今回は昇降演算子の交換関係について解説したいと思います。
量子力学では欠かせない存在の昇降演算子ですが、なんとなくよくわからない存在のように感じる方も多いかと思います。
さらに交換関係も昇降演算子を扱っていく過程で無視できない大切な関係です。
しっかりと理解して行きましょう。
昇降演算子
まず初めに昇降演算子とは何かをサラッと説明しておきます。
$$\hat{y} = \sqrt{\frac{m\omega}{\hbar}}x\ ,\ \hat{p} = -i\frac{\partial}{\partial y}$$
としたとき
$$\hat{a} = \frac{1}{\sqrt{2}}\left(\hat{y} + i\hat{p}\right),\;\;\; \hat{a}^\dagger=\frac{1}{\sqrt{2}}\left(\hat{y} – i\hat{p}\right)$$
この\(\hat{a},\;\hat{a}^\dagger\)を合わせて昇降演算子と呼びます。
では今度は実際に問題を解いてみましょう。
昇降演算子の交換法則
\([\hat{a},\hat{a}^\dagger] = 1\)を示せ
まず、[]の説明からすると、これはポアソン括弧といって$$[A,B] = AB -BA$$を表します。
なので
$$[\hat{a},\hat{a}^\dagger] = \hat{a}\hat{a}^\dagger – \hat{a}^\dagger\hat{a}$$
となります。
これを計算していくと、
$$\begin{eqnarray} \hat{a}\hat{a}^\dagger – \hat{a}^\dagger\hat{a} &=& \frac{1}{\sqrt{2}}\left(\hat{y} + i\hat{p}\right)\frac{1}{\sqrt{2}}\left(\hat{y} – i\hat{p}\right) – \frac{1}{\sqrt{2}}\left(\hat{y} – i\hat{p}\right)\frac{1}{\sqrt{2}}\left(\hat{y} + i\hat{p}\right) \\ &=& \frac{1}{2}\left(\left(\hat{y}\right)^2 -i\hat{y}\hat{p} + i\hat{p}\hat{y} + \left(\hat{p}\right)^2 – \left(\hat{y}\right)^2 -i\hat{y}\hat{p} +i\hat{p}\hat{y} -\left(\hat{p}\right)^2\right) \\ &=& \left(-i\hat{y}\hat{p} +i\hat{p}\hat{y}\right) \\ &=& -i[\hat{y},\hat{p}]\tag{1} \end{eqnarray}$$
さて、\([\hat{y},\;\hat{p}]\)はいくつでしょうか?計算してみましょう。
ここで\([\hat{y},\hat{p}]\phi = \underbrace{\hat{y}\hat{p}\phi}_{①} – \underbrace{\hat{p}\hat{y}\phi}_{②}\)を考えてみましょう。
$$ \begin{eqnarray} ① = \hat{y}\hat{p}\phi &=& \sqrt{\frac{m\omega}{\hbar}}x\left(-i\frac{\partial \phi}{\partial y}\right)\\ &=& -i\sqrt{\frac{m\omega}{\hbar}}x\frac{\partial \phi}{\partial y} \end{eqnarray} $$ $$ \begin{eqnarray} ② = \hat{p}\hat{y}\phi &=& \left(-i\frac{\partial}{\partial y}\right)\sqrt{\frac{m\omega}{\hbar}}x\phi \\ &=& -i\sqrt{\frac{m\omega}{\hbar}}\frac{\partial x}{\partial y}\phi – i\sqrt{\frac{m\omega}{\hbar}}x\frac{\partial \phi}{\partial y} \end{eqnarray} $$
なので、
$$ \begin{eqnarray} [\hat{y},\; \hat{p}]\phi &=& -i\sqrt{\frac{m\omega}{\hbar}}\frac{\partial x}{\partial y}\phi – i\sqrt{\frac{m\omega}{\hbar}}x\frac{\partial \phi}{\partial y} + i\sqrt{\frac{m\omega}{\hbar}}x\frac{\partial \phi}{\partial y} \\ &=& -i\sqrt{\frac{m\omega}{\hbar}}\frac{\partial x}{\partial y}\phi \\ &=& -i\sqrt{\frac{m\omega}{\hbar}}\frac{\partial}{\partial y}\left(\sqrt{\frac{\hbar}{m\omega}y}\right)\phi \\ &=& i\phi \end{eqnarray} $$
よって$$[\hat{y},\; \hat{p}] = i$$です。
これを(1)に代入すると、$$[\hat{a},\hat{a}^\dagger] = 1$$となります。
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