\(\)
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{equation}
M_{n\to m}=\sum_{k} \frac{V_{m k}U_{k n}+U_{m k}V_{k n}}{E_{k}-E_{n}}
\tag{6-111}
\end{equation}
M_{n\to m}=\sum_{k} \frac{V_{m k}U_{k n}+U_{m k}V_{k n}}{E_{k}-E_{n}}
\tag{6-111}
\end{equation}
Next, suppose both potentials are periodic in time but have different frequencies, \(\omega_{1}\) and \(\omega_{2}\). 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 \(\lambda_{mn}^{(2)}\) に相当する 2次の係数 \(c_m^{(2)}(T)\) は次である：
\begin{align}
&c_m^{(2)}(T)=\left(\frac{-i}{\hbar}\right)^{2}\sum_k \int_0^{T}dt_1 \int_0^{t_1}dt_2\,
e^{i\omega_{mk}t_1}\,\bigl[V_{mk}(t_1)+U_{mk}(t_1)\bigr]\,e^{i\omega_{kn}t_2}\,\bigl[V_{kn}(t_2)+U_{kn}(t_2)
\bigr]\\
&\quad =\left(\frac{-i}{\hbar}\right)^{2}\sum_k \int_0^{T}dt_1\,e^{i\omega_{mk}t_1} \int_0^{t_1}dt_2\,
e^{i\omega_{kn}t_2}\,\bigl[V_{mk}(t_1)+U_{mk}(t_1)\bigr]\,\bigl[V_{kn}(t_2)+U_{kn}(t_2)\bigr]
\tag{1}
\end{align}
&c_m^{(2)}(T)=\left(\frac{-i}{\hbar}\right)^{2}\sum_k \int_0^{T}dt_1 \int_0^{t_1}dt_2\,
e^{i\omega_{mk}t_1}\,\bigl[V_{mk}(t_1)+U_{mk}(t_1)\bigr]\,e^{i\omega_{kn}t_2}\,\bigl[V_{kn}(t_2)+U_{kn}(t_2)
\bigr]\\
&\quad =\left(\frac{-i}{\hbar}\right)^{2}\sum_k \int_0^{T}dt_1\,e^{i\omega_{mk}t_1} \int_0^{t_1}dt_2\,
e^{i\omega_{kn}t_2}\,\bigl[V_{mk}(t_1)+U_{mk}(t_1)\bigr]\,\bigl[V_{kn}(t_2)+U_{kn}(t_2)\bigr]
\tag{1}
\end{align}
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{align}
&\bigl[V_{mk}(t_1)+U_{mk}(t_1)\bigr]\,\bigl[V_{kn}(t_2)+U_{kn}(t_2)\bigr]\notag\\
&\quad =V_{mk}(t_1)V_{kn}(t_2)+V_{mk}(t_1)U_{kn}(t_2)+U_{mk}(t_1)V_{kn}(t_2)+U_{mk}(t_1)U_{kn}(t_2)\\
&\quad =V_{mk}(t_1)U_{kn}(t_2)+U_{mk}(t_1)V_{kn}(t_2)=V_{mk}U_{kn}+U_{mk}V_{kn}
\tag{2}
\end{align}
&\bigl[V_{mk}(t_1)+U_{mk}(t_1)\bigr]\,\bigl[V_{kn}(t_2)+U_{kn}(t_2)\bigr]\notag\\
&\quad =V_{mk}(t_1)V_{kn}(t_2)+V_{mk}(t_1)U_{kn}(t_2)+U_{mk}(t_1)V_{kn}(t_2)+U_{mk}(t_1)U_{kn}(t_2)\\
&\quad =V_{mk}(t_1)U_{kn}(t_2)+U_{mk}(t_1)V_{kn}(t_2)=V_{mk}U_{kn}+U_{mk}V_{kn}
\tag{2}
\end{align}
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{align}
