Introduction to Quantum Mechanics · Problem 11.24

Quantum Dynamics — David J. Griffiths

Physics — Quantum Mechanics · time-dependent-perturbation-theory-multilevel

Problem

In this problem we develop time-dependent perturbation theory for a multilevel system, starting with the generalization of Equations 11.5 and 11.6: H^0ψn=Enψn,⟨ψn∣ψm⟩=δnm.(11.113)\hat{H}_0 \psi_n = E_n \psi_n, \quad \langle\psi_n|\psi_m\rangle = \delta_{nm}. \tag{11.113} At time t=0t = 0 we turn on a perturbation H′(t)H'(t), so that the total Hamiltonian is H^=H^0+H^′(t).(11.114)\hat{H} = \hat{H}_0 + \hat{H}'(t). \tag{11.114} (a) Generalize Equation 11.10 to read Ψ(t)=∑ncn(t)ψne−iEnt/ℏ,(11.115)\Psi(t) = \sum_n c_n(t)\psi_n e^{-iE_n t/\hbar}, \tag{11.115} and show that c˙m=−iℏ∑ncnHmn′ei(Em−En)t/ℏ,(11.116)\dot{c}_m = -\frac{i}{\hbar}\sum_n c_n H'_{mn} e^{i(E_m - E_n)t/\hbar}, \tag{11.116} where Hmn′≡⟨ψm∣H^′∣ψn⟩.(11.117)H'_{mn} \equiv \langle\psi_m|\hat{H}'|\psi_n\rangle. \tag{11.117} (b) If the system starts out in the state ψN\psi_N, show that (in first-order perturbation theory) cN(t)≈1−iℏ∫0tHNN′(t′) dt′.(11.118)c_N(t) \approx 1 - \frac{i}{\hbar}\int_0^t H'_{NN}(t')\,dt'. \tag{11.118} and cm(t)≈−iℏ∫0tHmN′(t′)ei(Em−EN)t′/ℏ dt′,(m≠N).(11.119)c_m(t) \approx -\frac{i}{\hbar}\int_0^t H'_{mN}(t')e^{i(E_m - E_N)t'/\hbar}\,dt', \quad (m \neq N). \tag{11.119} (c) For example, suppose H^′\hat{H}' is constant (except that it was turned on at t=0t = 0, and switched off again at some later time TT). Find the probability of transition from state NN to state MM (M≠NM \neq N), as a function of TT. *Answer:* 4∣HMN′∣2sin⁡2[(EN−EM)T/2ℏ](EN−EM)2.(11.120)4\left|H'_{MN}\right|^2 \frac{\sin^2\left[(E_N - E_M)T/2\hbar\right]}{(E_N - E_M)^2}. \tag{11.120} (d) Now suppose H^′\hat{H}' is a sinusoidal function of time: H^′=Vcos⁡(ωt)\hat{H}' = V\cos(\omega t). Making the usual assumptions, show that transitions occur only to states with energy EM=EN±ℏωE_M = E_N \pm \hbar\omega, and the transition probability is PN→M=∣VMN∣2sin⁡2[(EN−EM±ℏω)T/2ℏ](EN−EM±ℏω)2.(11.121)P_{N\to M} = |V_{MN}|^2 \frac{\sin^2\left[(E_N - E_M \pm \hbar\omega)T/2\hbar\right]}{(E_N - E_M \pm \hbar\omega)^2}. \tag{11.121} (e) Suppose a multi-level system is immersed in incoherent electromagnetic radiation. Using Section 11.2.3 as a guide, show that the transition rate for stimulated emission is given by the same formula (Equation 11.54) as for a two-level system.

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