Fermi's golden rule: why delta function instead of density states?

[yucheng]

TL;DR Summary

See bolded text!

Sakurai, in $\S$ 5.7.3 Constant Perturbation mentions that the transition rate can be written in both ways:

$$w_{i \to [n]} = \frac{2 \pi}{\hbar} |V_{ni}|^2 \rho(E_n)$$
and
$$w_{i \to n} = \frac{2 \pi}{\hbar} |V_{ni}|^2 \delta(E_n - E_i)$$
where it must be understood that this expression is integrated with $\int dE_n \rho(E_n)$

My question is, what is the advantage of the delta function representation? Which one is actually measured/quoted in experiments?

My guess: we know how to calculate $|V_{ni}|$, but we need not know $\rho(E_n)$! Hence it is more convenient to just quote the expression with the delta function.

P.S. does anyone have useful references/reading material of when Fermi's Golden Rule is useful, in experiments, how the transition rates determined experimentally are related to FGR?

Thanks in advance!

(To be frank, I believe all of the confusion regarding FGR is due to the fact it's introduced out of context, i.e. from where it is applied...)

 

[vanhees71]

It's a bit unclear stated, what's meant by the 2nd expression. It's the (average) transition-probability rate for the transition from an asymptotic free energy-eigen-in-state to an asymptotic free energy-eigen-out state. If either state is a scattering state, i.e., in the continuous part of the Hamiltonian's spectrum, the corresponding "eigen state" is a generalized function (distribution) rather than a true square-integrable state, and thus the system cannot be really in such an energy eigenstate. In scattering theory you have to take always asymptotic free "wave-packet states" to get finite results for cross sections and then take the limit to exact (generalized) energy eigen states. For a careful treatment of this somewhat delicate issue, see

M. Peskin and D. V. Schroeder, An Introduction to Quantum
Field Theory, Addison-Wesley Publ. Comp., Reading,
Massachusetts (1995).

Although there it's discussed for relativistic QFT, the same arguments hold true in non-relativistic QM scattering theory.

 

[WernerQH]

My question is, what is the advantage of the delta function representation? Which one is actually measured/quoted in experiments?

Integrating the delta function representation over any finite energy interval gives (by definition of the delta function!) the exact number of states in that interval. But instead of using this ragged function it is much more practical to use the smoothly varying average density of states. It is proportional to ${\mathbf k}^2$ or, in the case of photons, to $\nu^2$, and accounts for the fact that transition rates tend to grow quickly with increasing energy.

My guess: we know how to calculate $|V_{ni}|$, but we need not know $\rho(E_n)$! Hence it is more convenient to just quote the expression with the delta function.

No. The density of states is just as important as the squared modulus of the transition matrix element. Fermi's golden rule wouldn't make sense, be dimensionally incorrect without it!

P.S. does anyone have useful references/reading material of when Fermi's Golden Rule is useful, in experiments, how the transition rates determined experimentally are related to FGR?

The discussion I like best is in chapter 12 (Time-Dependent Perturbation Theory) in Gordon Baym's "Lectures on Quantum Mechanics" (Benjamin/Cummings Publishing Company, 1969). It's worth studying carefully. It is especially noteworthy that the transition probability, which at first sight would appear to be proportional to $t^2$ (square of the matrix element in first order perturbation theory) ends up proportional to $t$, as appropriate for a transition rate. The growth is diminished because the distribution over final states becomes ever narrower with time, the spread being proportional to $t^{-1}$.

(To be frank, I believe all of the confusion regarding FGR is due to the fact it's introduced out of context, i.e. from where it is applied...)

I don't understand what "confusion" you are referring to. For me the usefulness of Fermi's golden rule is beyond doubt. The tricky part is its derivation, because it is an amalgamation of Schrödinger's time-dependent equation with the Born rule. Some people think that application of the Born rule is related to "measurement" and must be strictly separated from unitary evolution according to Schrödinger's equation, but I think they are closely intertwined and cannot be separated.

 

[vanhees71]

It's just time-dependent first-order perturbation theory. The devil is in the detail, how to deal with distributions! The treatment in Baym's book is, of course, very clear.

 

[WernerQH]

The devil is in the detail, how to deal with distributions!

I don't have any qualms with distributions. As a physicist, I even find them quite intuitive.

It's just time-dependent first-order perturbation theory.

It's a bit more than that, because it is not merely computing a first order correction to a matrix element. Fermi's golden rule is an expression for an observable quantity, a transition rate, and cannot be derived without using the Born rule.

 

[vanhees71]

Of course not. I don't know, why one should avoid Born's rule at all. It's the only "interpretation" of the quantum formalism needed to make contact with observable phenomena.