Q15P Develop time-dependent perturbat... [FREE SOLUTION]

Chapter 9: Q15P (page 363)

Develop time-dependent perturbation theory for a multi-level system, starting with the generalization of Equations 9.1 and 9.2:
${\hat{H}}_{0} 蠄_{n} = E_{n} 蠄_{n}, 鈥夆赌夆赌� 鉄� 蠄_{n} 鈭� 蠄_{m} 鉄� = 未_{n m}$ (9.79)
At time t = 0 we turn on a perturbation $H^{'} (t)$ so that the total Hamiltonian is
$\hat{H} = {\hat{H}}_{0} + {\hat{H}}^{'} (t)$ (9.80).
(a) Generalize Equation 9.6 to read
$唯 (t) = c_{a} (t) 蠄_{a} e^{鈭� i E_{a} t / 鈩�} + c_{b} (t) 蠄_{b} e^{鈭� i E_{b} t / 鈩�}$ (9.81).
and show that
${\overset{藱}{c}}_{m} = 鈭� \frac{i}{鈩�} \underset{n}{鈭�} c_{n} H_{m n}^{'} e^{i (E_{m} 鈭� E_{n}) t / 鈩�}$ (9.82).
Where
$H_{m n}^{'} 鈮� 蠄_{m} | {\hat{H}}^{'} | 蠄_{n}$ (9.83).
(b) If the system starts out in the state $蠄_{N}$ , show that (in first-order perturbation theory)
$c_{N} (t) 鈮� 1 鈭� \frac{i}{鈩�} {鈭�}_{0}^{t} H_{N N}^{'} (t^{'}) d t^{'}$ (9.84).
and
$c_{m} (t) 鈮� 鈭� \frac{i}{鈩�} {鈭�}_{0}^{t} H_{m N}^{'} (t^{'}) e^{i (E_{m} 鈭� E_{N}) t^{'} / 鈩�} d t^{'}, 鈥夆赌夆赌� (m 鈮� N)$ (9.85).
(c) For example, suppose ${\hat{H}}^{'}$ 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 $(M 鈮� N),$ as a function of T. Answer:
$4 {| H_{M N}^{'} |}^{2} \frac{s i n^{2} [(E_{N} 鈭� E_{M}) T / 2 鈩�]}{{(E_{N} 鈭� E_{M})}^{2}}$ (9.86).
(d) Now suppose ${\hat{H}}^{'}$ is a sinusoidal function of time ${\hat{H}}^{'} = V c o s (蝇 t)$ : Making the usual assumptions, show that transitions occur only to states with energy $E_{M} = E_{N} 卤鈩�$ , and the transition probability is
$P_{N 鈫� M} = {| V_{M N} |}^{2} \frac{s i n^{2} [(E_{N} 鈭� E_{M} 卤鈩� 蝇) T / 2 鈩�]}{{(E_{N} 鈭� E_{M} 卤鈩� 蝇)}^{2}}$ (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.
$R_{b 鈫� a} = \frac{蟺}{3 系_{0} {鈩�}^{2}} | 鈩� |^{2} 蚁 (蝇_{0}) R b$ (9.47).

Short Answer

Expert verified

(a) $鈭� c_{n} e^{鈭� i E_{n} t / 鈩�} H_{m n}^{'} = i 鈩� {\overset{藱}{c}}_{m} e^{鈭� i E_{m} t / 鈩�}, o r {\overset{藱}{c}}_{m} = 鈭� \frac{i}{鈩�} \underset{n}{鈭�} c_{n} H_{m n}^{'} e^{i (E_{m} 鈭� E_{n}) t / 鈩�}$

(b) ${\overset{藱}{c}}_{m} = 鈭� \frac{i}{鈩�} H_{m N}^{'} e^{i (E_{m} 鈭� E_{N}) t / 鈩�}, 鈥夆赌夆赌塷r鈥夆赌夆赌� c_{m} (t) = 鈭� \frac{i}{鈩�} {鈭�}_{0}^{t} H_{m N}^{'} (t^{'}) e^{i (E_{m} 鈭� E_{N}) t^{'} / 鈩�} d t^{'}$

(c) $P_{N 鈫� M} = {| c_{M} |}^{2} = \frac{4 {| H_{M N}^{'} |}^{2}}{{(E_{M} 鈭� E_{N})}^{2}} \sin^{2} (\frac{E_{M} 鈭� E_{N}}{2 鈩�} t)$

(d) $P_{N 鈫� M} = \frac{{| V_{M N} |}^{2}}{{(E_{M} 鈭� E_{N} 卤鈩� 蝇)}^{2}} \sin^{2} (\frac{E_{M} 鈭� E_{N} 卤鈩� 蝇}{2 鈩�} t)$

(e) $R_{N 鈫� M} = \frac{蟺}{3 系_{0} {鈩�}^{2}} | 鈩� |^{2} 蚁 (蝇), with 蝇 = 卤 (E_{M} 鈭� E_{N})$

