Principles of Quantum Mechanics · Problem 18.3

Time-Dependent Perturbation Theory — R. Shankar

Physics — Quantum Mechanics · sinusoidal-perturbation

Problem

For the perturbation H′=V(r)cos⁡(ωt)H' = V(\mathbf{r})\cos(\omega t), show that the transition probability (Equation 18.30) can be written Pi→f=∣Vfi∣22ℏ2[sin⁡2[(ωfi+ω)t/2](ωfi+ω)2+sin⁡2[(ωfi−ω)t/2](ωfi−ω)2+cross terms]P_{i\to f} = \frac{|V_{fi}|^2}{2\hbar^2} \left[ \frac{\sin^2[(\omega_{fi}+\omega)t/2]}{(\omega_{fi}+\omega)^2} + \frac{\sin^2[(\omega_{fi}-\omega)t/2]}{(\omega_{fi}-\omega)^2} + \text{cross terms} \right] If we are looking at the case ω≈ωfi>0\omega \approx \omega_{fi} > 0 (stimulated emission is not relevant here), then the first term and the cross terms are negligible. Show that the transition rate (Equation 18.32) is Ri→f=π∣Vfi∣22ℏ2δ(ωfi−ω).R_{i\to f} = \frac{\pi |V_{fi}|^2}{2\hbar^2} \delta(\omega_{fi} - \omega).

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