問題 6-29 の解答例

Problem 6-29
Suppose two perturbations, $V (x, t)$ and $U (x, t)$ , are acting. (Examples include a combination of DC and AC electronic fields or a combination of electric and magnetic fields.) Suppose further that a certain transition cannot occur with either $V$ or $U$ alone, but can occur only when both act together. Under the special assumption that both $V$ and $U$ are constant in time, show that the matrix element determining the transition element is given by

\begin{matrix} (6-111) & M_{n \to m} = \sum_{k} \frac{V_{m k} U_{k n} + U_{m k} V_{k n}}{E_{k} - E_{n}} \end{matrix}

Next, suppose both potentials are periodic in time but have different frequencies,

ω_{1}

and

ω_{2}

. What then is the matrix element?

(解答)　仮定から行列要素は $V_{I} (t) + U_{I} (t)$ と書ける．従って， 2次の遷移振幅 $λ_{m n}^{(2)}$ に相当する 2次の係数 $c_{m}^{(2)} (T)$ は次である：

\begin{aligned} c_{m}^{(2)} (T) = {(\frac{- i}{ℏ})}^{2} \sum_{k} \int_{0}^{T} d t_{1} \int_{0}^{t_{1}} d t_{2} e^{i ω_{m k} t_{1}} [V_{m k} (t_{1}) + U_{m k} (t_{1})] e^{i ω_{k n} t_{2}} [V_{k n} (t_{2}) + U_{k n} (t_{2})] \\ (1) & = {(\frac{- i}{ℏ})}^{2} \sum_{k} \int_{0}^{T} d t_{1} e^{i ω_{m k} t_{1}} \int_{0}^{t_{1}} d t_{2} e^{i ω_{k n} t_{2}} [V_{m k} (t_{1}) + U_{m k} (t_{1})] [V_{k n} (t_{2}) + U_{k n} (t_{2})] \end{aligned}

ここでポテンシャルは

V

或は

U

の一方だけでは遷移は起こらないと仮定し, かつそれらは時間に依存しないとすれば, 上式中の行列要素は次に書ける：

\begin{aligned} [V_{m k} (t_{1}) + U_{m k} (t_{1})] [V_{k n} (t_{2}) + U_{k n} (t_{2})] \\ = V_{m k} (t_{1}) V_{k n} (t_{2}) + V_{m k} (t_{1}) U_{k n} (t_{2}) + U_{m k} (t_{1}) V_{k n} (t_{2}) + U_{m k} (t_{1}) U_{k n} (t_{2}) \\ (2) & = V_{m k} (t_{1}) U_{k n} (t_{2}) + U_{m k} (t_{1}) V_{k n} (t_{2}) = V_{m k} U_{k n} + U_{m k} V_{k n} \end{aligned}

すると式 (1) は, 式 (6-98) のときと同様にして次のようになる：

\begin{aligned} c_{m}^{(2)} (T) & = {(\frac{- i}{ℏ})}^{2} \sum_{k} (V_{m k} U_{k n} + U_{m k} V_{k n}) \int_{0}^{T} d t_{1} e^{i ω_{m k} t_{1}} \int_{0}^{t_{1}} d t_{2} e^{i ω_{k n} t_{2}} \\ (3) & = \sum_{k} \frac{V_{m k} U_{k n} + U_{m k} V_{k n}}{E_{k} - E_{n}} (\frac{e^{i ω_{m n} T} - 1}{E_{m} - E_{n}} - \frac{e^{i ω_{m k} T} - 1}{E_{m} - E_{k}}) \end{aligned}

よってこの場合の「遷移の行列要素」(matrix element for the transition) は, 式 (6-99) のときと同様な議論により次に書ける：

\begin{matrix} (4) & M_{n \to m} = \sum_{k} \frac{V_{m k} U_{k n} + U_{m k} V_{k n}}{E_{k} - E_{n}} \end{matrix}

次に, 両方のポテンシャルは周期的に時間変化する場合を考える．この場合, 問題 6-24 に倣ってポテンシャルを次のように表わすことにする：

\begin{aligned} V (t) + U (t) & = V (x) (e^{i ω_{1} t} + e^{- i ω_{1} t}) + U (x) (e^{i ω_{2} t} + e^{- i ω_{2} t}) \\ (5) & = 2 V (x) \cos ω_{1} t + 2 U (x) \cos ω_{2} t \end{aligned}

すると, 問題 6-24 と同様な手順により

c_{m}^{(2)} (T)

