Develop time-dependent perturbation theory for a multi-level system, starting with the generalization of Equations 9.1 and 9.2:

${\widehat{\mathbf{H}}}_{\mathbf{0}}{\mathit{\psi }}_{\mathbf{n}}\mathbf{=}{\mathit{E}}_{\mathbf{n}}{\mathit{\psi }}_{\mathbf{n}}\mathbf{,}\mathbf{⟨}{\mathbf{\psi }}_{\mathbf{n}}\mathbf{\mid }{\mathbf{\psi }}_{\mathbf{m}}\mathbf{⟩}\mathbf{=}{\mathit{\delta }}_{\mathbf{n}\mathbf{m}}$ (9.79)

At time t = 0 we turn on a perturbation ${\mathit{H}}^{\mathbf{\text{'}}}\mathbf{\left(}\mathit{t}\mathbf{\right)}$so that the total Hamiltonian is

$\widehat{\mathbf{H}}\mathbf{=}{\widehat{\mathbf{H}}}_{\mathbf{0}}\mathbf{+}{\widehat{\mathbf{H}}}^{\mathbf{\text{'}}}\mathbf{\left(}\mathit{t}\mathbf{\right)}$(9.80).

(a) Generalize Equation 9.6 to read

$\mathit{\Psi }\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{=}{\mathit{c}}_{\mathbf{a}}\mathbf{\left(}\mathit{t}\mathbf{\right)}{\mathit{\psi }}_{\mathbf{a}}{\mathit{e}}^{\mathbf{-}\mathbf{i}{\mathbf{E}}_{\mathbf{a}}\mathbf{t}\mathbf{/}\mathbf{\hslash }}\mathbf{+}{\mathit{c}}_{\mathbf{b}}\mathbf{\left(}\mathit{t}\mathbf{\right)}{\mathit{\psi }}_{\mathbf{b}}{\mathit{e}}^{\mathbf{-}\mathbf{i}{\mathbf{E}}_{\mathbf{b}}\mathbf{t}\mathbf{/}\mathbf{\hslash }}$(9.81).

and show that

${\dot{\mathbf{c}}}_{\mathbf{m}}\mathbf{=}\mathbf{-}\frac{\mathbf{i}}{\mathbf{\hslash }}\sum _n{c}_n{\mathit{H}}_{\mathbf{m}\mathbf{n}}^{\mathbf{\text{'}}}{\mathit{e}}^{\mathbf{i}\mathbf{(}{\mathbf{E}}_{\mathbf{m}}\mathbf{-}{\mathbf{E}}_{\mathbf{n}}\mathbf{)}\mathbf{t}\mathbf{/}\mathbf{\hslash }}$ (9.82).

Where

${\mathit{H}}_{\mathbf{m}\mathbf{n}}^{\mathbf{\text{'}}}\mathbf{\equiv }{\mathit{\psi }}_{\mathbf{m}}\mathbf{\left|}{\widehat{\mathbf{H}}}^{\mathbf{\text{'}}}\mathbf{\right|}{\mathit{\psi }}_{\mathbf{n}}$ (9.83).

(b) If the system starts out in the state ${\psi }_N$, show that (in first-order perturbation theory)

${\mathit{c}}_{\mathbf{N}}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{\approx }\mathbf{1}\mathbf{-}\frac{\mathbf{i}}{\mathbf{\hslash }}{\int }_0^t{H}_{NN}^{\text{'}}\mathbf{\left(}{\mathbf{t}}^{\mathbf{\text{'}}}\mathbf{\right)}\mathit{d}{\mathit{t}}^{\mathbf{\text{'}}}$(9.84).

and

${\mathit{c}}_{\mathbf{m}}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{\approx }\mathbf{-}\frac{\mathbf{i}}{\mathbf{\hslash }}{\int }_0^t{H}_{mN}^{\text{'}}\mathbf{\left(}{\mathbf{t}}^{\mathbf{\text{'}}}\mathbf{\right)}{\mathit{e}}^{\mathbf{i}\mathbf{(}{\mathbf{E}}_{\mathbf{m}}\mathbf{-}{\mathbf{E}}_{\mathbf{N}}\mathbf{)}{\mathbf{t}}^{\mathbf{\text{'}}}\mathbf{/}\mathbf{\hslash }}\mathit{d}{\mathit{t}}^{\mathbf{\text{'}}}\mathbf{,}\mathbf{(}\mathit{m}\mathbf{\ne }\mathit{N}\mathbf{)}$(9.85).

