Time dependent perturbation theory of the harmonic oscilator

[CAF123]

Homework Statement

A 1-d harmonic oscillator of charge $q$ is acted upon by a uniform electric field which may be considered to be a perturbation and which has time dependence of the form $E(t) = \frac{K }{\sqrt{\pi} \tau} \exp (−(t/\tau)^2)$. Assuming that when $t = -\infty$, the oscillator is in its ground state, it can be shown that the probability that it is in its first excited state at $t = +\infty$ using time-dependent perturbation theory is $$P_{t} = \frac{q^2 K^2}{2 \hbar m \omega} e^{-\frac{\tau^2 \omega^2}{4}}$$

Discuss the behaviour of the transition probability and the applicability of the perturbation theory result when a) $\tau \ll 1/\omega$ and b) $\tau \gg 1/\omega$.

Homework Equations

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(Given in question)

The Attempt at a Solution

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The given conditions imply that either $\tau^2 \omega^2 \ll 1$ or $\tau^2 \omega^2 \gg 1$.In the former case, I can expand the exp factor for small argument and for the latter case, the exp term dies away to zero quickly giving a transition prob of zero. I am just wondering what further comments can be made, in particular regarding the applicability of the perturbation theory result? Thanks!

 

[mark726]

In the case of $\tau \ll 1/\omega$, the transition probability is dominated by the factor $e^{-\frac{\tau^2 \omega^2}{4}}$, which is close to 1. This indicates a high probability of the oscillator being in its first excited state at $t = +\infty$. This behavior is consistent with the fact that when $\tau \ll 1/\omega$, the time dependence of the electric field is rapid compared to the natural frequency of the oscillator, so it effectively "pumps" energy into the system. Therefore, the perturbation theory result is applicable in this case.

On the other hand, in the case of $\tau \gg 1/\omega$, the transition probability is dominated by the factor $e^{-\frac{\tau^2 \omega^2}{4}}$, which is close to 0. This indicates a low probability of the oscillator being in its first excited state at $t = +\infty$. This behavior is consistent with the fact that when $\tau \gg 1/\omega$, the time dependence of the electric field is slow compared to the natural frequency of the oscillator, so it does not significantly affect the energy of the system. Therefore, the perturbation theory result may not be applicable in this case, as the perturbation is not strong enough to induce a significant change in the system.

In general, the perturbation theory result is most applicable when the perturbation is small compared to the natural frequency of the system. In this case, the perturbation can be treated as a small correction to the unperturbed system, and the resulting transition probability can be accurately calculated using perturbation theory. However, when the perturbation is large compared to the natural frequency, the perturbation theory result may not be accurate and other techniques may need to be used to calculate the transition probability.