c_m^{(2)}(T)&=\left(\frac{-i}{\hbar}\right)^{2}\sum_k (V_{mk}U_{kn}+U_{mk}V_{kn})\int_0^{T}dt_1
\,e^{i\omega_{mk}t_1} \int_0^{t_1}dt_2\,e^{i\omega_{kn}t_2}\\
&=\sum_k\frac{V_{mk}U_{kn}+U_{mk}V_{kn}}{E_k-E_n}\left(\frac{e^{i\omega_{mn}T}-1}{E_m-E_n}
-\frac{e^{i\omega_{mk}T}-1}{E_m-E_k}\right)
\tag{3}
\end{align}
c_m^{(2)}(T)&=\left(\frac{-i}{\hbar}\right)^{2}\sum_k (V_{mk}U_{kn}+U_{mk}V_{kn})\int_0^{T}dt_1
\,e^{i\omega_{mk}t_1} \int_0^{t_1}dt_2\,e^{i\omega_{kn}t_2}\\
&=\sum_k\frac{V_{mk}U_{kn}+U_{mk}V_{kn}}{E_k-E_n}\left(\frac{e^{i\omega_{mn}T}-1}{E_m-E_n}
-\frac{e^{i\omega_{mk}T}-1}{E_m-E_k}\right)
\tag{3}
\end{align}
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{equation}
M_{n\to m}=\sum_k \frac{V_{mk}U_{kn}+U_{mk}V_{kn}}{E_k-E_n}
\tag{4}
\end{equation}
M_{n\to m}=\sum_k \frac{V_{mk}U_{kn}+U_{mk}V_{kn}}{E_k-E_n}
\tag{4}
\end{equation}
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{align}
V(t)+U(t)&=V(x)(e^{i\omega_1t}+e^{-i\omega_1t})+U(x)(e^{i\omega_2t}+e^{-i\omega_2t})\\
&=2V(x)\cos\omega_1t +2U(x)\cos\omega_2t
\tag{5}
\end{align}
V(t)+U(t)&=V(x)(e^{i\omega_1t}+e^{-i\omega_1t})+U(x)(e^{i\omega_2t}+e^{-i\omega_2t})\\
&=2V(x)\cos\omega_1t +2U(x)\cos\omega_2t
\tag{5}
\end{align}
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{align}
&c_m^{(2)}(T)\\
&\quad =\left(\frac{-i}{\hbar}\right)^{2}\sum_k \int_0^{T}dt_1\int_0^{t_1}dt_2\,
e^{i\omega_{mk}t_1}\,\{V_{mk}(t_1)+U_{mk}(t_1)\}\,e^{i\omega_{kn}t_2}\,\{V_{kn}(t_2)+U_{kn}(t_2)\}\\
&\quad =\left(\frac{-i}{\hbar}\right)^{2}\sum_k \int_0^{T}dt_1\,e^{i\omega_{mk}t_1}\,
\{V_{mk}(t_1)+U_{mk}(t_1)\}\int_0^{t_1}dt_2\,e^{i\omega_{kn}t_2}\,\{V_{kn}(t_2)+U_{kn}(t_2)\}
\tag{6}
\end{align}
&c_m^{(2)}(T)\\
&\quad =\left(\frac{-i}{\hbar}\right)^{2}\sum_k \int_0^{T}dt_1\int_0^{t_1}dt_2\,
e^{i\omega_{mk}t_1}\,\{V_{mk}(t_1)+U_{mk}(t_1)\}\,e^{i\omega_{kn}t_2}\,\{V_{kn}(t_2)+U_{kn}(t_2)\}\\
&\quad =\left(\frac{-i}{\hbar}\right)^{2}\sum_k \int_0^{T}dt_1\,e^{i\omega_{mk}t_1}\,
\{V_{mk}(t_1)+U_{mk}(t_1)\}\int_0^{t_1}dt_2\,e^{i\omega_{kn}t_2}\,\{V_{kn}(t_2)+U_{kn}(t_2)\}
\tag{6}
\end{align}
まず, \(t_2\) についての積分は次となる：
\begin{align}