は次となる：

\begin{aligned} c_{m}^{(2)} (T) \\ = {(\frac{- i}{ℏ})}^{2} \sum_{k} \int_{0}^{T} d t_{1} \int_{0}^{t_{1}} d t_{2} e^{i ω_{m k} t_{1}} {V_{m k} (t_{1}) + U_{m k} (t_{1})} e^{i ω_{k n} t_{2}} {V_{k n} (t_{2}) + U_{k n} (t_{2})} \\ (6) & = {(\frac{- i}{ℏ})}^{2} \sum_{k} \int_{0}^{T} d t_{1} e^{i ω_{m k} t_{1}} {V_{m k} (t_{1}) + U_{m k} (t_{1})} \int_{0}^{t_{1}} d t_{2} e^{i ω_{k n} t_{2}} {V_{k n} (t_{2}) + U_{k n} (t_{2})} \end{aligned}

まず,

t_{2}

についての積分は次となる：

\begin{aligned} (\frac{- i}{ℏ}) & \int_{0}^{t_{1}} d t_{2} e^{i ω_{k n} t_{2}} {V_{k n} (t_{2}) + U_{k n} (t_{2})} \\ = (\frac{- i}{ℏ}) \int_{0}^{t_{1}} d t_{2} e^{i ω_{k n} t_{2}} {V_{k n} (e^{i ω_{1} t_{2}} + e^{- i ω_{1} t_{2}}) + U_{k n} (e^{i ω_{2} t_{2}} + e^{- i ω_{2} t_{2}})} \\ = \frac{1}{ℏ} [\frac{V_{k n}}{ω_{k n} + ω_{1}} {1 - e^{i (ω_{k n} + ω_{1}) t_{1}}} + \frac{V_{k n}}{ω_{k n} - ω_{1}} {1 - e^{i (ω_{k n} - ω_{1}) t_{1}}} \\ (7) & + \frac{U_{k n}}{ω_{k n} + ω_{2}} {1 - e^{i (ω_{k n} + ω_{2}) t_{1}}} + \frac{U_{k n}}{ω_{k n} - ω_{2}} {1 - e^{i (ω_{k n} - ω_{2}) t_{1}}}] \end{aligned}