(c) For example, suppose${\widehat{\mathbf{H}}}^{\mathbf{\text{'}}}$is constant (except that it was turned on at t = 0 , and switched off again at some later time . Find the probability of transition from state N to state M $\mathbf{(}\mathit{M}\mathbf{\ne }\mathit{N}\mathbf{)}\mathbf{,}$as a function of T. Answer:

$\mathbf{4}{\mathbf{\left|}{\mathbf{H}}_{\mathbf{M}\mathbf{N}}^{\mathbf{\text{'}}}\mathbf{\right|}}^{\mathbf{2}}\frac{\mathbf{s}\mathbf{i}{\mathbf{n}}^{\mathbf{2}}\mathbf{[}\mathbf{(}{\mathbf{E}}_{\mathbf{N}}\mathbf{-}{\mathbf{E}}_{\mathbf{M}}\mathbf{)}\mathbf{T}\mathbf{/}\mathbf{2}\mathbf{\hslash }\mathbf{]}}{{\mathbf{(}{\mathbf{E}}_{\mathbf{N}}\mathbf{-}{\mathbf{E}}_{\mathbf{M}}\mathbf{)}}^{\mathbf{2}}}$ (9.86).

(d) Now suppose${\widehat{\mathbf{H}}}^{\mathbf{\text{'}}}$is a sinusoidal function of time${\widehat{\mathbf{H}}}^{\mathbf{\text{'}}}\mathbf{=}\mathit{V}\mathit{c}\mathit{o}\mathit{s}\mathbf{\left(}\mathit{\omega }\mathit{t}\mathbf{\right)}$: Making the usual assumptions, show that transitions occur only to states with energy ${\mathit{E}}_{\mathbf{M}}\mathbf{=}{\mathit{E}}_{\mathbf{N}}\mathbf{\pm }\mathit{\hslash }$, and the transition probability is

${\mathit{P}}_{\mathbf{N}\mathbf{\to }\mathbf{M}}\mathbf{=}{\mathbf{\left|}{\mathbf{V}}_{\mathbf{M}\mathbf{N}}\mathbf{\right|}}^{\mathbf{2}}\frac{\mathbf{s}\mathbf{i}{\mathbf{n}}^{\mathbf{2}}\mathbf{[}\mathbf{(}{\mathbf{E}}_{\mathbf{N}}\mathbf{-}{\mathbf{E}}_{\mathbf{M}}\mathbf{\pm }\mathbf{\hslash }\mathbf{\omega }\mathbf{)}\mathbf{T}\mathbf{/}\mathbf{2}\mathbf{\hslash }\mathbf{]}}{{\mathbf{(}{\mathbf{E}}_{\mathbf{N}}\mathbf{-}{\mathbf{E}}_{\mathbf{M}}\mathbf{\pm }\mathbf{\hslash }\mathbf{\omega }\mathbf{)}}^{\mathbf{2}}}$ (9.87).

(e) Suppose a multi-level system is immersed in incoherent electromagnetic radiation. Using Section 9.2.3 as a guide, show that the transition rate for stimulated emission is given by the same formula (Equation 9.47) as for a two-level system.

${\mathit{R}}_{\mathbf{b}\mathbf{\to }\mathbf{a}}\mathbf{=}\frac{\mathbf{\pi }}{\mathbf{3}{\mathbf{ϵ}}_{\mathbf{0}}{\mathbf{\hslash }}^{\mathbf{2}}}\mathbf{|}\mathit{\wp }{\mathbf{|}}^{\mathbf{2}}\mathit{\rho }\mathbf{\left(}{\mathbf{\omega }}_{\mathbf{0}}\mathbf{\right)}\mathit{R}\mathit{b}$ (9.47).