\left(\frac{-i}{\hbar}\right)&\int_0^{t_1}dt_2\,e^{i\omega_{kn}t_2}\,\{V_{kn}(t_2)+U_{kn}(t_2)\}\\
&=\left(\frac{-i}{\hbar}\right)\int_0^{t_1}dt_2\,e^{i\omega_{kn}t_2}\,\left\{V_{kn}(e^{i\omega_1t_2}
+e^{-i\omega_1t_2})+U_{kn}(e^{i\omega_2t_2}+e^{-i\omega_2t_2})\right\}\\
&=\frac{1}{\hbar}\left[\frac{V_{kn}}{\omega_{kn}+\omega_1}\left\{1-e^{i(\omega_{kn}+\omega_1)t_1}\right\}
+\frac{V_{kn}}{\omega_{kn}-\omega_1}\left\{1-e^{i(\omega_{kn}-\omega_1)t_1}\right\}\right.\\
&\qquad\quad\left. +\frac{U_{kn}}{\omega_{kn}+\omega_2}\left\{1-e^{i(\omega_{kn}+\omega_2)t_1}\right\}
+\frac{U_{kn}}{\omega_{kn}-\omega_2}\left\{1-e^{i(\omega_{kn}-\omega_2)t_1}\right\}\right]
\tag{7}
\end{align}
\left(\frac{-i}{\hbar}\right)&\int_0^{t_1}dt_2\,e^{i\omega_{kn}t_2}\,\{V_{kn}(t_2)+U_{kn}(t_2)\}\\
&=\left(\frac{-i}{\hbar}\right)\int_0^{t_1}dt_2\,e^{i\omega_{kn}t_2}\,\left\{V_{kn}(e^{i\omega_1t_2}
+e^{-i\omega_1t_2})+U_{kn}(e^{i\omega_2t_2}+e^{-i\omega_2t_2})\right\}\\
&=\frac{1}{\hbar}\left[\frac{V_{kn}}{\omega_{kn}+\omega_1}\left\{1-e^{i(\omega_{kn}+\omega_1)t_1}\right\}
+\frac{V_{kn}}{\omega_{kn}-\omega_1}\left\{1-e^{i(\omega_{kn}-\omega_1)t_1}\right\}\right.\\
&\qquad\quad\left. +\frac{U_{kn}}{\omega_{kn}+\omega_2}\left\{1-e^{i(\omega_{kn}+\omega_2)t_1}\right\}
+\frac{U_{kn}}{\omega_{kn}-\omega_2}\left\{1-e^{i(\omega_{kn}-\omega_2)t_1}\right\}\right]
\tag{7}
\end{align}
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{align}
c_m^{(2)}(T)&=\frac{1}{\hbar^{2}}\sum_k\left[\frac{V_{mk}V_{kn}}{(\omega_{kn}+\omega_1)}
\frac{e^{i(\omega_{mn}+2\omega_1)T}-1}{(\omega_{mn}+2\omega_1)}
-\frac{V_{mk}V_{kn}}{(\omega_{kn}+\omega_1)}\frac{e^{i(\omega_{mk}+\omega_1)T}-1}{(\omega_{mk}+\omega_1)}\right.\\
&\quad+\frac{V_{mk}V_{kn}}{(\omega_{kn}+\omega_1)}\frac{e^{i\omega_{mn}T}-1}{\omega_{mn}}
-\frac{V_{mk}V_{kn}}{(\omega_{kn}+\omega_1)}\frac{e^{i(\omega_{mk}-\omega_1)T}-1}{(\omega_{mk}-\omega_1)}\\
&\quad+\frac{U_{mk}V_{kn}}{(\omega_{kn}+\omega_1)}\frac{e^{i(\omega_{mn}+\omega_1+\omega_2)T}-1}{(\omega_{mn}+\omega_1+\omega_2)}
-\frac{U_{mk}V_{kn}}{(\omega_{kn}+\omega_1)}\frac{e^{i(\omega_{mk}+\omega_2)T}-1}{(\omega_{mk}+\omega_2)}\\
