in the figure (1) the function F (t, ω) has a sharp peak about the central angular frequency ω. Do to this behavior we say that F, and thus the probability Pif (t; ω), shows a resonant nature. The distance between the first two zeros are defined as the resonance width ∆ω. The area underlined by Pif (t; ω) at this interval is over 95% of the total area. For ∆ω we have:

∆ω '

4 π t

The function F (t, ω) appears in equation (17) with an offset. As a result, the resonant point is located at ω = ωf i. The modulus of the resonant term in equation (13), |A−|^2 , behaves in the same manner. In the antiresonant case, |A+|^2 , the resonant point is located at ω = −ωf i. Placing both of these functions on the same graph it becomes clear that the part of |A+|^2

be less than 1, which we have when:

t ø

¯h |Wf i|

and using, (20):

1 |ωf i| ø

¯h |Wf i|

which we can read as: the matrix element of the perturbation must be much smaller than the energy separation between the initial and final states.

4. Quantum jumps to the continuum

When the final state is part of a continuum of states (i.e. when the energy belongs to a continuous part of the spectrum of H 0 ) we must account for all the states to which the system can jump to. This is done by integrating the probability as given by equation (17) with the density of states ρ(E) as weights:

P(t) =

{Eacc}

Pif (t)ρ(E)dE (24)

,where {Eacc} denotes all the states that the system can jump to under the influence of the perturbation. (What we have actually done is create a probability density from the prob- ability equation).

The probability function is sharply peaked at ω = ωf i and as a result acts as a delta function in the integral and thus selects the value for the density function at ω = ωf i. By substituting equation (17) in (24) we have:

P(t) =

{Eacc}

W (^) f i^2 ¯h^2

F (t, ω − ωf i)ρ(E)dE

{Eacc}

W (^) f i^2 ¯h^2

sin[(ωf i − ω)t/2] (ωf i − ω)/ 2

ρ(E)dE

as shown in the figure (2) the range of energies is very narrow and as a result the matrix element Wf i and the density of states ρ(E) can be considered as constant:

Figure 2: The function F (t, ω) acts as a delta function.

P(t) =

W (^) f i^2 ¯h^2

ρ(Ef i)

{Eacc}

sin[(ωf i − ω)t/2] (ωf i − ω)/ 2

dE

W (^) f i^2 ¯h^2

ρ(Ef i)

{Eacc}

sin[(ωf i − ω)t/2] (ωf i − ω)/ 2

¯hdω

W (^) f i^2 ¯h^2

ρ(Ef i)¯h

t

t^2

−∞

sin^2 x x^2

dx

where we substituted E = ¯hω, x = (ωf i − ω)t/2 and used the fact that for frequencies far from ωf i the function sin^2 x/x^2 is negligible so we can extend the limits to infinity. The value for the integral is well known and equal to π, thus we obtain:

Bibliography

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