これを式 (6) に代入し,

t_{1}

について積分すると次となる：

\begin{aligned} c_{m}^{(2)} (T) & = \frac{1}{ℏ^{2}} \sum_{k} [\frac{V_{m k} V_{k n}}{(ω_{k n} + ω_{1})} \frac{e^{i (ω_{m n} + 2 ω_{1}) T} - 1}{(ω_{m n} + 2 ω_{1})} - \frac{V_{m k} V_{k n}}{(ω_{k n} + ω_{1})} \frac{e^{i (ω_{m k} + ω_{1}) T} - 1}{(ω_{m k} + ω_{1})} \\ + \frac{V_{m k} V_{k n}}{(ω_{k n} + ω_{1})} \frac{e^{i ω_{m n} T} - 1}{ω_{m n}} - \frac{V_{m k} V_{k n}}{(ω_{k n} + ω_{1})} \frac{e^{i (ω_{m k} - ω_{1}) T} - 1}{(ω_{m k} - ω_{1})} \\ + \frac{U_{m k} V_{k n}}{(ω_{k n} + ω_{1})} \frac{e^{i (ω_{m n} + ω_{1} + ω_{2}) T} - 1}{(ω_{m n} + ω_{1} + ω_{2})} - \frac{U_{m k} V_{k n}}{(ω_{k n} + ω_{1})} \frac{e^{i (ω_{m k} + ω_{2}) T} - 1}{(ω_{m k} + ω_{2})} \\ + \frac{U_{m k} V_{k n}}{(ω_{k n} + ω_{1})} \frac{e^{i (ω_{m n} + ω_{1} - ω_{2}) T} - 1}{(ω_{m n} + ω_{1} - ω_{2})} - \frac{U_{m k} V_{k n}}{(ω_{k n} + ω_{1})} \frac{e^{i (ω_{m k} - ω_{2}) T} - 1}{(ω_{m k} - ω_{2})} \\ + \frac{V_{m k} V_{k n}}{(ω_{k n} - ω_{1})} \frac{e^{i ω_{m n} T} - 1}{ω_{m n}} - \frac{V_{m k} V_{k n}}{(ω_{k n} - ω_{1})} \frac{e^{i (ω_{m k} + ω_{1}) T} - 1}{(ω_{m k} + ω_{1})} \\ + \frac{V_{m k} V_{k n}}{(ω_{k n} - ω_{1})} \frac{e^{i (ω_{m n} - 2 ω_{1}) T} - 1}{(ω_{m n} - 2 ω_{1})} - \frac{V_{m k} V_{k n}}{(ω_{k n} - ω_{1})} \frac{e^{i (ω_{m k} - ω_{1}) T} - 1}{(ω_{m k} - ω_{1})} \\ + \frac{U_{m k} V_{k n}}{(ω_{k n} - ω_{1})} \frac{e^{i (ω_{m m} - ω_{1} + ω_{2}) T} - 1}{(ω_{m n} - ω_{1} + ω_{2})} - \frac{U_{m k} V_{k n}}{(ω_{k n} - ω_{1})} \frac{e^{i (ω_{m k} + ω_{2}) T} - 1}{(ω_{m k} + ω_{2})} \\ + \frac{U_{m k} V_{k n}}{(ω_{k n} - ω_{1})} \frac{e^{i (ω_{m n} - ω_{1} - ω_{2}) T} - 1}{(ω_{m n} - ω_{1} - ω_{2})} - \frac{U_{m k} V_{k n}}{(ω_{k n} - ω_{1})} \frac{e^{i (ω_{m k} - ω_{2}) T} - 1}{(ω_{m k} - ω_{2})} \\ + \frac{V_{m k} U_{k n}}{(ω_{k n} + ω_{2})} \frac{e^{i (ω_{m n} + ω_{1} + ω_{2}) T} - 1}{(ω_{m n} + ω_{1} + ω_{2})} - \frac{V_{m k} U_{k n}}{(ω_{k n} + ω_{2})} \frac{e^{i (ω_{m k} + ω_{1}) T} - 1}{(ω_{m k} + ω_{1})} \\ + \frac{V_{m k} U_{k n}}{(ω_{k n} + ω_{2})} \frac{e^{i (ω_{m n} - ω_{1} + ω_{2}) T} - 1}{(ω_{m n} - ω_{1} + ω_{2})} - \frac{V_{m k} U_{k n}}{(ω_{k n} + ω_{2})} \frac{e^{i (ω_{m k} - ω_{1}) T} - 1}{(ω_{m k} - ω_{1})} \\ + \frac{U_{m k} U_{k n}}{(ω_{k n} + ω_{2})} \frac{e^{i (ω_{m n} + 2 ω_{2}) T} - 1}{(ω_{m n} + 2 ω_{2})} - \frac{U_{m k} U_{k n}}{(ω_{k n} + ω_{2})} \frac{e^{i (ω_{m k} + ω_{2}) T} - 1}{(ω_{m k} + ω_{2})} \\ + \frac{U_{m k} U_{k n}}{(ω_{k n} + ω_{2})} \frac{e^{i ω_{m n} T} - 1}{ω_{m n}} - \frac{U_{m k} U_{k n}}{(ω_{k n} + ω_{2})} \frac{e^{i (ω_{m k} - ω_{2}) T} - 1}{(ω_{m k} - ω_{2})} \\ + \frac{V_{m k} U_{k n}}{(ω_{k n} - ω_{2})} \frac{e^{i (ω_{m n} + ω_{1} - ω_{2}) T} - 1}{(ω_{m n} + ω_{1} - ω_{2})} - \frac{V_{m k} U_{k n}}{(ω_{k n} - ω_{2})} \frac{e^{i (ω_{m k} + ω_{1}) T} - 1}{(ω_{m k} + ω_{1})} \\ + \frac{V_{m k} U_{k n}}{(ω_{k n} - ω_{2})} \frac{e^{i (ω_{m n} - ω_{1} - ω_{2}) T} - 1}{(ω_{m n} - ω_{1} - ω_{2})} - \frac{V_{m k} U_{k n}}{(ω_{k n} - ω_{2})} \frac{e^{i (ω_{m k} - ω_{1}) T} - 1}{(ω_{m k} - ω_{1})} \\ + \frac{U_{m k} U_{k n}}{(ω_{k n} - ω_{2})} \frac{e^{i ω_{m n} T} - 1}{ω_{m n}} - \frac{U_{m k} U_{k n}}{(ω_{k n} - ω_{2})} \frac{e^{i (ω_{m k} + ω_{2}) T} - 1}{(ω_{m k} + ω_{2})} \\ (8) & + \frac{U_{m k} U_{k n}}{(ω_{k n} - ω_{2})} \frac{e^{i (ω_{m n} - 2 ω_{2}) T} - 1}{(ω_{m n} - 2 ω_{2})} - \frac{U_{m k} U_{k n}}{(ω_{k n} - ω_{2})} \frac{e^{i (ω_{m k} - ω_{2}) T} - 1}{(ω_{m k} - ω_{2})} \end{aligned}