&\quad+\frac{U_{mk}V_{kn}}{(\omega_{kn}+\omega_1)}\frac{e^{i(\omega_{mn}+\omega_1-\omega_2)T}-1}
{(\omega_{mn}+\omega_1-\omega_2)}-\frac{U_{mk}V_{kn}}{(\omega_{kn}+\omega_1)}
\frac{e^{i(\omega_{mk}-\omega_2)T}-1}{(\omega_{mk}-\omega_2)}\\
&\quad+\frac{V_{mk}V_{kn}}{(\omega_{kn}-\omega_1)}\frac{e^{i\omega_{mn}T}-1}{\omega_{mn}}
-\frac{V_{mk}V_{kn}}{(\omega_{kn}-\omega_1)}\frac{e^{i(\omega_{mk}+\omega_1)T}-1}{(\omega_{mk}+\omega_1)}\\
&\quad+\frac{V_{mk}V_{kn}}{(\omega_{kn}-\omega_1)}\frac{e^{i(\omega_{mn}-2\omega_1)T}-1}{(\omega_{mn}-2\omega_1)}
-\frac{V_{mk}V_{kn}}{(\omega_{kn}-\omega_1)}\frac{e^{i(\omega_{mk}-\omega_1)T}-1}{(\omega_{mk}-\omega_1)}\\
&\quad+\frac{U_{mk}V_{kn}}{(\omega_{kn}-\omega_1)}\frac{e^{i(\omega_{mm}-\omega_1+\omega_2)T}-1}
{(\omega_{mn}-\omega_1+\omega_2)}-\frac{U_{mk}V_{kn}}{(\omega_{kn}-\omega_1)}
\frac{e^{i(\omega_{mk}+\omega_2)T}-1}{(\omega_{mk}+\omega_2)}\\
&\quad+\frac{U_{mk}V_{kn}}{(\omega_{kn}-\omega_1)}\frac{e^{i(\omega_{mn}-\omega_1-\omega_2)T}-1}
{(\omega_{mn}-\omega_1-\omega_2)}-\frac{U_{mk}V_{kn}}{(\omega_{kn}-\omega_1)}
\frac{e^{i(\omega_{mk}-\omega_2)T}-1}{(\omega_{mk}-\omega_2)}\\
&\quad+\frac{V_{mk}U_{kn}}{(\omega_{kn}+\omega_2)}\frac{e^{i(\omega_{mn}+\omega_1+\omega_2)T}-1}
{(\omega_{mn}+\omega_1+\omega_2)}-\frac{V_{mk}U_{kn}}{(\omega_{kn}+\omega_2)}
\frac{e^{i(\omega_{mk}+\omega_1)T}-1}{(\omega_{mk}+\omega_1)}\\
&\quad+\frac{V_{mk}U_{kn}}{(\omega_{kn}+\omega_2)}\frac{e^{i(\omega_{mn}-\omega_1+\omega_2)T}-1}
{(\omega_{mn}-\omega_1+\omega_2)}-\frac{V_{mk}U_{kn}}{(\omega_{kn}+\omega_2)}
\frac{e^{i(\omega_{mk}-\omega_1)T}-1}{(\omega_{mk}-\omega_1)}\\
&\quad+\frac{U_{mk}U_{kn}}{(\omega_{kn}+\omega_2)}\frac{e^{i(\omega_{mn}+2\omega_2)T}-1}
{(\omega_{mn}+2\omega_2)}-\frac{U_{mk}U_{kn}}{(\omega_{kn}+\omega_2)}
\frac{e^{i(\omega_{mk}+\omega_2)T}-1}{(\omega_{mk}+\omega_2)}\\
&\quad+\frac{U_{mk}U_{kn}}{(\omega_{kn}+\omega_2)}\frac{e^{i\omega_{mn}T}-1}{\omega_{mn}}
-\frac{U_{mk}U_{kn}}{(\omega_{kn}+\omega_2)}\frac{e^{i(\omega_{mk}-\omega_2)T}-1}{(\omega_{mk}-\omega_2)}\\
