Oscillatory perturbation and Fermi’s golden rule

4.2 Oscillatory perturbation and Fermi’s golden rule

Summary: Fermi’s golden rule is the most important result of this chapter

Without specifying anything more about the system or perturbation, let us consider the case ${\hat{H}}^{(1)} (t) = {\hat{H}}^{(1)} e^{- i ω t}$ . (Note $ω$ is not now anything to do with the harmonic oscillator, and indeed if we wanted to apply this to that system we’d need to use labels to distinguish the oscillator frequency from the applied frequency).

At the outset we should note that with this problem we are heading towards the interaction of atoms with a radiation ﬁeld.

For deﬁniteness, let the perturbation apply from $- t ∕ 2$ to $t ∕ 2$ . Then

\begin{array}{rcl} d_{n} (t) & = & - \frac{i}{ℏ} ⟨ n | {\hat{H}}^{(1)} | i ⟩ \int_{- t ∕ 2}^{t ∕ 2} e^{i (ω_{n i} - ω) t^{'}} d t^{'} \\ = & - \frac{2 i}{ℏ (ω_{n i} - ω)} ⟨ n | {\hat{H}}^{(1)} | i ⟩ sin (\frac{1}{2} (ω_{n i} - ω) t) \\ \Rightarrow P_{i \to n} & = & \frac{4}{ℏ^{2} {(ω_{n i} - ω)}^{2}} {|⟨ n | {\hat{H}}^{(1)} | i ⟩|}^{2} {sin}^{2} (\frac{1}{2} (ω_{n i} - ω) t) \\ = & \frac{t^{2}}{ℏ^{2}} {|⟨ n | {\hat{H}}^{(1)} | i ⟩|}^{2} {sinc}^{2} (\frac{1}{2} (ω_{n i} - ω) t) \end{array}

where $sinc (x) \equiv sin (x) ∕ x$ .

This expression is unproblematic provided $ℏ ω$ doesn’t coincide with any excitation energy of the system. In that case, the transition probability just oscillates with time and remains very small.

The more interesting case is where we lift that restriction, but the expression then requires rather careful handling! The standard exposition goes as follows: as $t$ becomes very large, $t {sinc}^{2} (\frac{1}{2} (ω_{n i} - ω) t)$ is a sharper and sharper – and taller and taller – function of $ω_{n i} - ω$ , and in fact tends to $2 π δ (ω_{n i} - ω)$ . (The normalisation comes from comparing the integrals of each side with respect to $ω$ : $\int_{- \infty}^{\infty} {sinc}^{2} x d x = π$ . See section A.8 for more on $δ$ -functions.) Then we have

P_{i \to n} = \frac{2 π t}{ℏ^{2}} {|⟨ n | {\hat{H}}^{(1)} | i ⟩|}^{2} δ (ω_{n i} - ω) .

The need to take the average over a long time period to obtain the frequency-matching delta function is easily understood if we remember that any wave train only approaches monochromaticity in the limit that it is very long; any truncation induces a spread of frequencies.

So the probability increases linearly with time, which accords with our expectation if the perturbation has a certain chance of inducing the transition in any given time interval. Finally then, we write for the transition rate (probability per unit time):

R_{i \to n} = \frac{1}{t} P_{i \to n} = \frac{2 π}{ℏ} {|⟨ n | {\hat{H}}^{(1)} | i ⟩|}^{2} δ (E_{n i} - ℏ ω)

This result is commonly known as “Fermi’s golden rule”, and it is the main result of this section. We have written $ℏ ω = E$ to highlight the $δ$ -function as enforcing conservation of energy in the transition: the energy diﬀerence between the ﬁnal and initial state must be supplied by the perturbation. Although we have assumed nothing about radiation in the derivation, we have ended up with the prediction that only the correct frequency of perturbation can induce a transition, a result reminiscent of the photoelectric eﬀect which demonstrates empirically that energy is absorbed from ﬁelds in quanta of energy $ℏ ω$ .

Taking on trust its importance though, how are we to interpret it? Taken literally, it says that if $E \neq E_{n i}$ nothing happens, which is dull, and if $E = E_{n i}$ the transition rate is inﬁnite and ﬁrst-order perturbation theory would appear to have broken down! The resolution is that for actual applications, we do not have perfectly monochromatic radiation inducing transitions to single, absolutely sharp energy levels. Such a perfect resonance would indeed give inﬁnite transition rates, but is unphysical. In practice there is always an integration over energy, with a weighting function $ρ (E)$ which is a distribution function smearing the strength over a range of energies. Authors who don’t like playing with $δ$ functions leave the $t^{2} {sinc}^{2} (\dots)$ form in place till this is evident, but the bottom line is the same (see eg Mandl).

Had we started from ${\hat{H}}^{(1)} e^{i ω t}$ we would have ended up with $δ (E_{n i} + E)$ In this case, energy must be given up to the perturbing ﬁeld: this is emission rather than absorption. With a real ﬁeld $cos (ω t) = \frac{1}{2} (e^{i ω t} + e^{- i ω t})$ , both processes can occur.

Finally we must point out that though we’ve derived Fermi’s golden rule for oscillatory perturbations, the expression holds equally in the $ω \to 0$ limit i.e. for a constant perturbation acting from time $- t ∕ 2 \to t ∕ 2$ . It should be easy to see that exactly the same result is obtained, except that the energy-conservation $δ$ -function is simply $δ E_{n i}$ . This form is more appropriate if, instead of considering an external ﬁeld which can supply or absorb energy, one thinks in terms of photons and deﬁnes $| n ⟩$ and $| i ⟩$ to include not only the atomic state, but also any incoming or out-going photon. Viewed in that way the energies of the initial and ﬁnal state must be the same. Both derivations are valid but the one given here is more appropriate for our purposes given that we are not going to quantise the radiation ﬁeld. (Wait till next year for that!) However the constant perturbation form will be used in scattering in the next chapter.

Gasiorowicz ch 15.2
(Mandl ch 9.4)
Shankar ch 18.2
(Townsend ch 14.7)

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