# Fermi's Golden Rule: non-sinusoidal $t\to\infty$

• MisterX
In summary: It's "Fermi's Golden Rule for time-dependent perturbation." So the question is, how do you apply Fermi's Golden Rule in the case of a time-dependent perturbation? And the answer is, you can't really, because Fermi's Golden Rule is specifically for the case of a single frequency perturbation. However, you can still use the general formalism of time-dependent perturbation theory to calculate transition probabilities. In summary, Fermi's Golden Rule is typically applied to constant or sinusoidal time-varying potentials, but can also be applied to more complex time-varying potentials using the general formalism of time-dependent perturbation theory.
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 $a \in \mathbb{R}, a > 0$ ?

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##.

MisterX said:
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 said:
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.

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."

Avodyne said:
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.

## What is Fermi's Golden Rule?

Fermi's Golden Rule is a fundamental principle in quantum mechanics that describes the transition rate between energy states in a quantum system. It is named after physicist Enrico Fermi.

## What is meant by "non-sinusoidal $t\to\infty$" in Fermi's Golden Rule?

This phrase refers to the time dependence of a quantum system as it approaches infinity. It means that the system is no longer following a sinusoidal pattern, but instead has reached a steady state where the probability of transition between energy states remains constant over time.

## How is Fermi's Golden Rule used in quantum mechanics?

Fermi's Golden Rule is used to calculate the likelihood of a quantum system transitioning from one energy state to another. It is essential for understanding the behavior of particles at the atomic and subatomic level.

## What is the significance of the non-sinusoidal $t\to\infty$ limit in Fermi's Golden Rule?

The non-sinusoidal $t\to\infty$ limit represents the long-term behavior of a quantum system. It allows us to make predictions about the system's behavior over time and understand the stability of energy states within the system.

## Are there any limitations to Fermi's Golden Rule?

Like any scientific principle, Fermi's Golden Rule has its limitations. It assumes that the system is in a steady state and does not take into account any external influences or perturbations. It also does not apply to systems with strong interactions or when the energy states are not well-defined.

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