&\quad+\frac{V_{mk}U_{kn}}{(\omega_{kn}-\omega_2)}\frac{e^{i(\omega_{mn}+\omega_1-\omega_2)T}-1}
{(\omega_{mn}+\omega_1-\omega_2)}-\frac{V_{mk}U_{kn}}{(\omega_{kn}-\omega_2)}
\frac{e^{i(\omega_{mk}+\omega_1)T}-1}{(\omega_{mk}+\omega_1)}\\
&\quad+\frac{V_{mk}U_{kn}}{(\omega_{kn}-\omega_2)}\frac{e^{i(\omega_{mn}-\omega_1-\omega_2)T}-1}
{(\omega_{mn}-\omega_1-\omega_2)}-\frac{V_{mk}U_{kn}}{(\omega_{kn}-\omega_2)}
\frac{e^{i(\omega_{mk}-\omega_1)T}-1}{(\omega_{mk}-\omega_1)}\\
&\quad+\frac{U_{mk}U_{kn}}{(\omega_{kn}-\omega_2)}\frac{e^{i\omega_{mn}T}-1}{\omega_{mn}}
-\frac{U_{mk}U_{kn}}{(\omega_{kn}-\omega_2)}\frac{e^{i(\omega_{mk}+\omega_2)T}-1}{(\omega_{mk}+\omega_2)}\\
&\quad+\frac{U_{mk}U_{kn}}{(\omega_{kn}-\omega_2)}\frac{e^{i(\omega_{mn}-2\omega_2)T}-1}
{(\omega_{mn}-2\omega_2)}-\frac{U_{mk}U_{kn}}{(\omega_{kn}-\omega_2)}
\frac{e^{i(\omega_{mk}-\omega_2)T}-1}{(\omega_{mk}-\omega_2)}
\tag{8}
\end{align}
c_m^{(2)}(T)&=\frac{1}{\hbar^{2}}\sum_k\left[\frac{V_{mk}V_{kn}}{(\omega_{kn}+\omega_1)}
\frac{e^{i(\omega_{mn}+2\omega_1)T}-1}{(\omega_{mn}+2\omega_1)}
-\frac{V_{mk}V_{kn}}{(\omega_{kn}+\omega_1)}\frac{e^{i(\omega_{mk}+\omega_1)T}-1}{(\omega_{mk}+\omega_1)}\right.\\
&\quad+\frac{V_{mk}V_{kn}}{(\omega_{kn}+\omega_1)}\frac{e^{i\omega_{mn}T}-1}{\omega_{mn}}
-\frac{V_{mk}V_{kn}}{(\omega_{kn}+\omega_1)}\frac{e^{i(\omega_{mk}-\omega_1)T}-1}{(\omega_{mk}-\omega_1)}\\
&\quad+\frac{U_{mk}V_{kn}}{(\omega_{kn}+\omega_1)}\frac{e^{i(\omega_{mn}+\omega_1+\omega_2)T}-1}{(\omega_{mn}+\omega_1+\omega_2)}
-\frac{U_{mk}V_{kn}}{(\omega_{kn}+\omega_1)}\frac{e^{i(\omega_{mk}+\omega_2)T}-1}{(\omega_{mk}+\omega_2)}\\
&\quad+\frac{U_{mk}V_{kn}}{(\omega_{kn}+\omega_1)}\frac{e^{i(\omega_{mn}+\omega_1-\omega_2)T}-1}
{(\omega_{mn}+\omega_1-\omega_2)}-\frac{U_{mk}V_{kn}}{(\omega_{kn}+\omega_1)}
\frac{e^{i(\omega_{mk}-\omega_2)T}-1}{(\omega_{mk}-\omega_2)}\\
&\quad+\frac{V_{mk}V_{kn}}{(\omega_{kn}-\omega_1)}\frac{e^{i\omega_{mn}T}-1}{\omega_{mn}}
-\frac{V_{mk}V_{kn}}{(\omega_{kn}-\omega_1)}\frac{e^{i(\omega_{mk}+\omega_1)T}-1}{(\omega_{mk}+\omega_1)}\\
&\quad+\frac{V_{mk}V_{kn}}{(\omega_{kn}-\omega_1)}\frac{e^{i(\omega_{mn}-2\omega_1)T}-1}{(\omega_{mn}-2\omega_1)}
