Fermi's golden rule for non-sinusoidal potentials like e^{-at}

[MisterX]

From what I have seen, Fermi's Golden rule is applied to constant or sinusoidal time varying potentials. But what if the perturbation is of the form $V_0\left( \mathbf{x}\right)f\left(t\right)$, where $f(t)$ is not a constant or sinusoidal? I am not really familiar with the derivation of Fermi's golden rule, and the explanations I was given both seemed very hand-wavy. I know we can Fourier decompose $f(t)$ in time, but it's not clear to me how that might be related back to transition probabilities. In particular, what if $f(t) = e^{-a t}$ with $a \in \mathbb{R}, a > 0$ ?

 

[andrien]

why don't you take a look at method of variation of constant.To first order of perturbation theory,probability is just given by absolute square of $c_n^1$.

 

[Bill_K]

From what I have seen, Fermi's Golden rule is applied to constant or sinusoidal time varying potentials. But what if the perturbation is of the form $V_0\left( \mathbf{x}\right)f\left(t\right)$, where $f(t)$ is not a constant or sinusoidal? I am not really familiar with the derivation of Fermi's golden rule, and the explanations I was given both seemed very hand-wavy. I know we can Fourier decompose $f(t)$ in time, but it's not clear to me how that might be related back to transition probabilities. In particular, what if $f(t) = e^{-a t}$ with $a \in \mathbb{R}, a > 0$ ?

Fermi's Golden Rule applies specifically to the transition from a discrete initial state to a continuum of final states. One of the factors in it is ρ(E), the density of states at the final energy E. The reason the perturbing potential is restricted to a single frequency is so that E will be well-defined.

 

[Avodyne]

You can still apply the general formalism of time-dependent perturbation theory. Ooguri (now at Caltech) has some nice lecture notes on all of this from an old course he taught at Berkeley:
http://hitoshi.berkeley.edu/221a/timedependent.pdf

 

[Bill_K]

You can still apply the general formalism of time-dependent perturbation theory.

Of course you can. But that was not the question. The question, I believe, was about Fermi's Golden Rule.

 

[Avodyne]

The question was "what if the perturbation is of the form $V_0\left( \mathbf{x}\right)f\left(t\right)$, where $f(t)$ is not a constant or sinusoidal?" And the answer is, "apply the general formalism of time-dependent perturbation theory."

 

[Bill_K]

The question was "what if the perturbation is of the form $V_0\left( \mathbf{x}\right)f\left(t\right)$, where $f(t)$ is not a constant or sinusoidal?" And the answer is, "apply the general formalism of time-dependent perturbation theory."

Take a look at the title of the thread.