Step by step solution

(a) Generalize Equation 9.6

$\begin{array}{l} 唯 (t) = 鈭� c_{n} (t) e^{鈭� i E_{n t} / 鈩�} 蠄_{n} \\ 鈥夆赌� H 唯 = i 鈩� \frac{鈭� 唯}{鈭� t}; 鈥夆赌夆赌� H = H_{0} + H^{'} (t); 鈥夆赌夆赌� H_{0} 蠄_{n} = E_{n} 蠄_{n} \end{array}$

$\begin{array}{l} 鈭� c_{n} e^{鈭� i E_{n} t / 鈩�} E_{n} 蠄_{n} + 鈭� c_{n} e^{鈭� i E_{n} t / 鈩�} H^{'} 蠄_{n} \\ = i 鈩� 鈭� {\overset{藱}{c}}_{n} e^{鈭� i E_{n} t / 鈩�} 蠄_{n} + i 鈩� (鈭� \frac{i}{鈩�}) 鈭� c_{n} E_{n} e^{鈭� i E_{n} t / 鈩�} 蠄_{n} \end{array}$

The first and last terms cancel, so

$\begin{array}{l} 鈭� c_{n} e^{鈭� i E_{n} t / 鈩�} H^{'} 蠄_{n} = i 鈩� 鈭� {\overset{藱}{c}}_{n} e^{鈭� i E_{n} t / 鈩�} 蠄_{n} \\ 鈭� c_{n} e^{鈭� i E_{n} t / 鈩�} 鉄� 蠄_{m} | H^{'} | 蠄_{n} 鉄� = i 鈩� 鈭� {\overset{藱}{c}}_{n} e^{鈭� i E_{n} t / 鈩�} 鉄� 蠄_{m} 鈭� 蠄_{n} 鉄� \end{array}$

Assume orthonormality of the unperturbed states,

$鉄� 蠄_{m} 鈭� 蠄_{n} 鉄� = 未_{m n}, 鈥�$

and define

$H_{m n}^{'} 鈮� 鉄� 蠄_{m} | H^{'} | 蠄_{n} 鉄�$

$鈭� c_{n} e^{鈭� i E_{n} t / 鈩�} H_{m n}^{'} = i 鈩� {\overset{藱}{c}}_{m} e^{鈭� i E_{m} t / 鈩�}, o r {\overset{藱}{c}}_{m} = 鈭� \frac{i}{鈩�} \underset{n}{鈭�} c_{n} H_{m n}^{'} e^{i (E_{m} 鈭� E_{n}) t / 鈩�} .$

(b)showing in first-order perturbation theory

Zero order $: c_{N} (t) = 1, 鈥夆赌夆赌� c_{m} (t) = 0 鈥夆赌夆赌塮or m 鈮� N c N (t) = 1,$

Then in first order:

$\begin{array}{l} {\overset{藱}{c}}_{N} = 鈭� \frac{i}{鈩�} H_{N N}^{'}, 鈥夆赌夆赌塷r鈥夆赌夆赌� c_{N} (t) = 1 鈭� \frac{i}{鈩�} {鈭�}_{0}^{t} H_{N N}^{'} (t^{'}) d t^{'}, w h e r e a s f o r m 鈮� N : \\ {\overset{藱}{c}}_{m} = 鈭� \frac{i}{鈩�} H_{m N}^{'} e^{i (E_{m} 鈭� E_{N}) t / 鈩�}, 鈥夆赌夆赌塷r鈥夆赌夆赌� c_{m} (t) = 鈭� \frac{i}{鈩�} {鈭�}_{0}^{t} H_{m N}^{'} (t^{'}) e^{i (E_{m} 鈭� E_{N}) t^{'} / 鈩�} d t^{'} \end{array}$

(c) Finding the probability of transition from state N to state M (M≠N),as a function of T

$\begin{array}{l} c_{M} (t) = 鈭� \frac{i}{鈩�} H_{M N}^{'} {鈭�}_{0}^{t} e^{i (E_{M} 鈭� E_{N}) t^{'} / 鈩�} d t^{'} \\ = 鈭� {\frac{i}{鈩�} H_{M N}^{'} [\frac{e^{i (E_{M} 鈭� E_{N}) t^{'} / 鈩�}}{i (E_{M} 鈭� E_{N}) / 鈩�}] |}_{0}^{t} \\ = 鈭� H_{M N}^{'} [\frac{e^{i (E_{M} 鈭� E_{N}) t / 鈩�} 鈭� 1}{E_{M} 鈭� E_{N}}] \end{array}$

$= 鈭� \frac{H_{M N}^{'}}{(E_{M} 鈭� E_{N})} e^{i (E_{M} 鈭� E_{N}) t / 2 鈩�} 2 i \sin (\frac{E_{M} 鈭� E_{N}}{2 鈩�} t)$