式 (6-98) から式 (6-99) を求めたときの議論と, 問題6-24の式 (6-94) を求めたときの議論とから, この場合の遷移率は次のようになると類推される?． (ただし, これも類推したに過ぎないので, 後で検討を要するものである)：

\begin{aligned} w_{n \to m} & = \frac{P (n \to m)}{d t} = \frac{2 π}{ℏ} | M_{n \to m}^{(1)} |^{2} [δ (E_{m} - E_{n}) + δ (E_{m} - E_{n} + 2 ℏ ω_{1})] \\ + \frac{2 π}{ℏ} | M_{n \to m}^{(2)} |^{2} [δ (E_{m} - E_{n} + ℏ ω_{1} + ℏ ω_{2}) + δ (E_{m} - E_{n} + ℏ ω_{1} - ℏ ω_{2})] \\ + \frac{2 π}{ℏ} | M_{n \to m}^{(3)} |^{2} [δ (E_{m} - E_{n}) + δ (E_{m} - E_{n} - 2 ℏ ω_{1})] \\ + \frac{2 π}{ℏ} | M_{n \to m}^{(4)} |^{2} [δ (E_{m} - E_{n} - ℏ ω_{1} + ℏ ω_{2}) + δ (E_{m} - E_{n} - ℏ ω_{1} - ℏ ω_{2})] \\ + \frac{2 π}{ℏ} | M_{n \to m}^{(5)} |^{2} [δ (E_{m} - E_{n} + ℏ ω_{1} + ℏ ω_{2}) + δ (E_{m} - E_{n} - ℏ ω_{1} + ℏ ω_{2})] \\ + \frac{2 π}{ℏ} | M_{n \to m}^{(6)} |^{2} [δ (E_{m} - E_{n}) + δ (E_{m} - E_{n} + 2 ℏ ω_{2})] \\ + \frac{2 π}{ℏ} | M_{n \to m}^{(7)} |^{2} [δ (E_{m} - E_{n} + ℏ ω_{1} - ℏ ω_{2}) + δ (E_{m} - E_{n} - ℏ ω_{1} - ℏ ω_{2})] \\ (9) & + \frac{2 π}{ℏ} | M_{n \to m}^{(8)} |^{2} [δ (E_{m} - E_{n}) + δ (E_{m} - E_{n} - 2 ℏ ω_{2})] \end{aligned}

ただし, 各項に含まれる「遷移の行列要素」は次である：

\begin{aligned} M_{n \to m}^{(1)} & = \sum_{k} \frac{V_{m k} V_{k n}}{E_{k} - E_{n} + ℏ ω_{1} - i ε}, M_{n \to m}^{(2)} = \sum_{k} \frac{U_{m k} V_{k n}}{E_{k} - E_{n} + ℏ ω_{1} - i ε}, \\ M_{n \to m}^{(3)} & = \sum_{k} \frac{V_{m k} V_{k n}}{E_{k} - E_{n} - ℏ ω_{1} - i ε}, M_{n \to m}^{(4)} = \sum_{k} \frac{U_{m k} V_{k n}}{E_{k} - E_{n} - ℏ ω_{1} - i ε}, \\ M_{n \to m}^{(5)} & = \sum_{k} \frac{V_{m k} U_{k n}}{E_{k} - E_{n} + ℏ ω_{2} - i ε}, M_{n \to m}^{(6)} = \sum_{k} \frac{U_{m k} U_{k n}}{E_{k} - E_{n} + ℏ ω_{2} - i ε}, \\ (10) & M_{n \to m}^{(7)} & = \sum_{k} \frac{V_{m k} U_{k n}}{E_{k} - E_{n} - ℏ ω_{2} - i ε}, M_{n \to m}^{(8)} = \sum_{k} \frac{U_{m k} U_{k n}}{E_{k} - E_{n} - ℏ ω_{2} - i ε} \end{aligned}

月	火	水	木	金	土	日
	1	2	3	4	5	6
7	8	9	10	11	12	13
14	15	16	17	18	19	20
21	22	23	24	25	26	27
28	29	30