$\begin{array}{l}\Psi \left(t\right)=\sum c_n\left(t\right)e^{-i{E}_{nt}/\hslash }{\psi }_n\\ H\Psi =i\hslash \frac{\partial \Psi }{\partial t};H=H_0+H^{\text{'}}\left(t\right);H_0{\psi }_n=E_n{\psi }_n\end{array}$

so

$\begin{array}{l}\sum c_n{e}^{-i{E}_{n}t/\hslash }{E}_n{\psi }_n+\sum c_n{e}^{-i{E}_{n}t/\hslash }{H}^{\text{'}}{\psi }_n\\ =i\hslash \sum {\dot{c}}_n{e}^{-i{E}_{n}t/\hslash }{\psi }_n+i\hslash (-\frac{i}{\hslash })\sum c_n{E}_n{e}^{-i{E}_{n}t/\hslash }{\psi }_n\end{array}$

The first and last terms cancel, so

$\begin{array}{l}\sum c_n{e}^{-i{E}_{n}t/\hslash }{H}^{\text{'}}{\psi }_n=i\hslash \sum {\dot{c}}_n{e}^{-i{E}_{n}t/\hslash }{\psi }_n\\ \sum c_n{e}^{-i{E}_{n}t/\hslash }⟨{\psi }_m\left|H^{\text{'}}\right|{\psi }_n⟩=i\hslash \sum {\dot{c}}_n{e}^{-i{E}_{n}t/\hslash }⟨{\psi }_m\mid {\psi }_n⟩\end{array}$

Assume orthonormality of the unperturbed states,

$⟨{\psi }_m\mid {\psi }_n⟩={\delta }_{mn},$

and define

$H_{mn}^{\text{'}}\equiv ⟨{\psi }_m\left|H^{\text{'}}\right|{\psi }_n⟩$

$\sum c_n{e}^{-i{E}_{n}t/\hslash }{H}_{mn}^{\text{'}}=i\hslash {\dot{c}}_m{e}^{-i{E}_{m}t/\hslash },or{\dot{c}}_m=-\frac{i}{\hslash }\sum _n{c}_n{H}_{mn}^{\text{'}}{e}^{i(E_m-E_n)t/\hslash }.$

Zero order$:c_N\left(t\right)=1,c_m\left(t\right)=0\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}for}m\ne NcN\left(t\right)=1,$

Then in first order:

$\begin{array}{l}{\dot{c}}_N=-\frac{i}{\hslash }{H}_{NN}^{\text{'}},\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}or\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{c}_N\left(t\right)=1-\frac{i}{\hslash }{\int }_0^t{H}_{NN}^{\text{'}}\left(t^{\text{'}}\right)d{t}^{\text{'}},whereasform\ne N:\\ {\dot{c}}_m=-\frac{i}{\hslash }{H}_{mN}^{\text{'}}{e}^{i(E_m-E_N)t/\hslash },\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}or\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{c}_m\left(t\right)=-\frac{i}{\hslash }{\int }_0^t{H}_{mN}^{\text{'}}\left(t^{\text{'}}\right)e^{i(E_m-E_N)t^{\text{'}}/\hslash }d{t}^{\text{'}}\end{array}$

$\begin{array}{l}{c}_M\left(t\right)=-\frac{i}{\hslash }{H}_{MN}^{\text{'}}{\int }_0^t{e}^{i(E_M-E_N)t^{\text{'}}/\hslash }d{t}^{\text{'}}\\ =-{\frac{i}{\hslash }{H}_{MN}^{\text{'}}\left[\frac{e^{i(E_M-E_N)t^{\text{'}}/\hslash }}{i(E_M-E_N)/\hslash }\right]|}_0^t\\ =-H_{MN}^{\text{'}}\left[\frac{e^{i(E_M-E_N)t/\hslash }-1}{E_M-E_N}\right]\end{array}$