$P_{N 鈫� M} = {| c_{M} |}^{2} = \frac{4 {| H_{M N}^{'} |}^{2}}{{(E_{M} 鈭� E_{N})}^{2}} \sin^{2} (\frac{E_{M} 鈭� E_{N}}{2 鈩�} t)$

(d)showing that transitions occur only to states with energyEM=EN±ℏ

$\begin{array}{l} c_{M} (t) = 鈭� \frac{i}{鈩�} V_{M N} \frac{1}{2} {鈭�}_{0}^{t} (e^{i 蝇 t^{'}} + e^{鈭� i 蝇 t^{'}}) e^{i (E_{M} 鈭� E_{N}) t^{'} / 鈩�} d t^{'} \\ = 鈭� {\frac{i V_{M N}}{2 鈩�} [\frac{e^{i (鈩� 蝇 + E_{M} 鈭� E_{N}) t^{'} / 鈩�}}{i (鈩� 蝇 + E_{M} 鈭� E_{N}) / 鈩�} + \frac{e^{i (鈭� 鈩� 蝇 + E_{M} 鈭� E_{N}) t^{'} / 鈩�}}{i (鈭� 鈩� 蝇 + E_{M} 鈭� E_{N}) / 鈩�}] |}_{0}^{t} \end{array}$

$If E_{M} > E_{N}$ , the second term dominates, and transitions occur only for $蝇鈮� (E_{M} 鈭� E_{N}) / 鈩� :$

$c_{M} (t) 鈮� 鈭� \frac{i V_{M N}}{2 鈩�} \frac{1}{(i / 鈩�) (E_{M} 鈭� E_{N} 鈭� 鈩� 蝇)} e^{i (E_{M} 鈭� E_{N} 鈭� 鈩� 蝇) t / 2 鈩�} 2 i \sin (\frac{E_{M} 鈭� E_{N} 鈭� 鈩� 蝇}{2 鈩�} t)$

$P_{N 鈫� M} = {| c_{M} |}^{2} = \frac{{| V_{M N} |}^{2}}{{(E_{M} 鈭� E_{N} 鈭� 鈩� 蝇)}^{2}} \sin^{2} (\frac{E_{M} 鈭� E_{N} 鈭� 鈩� 蝇}{2 鈩�} t)$

$If E_{M} < E_{N}$ the first term dominates, and transitions occur only for $蝇鈮� (E_{N} 鈭� E_{M}) / 鈩�$

$c_{M} (t) 鈮� 鈭� \frac{i V_{M N}}{2 鈩�} \frac{1}{(i / 鈩�) (E_{M} 鈭� E_{N} + 鈩� 蝇)} e^{i (E_{M} 鈭� E_{N} + 鈩� 蝇) t / 2 鈩�} 2 i \sin (\frac{E_{M} 鈭� E_{N} + 鈩� 蝇}{2 鈩�} t)$

and hence

$P_{N 鈫� M} = \frac{{| V_{M N} |}^{2}}{{(E_{M} 鈭� E_{N} + 鈩� 蝇)}^{2}} \sin^{2} (\frac{E_{M} 鈭� E_{N} + 鈩� 蝇}{2 鈩�} t)$

Combining the two results, we conclude that transitions occur to states with energy $E_{M} 鈮� E_{N} 卤鈩�$ and

$P_{N 鈫� M} = \frac{{| V_{M N} |}^{2}}{{(E_{M} 鈭� E_{N} 卤鈩� 蝇)}^{2}} \sin^{2} (\frac{E_{M} 鈭� E_{N} 卤鈩� 蝇}{2 鈩�} t)$

(e)show that the transition rate for stimulated emission is given by the same formula (Equation 9.47) as for a two-level system.

For light $V_{b a} = 鈭� 鈩� E_{0}$ , (Eq. 9.34). The rest is as before (Section 11.2.3), leading to Eq. 9.47:

$V_{b a} = 鈭� 鈩� E_{0}$ (9.34).

$R_{b 鈫� a} = \frac{蟺}{3 系_{0} {鈩�}^{2}} | 鈩� |^{2} 蚁 (蝇_{0})$ (9.47).

$R_{N 鈫� M} = \frac{蟺}{3 系_{0} {鈩�}^{2}} | 鈩� |^{2} 蚁 (蝇), with 蝇 = 卤 (E_{M} 鈭� E_{N})$

$(+ sign 鈬� absorption, 鈭� sign 鈬� stimulatedemission)$

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Short Answer

Step by step solution

(a) Generalize Equation 9.6

(b)showing in first-order perturbation theory

(c) Finding the probability of transition from state N to state M (M≠N),as a function of T

(d)showing that transitions occur only to states with energyEM=EN±ℏ

(e)show that the transition rate for stimulated emission is given by the same formula (Equation 9.47) as for a two-level system.

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