-\frac{V_{mk}V_{kn}}{(\omega_{kn}-\omega_1)}\frac{e^{i(\omega_{mk}-\omega_1)T}-1}{(\omega_{mk}-\omega_1)}\\
&\quad+\frac{U_{mk}V_{kn}}{(\omega_{kn}-\omega_1)}\frac{e^{i(\omega_{mm}-\omega_1+\omega_2)T}-1}
{(\omega_{mn}-\omega_1+\omega_2)}-\frac{U_{mk}V_{kn}}{(\omega_{kn}-\omega_1)}
\frac{e^{i(\omega_{mk}+\omega_2)T}-1}{(\omega_{mk}+\omega_2)}\\
&\quad+\frac{U_{mk}V_{kn}}{(\omega_{kn}-\omega_1)}\frac{e^{i(\omega_{mn}-\omega_1-\omega_2)T}-1}
{(\omega_{mn}-\omega_1-\omega_2)}-\frac{U_{mk}V_{kn}}{(\omega_{kn}-\omega_1)}
\frac{e^{i(\omega_{mk}-\omega_2)T}-1}{(\omega_{mk}-\omega_2)}\\
&\quad+\frac{V_{mk}U_{kn}}{(\omega_{kn}+\omega_2)}\frac{e^{i(\omega_{mn}+\omega_1+\omega_2)T}-1}
{(\omega_{mn}+\omega_1+\omega_2)}-\frac{V_{mk}U_{kn}}{(\omega_{kn}+\omega_2)}
\frac{e^{i(\omega_{mk}+\omega_1)T}-1}{(\omega_{mk}+\omega_1)}\\
&\quad+\frac{V_{mk}U_{kn}}{(\omega_{kn}+\omega_2)}\frac{e^{i(\omega_{mn}-\omega_1+\omega_2)T}-1}
{(\omega_{mn}-\omega_1+\omega_2)}-\frac{V_{mk}U_{kn}}{(\omega_{kn}+\omega_2)}
\frac{e^{i(\omega_{mk}-\omega_1)T}-1}{(\omega_{mk}-\omega_1)}\\
&\quad+\frac{U_{mk}U_{kn}}{(\omega_{kn}+\omega_2)}\frac{e^{i(\omega_{mn}+2\omega_2)T}-1}
{(\omega_{mn}+2\omega_2)}-\frac{U_{mk}U_{kn}}{(\omega_{kn}+\omega_2)}
\frac{e^{i(\omega_{mk}+\omega_2)T}-1}{(\omega_{mk}+\omega_2)}\\
&\quad+\frac{U_{mk}U_{kn}}{(\omega_{kn}+\omega_2)}\frac{e^{i\omega_{mn}T}-1}{\omega_{mn}}
-\frac{U_{mk}U_{kn}}{(\omega_{kn}+\omega_2)}\frac{e^{i(\omega_{mk}-\omega_2)T}-1}{(\omega_{mk}-\omega_2)}\\
&\quad+\frac{V_{mk}U_{kn}}{(\omega_{kn}-\omega_2)}\frac{e^{i(\omega_{mn}+\omega_1-\omega_2)T}-1}
{(\omega_{mn}+\omega_1-\omega_2)}-\frac{V_{mk}U_{kn}}{(\omega_{kn}-\omega_2)}
\frac{e^{i(\omega_{mk}+\omega_1)T}-1}{(\omega_{mk}+\omega_1)}\\
&\quad+\frac{V_{mk}U_{kn}}{(\omega_{kn}-\omega_2)}\frac{e^{i(\omega_{mn}-\omega_1-\omega_2)T}-1}
{(\omega_{mn}-\omega_1-\omega_2)}-\frac{V_{mk}U_{kn}}{(\omega_{kn}-\omega_2)}
\frac{e^{i(\omega_{mk}-\omega_1)T}-1}{(\omega_{mk}-\omega_1)}\\
&\quad+\frac{U_{mk}U_{kn}}{(\omega_{kn}-\omega_2)}\frac{e^{i\omega_{mn}T}-1}{\omega_{mn}}