$=-\frac{H_{MN}^{\text{'}}}{(E_M-E_N)}{e}^{i(E_M-E_N)t/2\hslash }2i\sin \left(\frac{E_M-E_N}{2\hslash }t\right)$

$P_{N\to M}={\left|c_M\right|}^2=\frac{4{\left|H_{MN}^{\text{'}}\right|}^2}{{(E_M-E_N)}^2}{\sin }^2\left(\frac{E_M-E_N}{2\hslash }t\right)$

$\begin{array}{l}{c}_M\left(t\right)=-\frac{i}{\hslash }{V}_{MN}\frac{1}{2}{\int }_0^t(e^{i\omega t^{\text{'}}}+e^{-i\omega t^{\text{'}}})e^{i(E_M-E_N)t^{\text{'}}/\hslash }d{t}^{\text{'}}\\ =-{\frac{i{V}_{MN}}{2\hslash }[\frac{e^{i(\hslash \omega +E_M-E_N)t^{\text{'}}/\hslash }}{i(\hslash \omega +E_M-E_N)/\hslash }+\frac{e^{i(-\hslash \omega +E_M-E_N)t^{\text{'}}/\hslash }}{i(-\hslash \omega +E_M-E_N)/\hslash }]|}_0^t\end{array}$

$\text{If}{E}_M>E_N$, the second term dominates, and transitions occur only for$\omega \approx (E_M-E_N)/\hslash :$

$c_M\left(t\right)\approx -\frac{i{V}_{MN}}{2\hslash }\frac{1}{(i/\hslash )(E_M-E_N-\hslash \omega )}{e}^{i(E_M-E_N-\hslash \omega )t/2\hslash }2i\sin \left(\frac{E_M-E_N-\hslash \omega }{2\hslash }t\right)$

so

$P_{N\to M}={\left|c_M\right|}^2=\frac{{\left|V_{MN}\right|}^2}{{(E_M-E_N-\hslash \omega )}^2}{\sin }^2\left(\frac{E_M-E_N-\hslash \omega }{2\hslash }t\right)$

$\text{If}{E}_M<E_N$ the first term dominates, and transitions occur only for$\omega \approx (E_N-E_M)/\hslash$

$c_M\left(t\right)\approx -\frac{i{V}_{MN}}{2\hslash }\frac{1}{(i/\hslash )(E_M-E_N+\hslash \omega )}{e}^{i(E_M-E_N+\hslash \omega )t/2\hslash }2i\sin \left(\frac{E_M-E_N+\hslash \omega }{2\hslash }t\right)$

and hence

$P_{N\to M}=\frac{{\left|V_{MN}\right|}^2}{{(E_M-E_N+\hslash \omega )}^2}{\sin }^2\left(\frac{E_M-E_N+\hslash \omega }{2\hslash }t\right)$

Combining the two results, we conclude that transitions occur to states with energy $E_M\approx E_N\pm \hslash$and

$P_{N\to M}=\frac{{\left|V_{MN}\right|}^2}{{(E_M-E_N\pm \hslash \omega )}^2}{\sin }^2\left(\frac{E_M-E_N\pm \hslash \omega }{2\hslash }t\right)$

For light$V_{ba}=-\wp E_0$, (Eq. 9.34). The rest is as before (Section 11.2.3), leading to Eq. 9.47:

$V_{ba}=-\wp E_0$ (9.34).

$R_{b\to a}=\frac{\pi }{3{ϵ}_0{\hslash }^2}|\wp {|}^2\rho \left({\omega }_0\right)$ (9.47).

$R_{N\to M}=\frac{\pi }{3{ϵ}_0{\hslash }^2}\left|\wp {|}^2\rho \right(\omega ),\text{with}\omega =\pm (E_M-E_N)$

$(+\mathrm{sign}\Rightarrow \text{absorption,}-\mathrm{sign}\Rightarrow \text{stimulatedemission)}$