-\frac{U_{mk}U_{kn}}{(\omega_{kn}-\omega_2)}\frac{e^{i(\omega_{mk}+\omega_2)T}-1}{(\omega_{mk}+\omega_2)}\\
&\quad+\frac{U_{mk}U_{kn}}{(\omega_{kn}-\omega_2)}\frac{e^{i(\omega_{mn}-2\omega_2)T}-1}
{(\omega_{mn}-2\omega_2)}-\frac{U_{mk}U_{kn}}{(\omega_{kn}-\omega_2)}
\frac{e^{i(\omega_{mk}-\omega_2)T}-1}{(\omega_{mk}-\omega_2)}
\tag{8}
\end{align}
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{align}
w_{n\to m}&=\frac{P(n\to m)}{dt}=\frac{2\pi}{\hbar}\bigl|M_{n\to m}^{(1)}\bigr|^{2}\Bigl[\delta(E_m-E_n)
+\delta(E_m-E_n+2\hbar\omega_1)\Bigr]\\
&\quad+\frac{2\pi}{\hbar}\bigl|M_{n\to m}^{(2)}\bigr|^{2}\Bigl[\delta(E_m-E_n+\hbar\omega_1
+\hbar\omega_2)+\delta(E_m-E_n+\hbar\omega_1-\hbar\omega_2)\Bigr]\\
&\quad+\frac{2\pi}{\hbar}\bigl|M_{n\to m}^{(3)}\bigr|^{2}\Bigl[\delta(E_m-E_n)+\delta(E_m-E_n-2\hbar\omega_1)
\Bigr]\\
&\quad+\frac{2\pi}{\hbar}\bigl|M_{n\to m}^{(4)}\bigr|^{2}\Bigl[\delta(E_m-E_n-\hbar\omega_1
+\hbar\omega_2)+\delta(E_m-E_n-\hbar\omega_1-\hbar\omega_2)\Bigr]\\
&\quad+\frac{2\pi}{\hbar}\bigl|M_{n\to m}^{(5)}\bigr|^{2}\Bigl[\delta(E_m-E_n+\hbar\omega_1
+\hbar\omega_2)+\delta(E_m-E_n-\hbar\omega_1+\hbar\omega_2)\Bigr]\\
&\quad+\frac{2\pi}{\hbar}\bigl|M_{n\to m}^{(6)}\bigr|^{2}\Bigl[\delta(E_m-E_n)
+\delta(E_m-E_n+2\hbar\omega_2)\Bigr]\\
&\quad+\frac{2\pi}{\hbar}\bigl|M_{n\to m}^{(7)}\bigr|^{2}\Bigl[\delta(E_m-E_n+\hbar\omega_1
-\hbar\omega_2)+\delta(E_m-E_n-\hbar\omega_1-\hbar\omega_2)\Bigr]\\
&\quad+\frac{2\pi}{\hbar}\bigl|M_{n\to m}^{(8)}\bigr|^{2}\Bigl[\delta(E_m-E_n)
+\delta(E_m-E_n-2\hbar\omega_2)\Bigr]
\tag{9}
\end{align}
w_{n\to m}&=\frac{P(n\to m)}{dt}=\frac{2\pi}{\hbar}\bigl|M_{n\to m}^{(1)}\bigr|^{2}\Bigl[\delta(E_m-E_n)
+\delta(E_m-E_n+2\hbar\omega_1)\Bigr]\\
&\quad+\frac{2\pi}{\hbar}\bigl|M_{n\to m}^{(2)}\bigr|^{2}\Bigl[\delta(E_m-E_n+\hbar\omega_1
+\hbar\omega_2)+\delta(E_m-E_n+\hbar\omega_1-\hbar\omega_2)\Bigr]\\
&\quad+\frac{2\pi}{\hbar}\bigl|M_{n\to m}^{(3)}\bigr|^{2}\Bigl[\delta(E_m-E_n)+\delta(E_m-E_n-2\hbar\omega_1)
\Bigr]\\
&\quad+\frac{2\pi}{\hbar}\bigl|M_{n\to m}^{(4)}\bigr|^{2}\Bigl[\delta(E_m-E_n-\hbar\omega_1
+\hbar\omega_2)+\delta(E_m-E_n-\hbar\omega_1-\hbar\omega_2)\Bigr]\\
&\quad+\frac{2\pi}{\hbar}\bigl|M_{n\to m}^{(5)}\bigr|^{2}\Bigl[\delta(E_m-E_n+\hbar\omega_1
+\hbar\omega_2)+\delta(E_m-E_n-\hbar\omega_1+\hbar\omega_2)\Bigr]\\
&\quad+\frac{2\pi}{\hbar}\bigl|M_{n\to m}^{(6)}\bigr|^{2}\Bigl[\delta(E_m-E_n)
+\delta(E_m-E_n+2\hbar\omega_2)\Bigr]\\
&\quad+\frac{2\pi}{\hbar}\bigl|M_{n\to m}^{(7)}\bigr|^{2}\Bigl[\delta(E_m-E_n+\hbar\omega_1
-\hbar\omega_2)+\delta(E_m-E_n-\hbar\omega_1-\hbar\omega_2)\Bigr]\\
&\quad+\frac{2\pi}{\hbar}\bigl|M_{n\to m}^{(8)}\bigr|^{2}\Bigl[\delta(E_m-E_n)
+\delta(E_m-E_n-2\hbar\omega_2)\Bigr]
\tag{9}
\end{align}
ただし, 各項に含まれる「遷移の行列要素」は次である：
\begin{align}
M_{n\to m}^{(1)}&=\sum_k \frac{V_{mk}V_{kn}}{E_k-E_n+\hbar\omega_1-i\varepsilon},\qquad
M_{n\to m}^{(2)}=\sum_k \frac{U_{mk}V_{kn}}{E_k-E_n+\hbar\omega_1-i\varepsilon},\\
M_{n\to m}^{(3)}&=\sum_k \frac{V_{mk}V_{kn}}{E_k-E_n-\hbar\omega_1-i\varepsilon},\qquad
M_{n\to m}^{(4)}=\sum_k \frac{U_{mk}V_{kn}}{E_k-E_n-\hbar\omega_1-i\varepsilon},\\
M_{n\to m}^{(5)}&=\sum_k \frac{V_{mk}U_{kn}}{E_k-E_n+\hbar\omega_2-i\varepsilon},\qquad
M_{n\to m}^{(6)}=\sum_k \frac{U_{mk}U_{kn}}{E_k-E_n+\hbar\omega_2-i\varepsilon},\\
M_{n\to m}^{(7)}&=\sum_k \frac{V_{mk}U_{kn}}{E_k-E_n-\hbar\omega_2-i\varepsilon},\qquad
M_{n\to m}^{(8)}=\sum_k \frac{U_{mk}U_{kn}}{E_k-E_n-\hbar\omega_2-i\varepsilon}
\tag{10}
\end{align}
M_{n\to m}^{(1)}&=\sum_k \frac{V_{mk}V_{kn}}{E_k-E_n+\hbar\omega_1-i\varepsilon},\qquad
M_{n\to m}^{(2)}=\sum_k \frac{U_{mk}V_{kn}}{E_k-E_n+\hbar\omega_1-i\varepsilon},\\
M_{n\to m}^{(3)}&=\sum_k \frac{V_{mk}V_{kn}}{E_k-E_n-\hbar\omega_1-i\varepsilon},\qquad
M_{n\to m}^{(4)}=\sum_k \frac{U_{mk}V_{kn}}{E_k-E_n-\hbar\omega_1-i\varepsilon},\\
M_{n\to m}^{(5)}&=\sum_k \frac{V_{mk}U_{kn}}{E_k-E_n+\hbar\omega_2-i\varepsilon},\qquad
M_{n\to m}^{(6)}=\sum_k \frac{U_{mk}U_{kn}}{E_k-E_n+\hbar\omega_2-i\varepsilon},\\
M_{n\to m}^{(7)}&=\sum_k \frac{V_{mk}U_{kn}}{E_k-E_n-\hbar\omega_2-i\varepsilon},\qquad
M_{n\to m}^{(8)}=\sum_k \frac{U_{mk}U_{kn}}{E_k-E_n-\hbar\omega_2-i\varepsilon}
\tag{10}
\end{align}
