第5章　中心力場のSchrödinger方程式②

##5.2 極座標による3次元のSchrödinger方程式(つづき)

[(5.14)](#tag{5.14})が正しいSchrödinger方程式と同等であることを確かめる．
古典論では$\mathbf{L}=\mathbf{r}\times\mathbf{p}$なので，量子論でも

<a name ="tag{5.15}">

```math
\begin{align}
\hat{L}_x
&=
\hat{y}\hat{p}_z-\hat{z}\hat{p}_y
=
\frac{\hbar}{i}
\left(y\frac{\partial}{\partial z}-z\frac{\partial}{\partial y}\right)
\\
\hat{L}_y
&=
\hat{z}\hat{p}_x-\hat{x}\hat{p}_z
=
\frac{\hbar}{i}
\left(z\frac{\partial}{\partial x}-x\frac{\partial}{\partial z}\right)
\\
\hat{L}_z
&=
\hat{x}\hat{p}_y-\hat{y}\hat{p}_x
=
\frac{\hbar}{i}
\left(x\frac{\partial}{\partial y}-y\frac{\partial}{\partial x}\right)
\end{align}

\tag{5.15}
```

とする．[(5.15)](#tag{5.15})は[(5.8)](#tag{5.8})を使って極座標に変換できる．
[(5.8)](#tag{5.8})より，

<a name ="tag{5.15.1}">

```math
\frac{\partial x}{\partial r}=\sin\theta\cos\varphi, \quad
\frac{\partial x}{\partial \theta}=r\cos\theta\cos\varphi, \quad
\frac{\partial x}{\partial \varphi}=-r\sin\theta\sin\varphi
\tag{5.15.1}
```

<a name ="tag{5.15.2}">

```math
\frac{\partial y}{\partial r}=\sin\theta\sin\varphi, \quad
\frac{\partial y}{\partial \theta}=r\cos\theta\sin\varphi, \quad
\frac{\partial y}{\partial \varphi}=r\sin\theta\cos\varphi
\tag{5.15.2}
```

<a name ="tag{5.15.3}">

```math
\frac{\partial z}{\partial r}=\cos\theta, \quad
\frac{\partial z}{\partial \theta}=-r\sin\theta, \quad
\frac{\partial z}{\partial \varphi}=0
\tag{5.15.3}
```

一階の偏微分演算子は，

<a name ="tag{5.15.4}">

```math
\frac{\partial}{\partial r}
=\frac{\partial x}{\partial r}\frac{\partial}{\partial x}
+\frac{\partial y}{\partial r}\frac{\partial}{\partial y}
+\frac{\partial z}{\partial r}\frac{\partial}{\partial z}
=\sin\theta\cos\varphi\frac{\partial}{\partial x}
+\sin\theta\sin\varphi\frac{\partial}{\partial y}
+\cos\theta\frac{\partial}{\partial z}

\tag{5.15.4}
```

<a name ="tag{5.15.5}">

```math
\frac{\partial}{\partial \theta}
=\frac{\partial x}{\partial \theta}\frac{\partial}{\partial x}
+\frac{\partial y}{\partial \theta}\frac{\partial}{\partial y}
+\frac{\partial z}{\partial \theta}\frac{\partial}{\partial z}
=r\cos\theta\cos\varphi\frac{\partial}{\partial x}
+r\cos\theta\sin\varphi\frac{\partial}{\partial y}
-r\sin\theta\frac{\partial}{\partial z}

\tag{5.15.5}
```

<a name ="tag{5.15.6}">

```math
\frac{\partial}{\partial \varphi}
=\frac{\partial x}{\partial \varphi}\frac{\partial}{\partial x}
+\frac{\partial y}{\partial \varphi}\frac{\partial}{\partial y}
+\frac{\partial z}{\partial \varphi}\frac{\partial}{\partial z}
=-r\sin\theta\sin\varphi\frac{\partial}{\partial x}
+r\sin\theta\cos\varphi\frac{\partial}{\partial y}

\tag{5.15.6}
```

[(5.15.4)](#tag{5.15.4})×$r\sin\theta$より

```math
r\sin\theta\frac{\partial}{\partial r}
=r\sin^2\theta\cos\varphi\frac{\partial}{\partial x}
+r\sin^2\theta\sin\varphi\frac{\partial}{\partial y}
+r\sin\theta\cos\theta\frac{\partial}{\partial z}
```

[(5.15.5)](#tag{5.15.5})×$\cos\theta$より

```math
\cos\theta\frac{\partial}{\partial\theta}
=r\cos^2\theta\cos\varphi\frac{\partial}{\partial x}
+r\cos^2\theta\sin\varphi\frac{\partial}{\partial y}
-r\sin\theta\cos\theta\frac{\partial}{\partial z}
```

両式を加えると，

<a name ="tag{5.15.7}">

```math
r\sin\theta\frac{\partial}{\partial r}+\cos\theta\frac{\partial}{\partial\theta}
=r\cos\varphi\frac{\partial}{\partial x}+r\sin\varphi\frac{\partial}{\partial y}
\tag{5.15.7}
```

[(5.15.6)](#tag{5.15.6})×$\sin\varphi$より

```math
\sin\varphi\frac{\partial}{\partial\varphi}
=-r\sin\theta\sin^2\varphi\frac{\partial}{\partial x}
+r\sin\theta\sin\varphi\cos\varphi\frac{\partial}{\partial y}
```

[(5.15.7)](#tag{5.15.7})×$\sin\theta\cos\varphi$より

```math
r\sin^2\theta\cos\varphi\frac{\partial}{\partial r}
+\sin\theta\cos\theta\cos\varphi\frac{\partial}{\partial \theta}
=r\sin\theta\cos^2\varphi\frac{\partial}{\partial x}
+r\sin\theta\sin\varphi\cos\varphi\frac{\partial}{\partial y}
```

両式を加えると，

```math
r\sin^2\theta\cos\varphi\frac{\partial}{\partial r}
+\sin\theta\cos\theta\cos\varphi\frac{\partial}{\partial \theta}
-\sin\varphi\frac{\partial}{\partial\varphi}
=r\sin\theta\frac{\partial}{\partial x}
```

すなわち

<a name ="tag{5.15.8}">

```math
\frac{\partial}{\partial x}=
\sin\theta\cos\varphi\frac{\partial}{\partial r}
+\frac{\cos\theta\cos\varphi}{r}\frac{\partial}{\partial\theta}
-\frac{\sin\varphi}{r\sin\theta}\frac{\partial}{\partial\varphi}
\tag{5.15.8}
```

また[(5.15.6)](#tag{5.15.6})×$\cos\varphi$より

```math
\cos\varphi\frac{\partial}{\partial \varphi}
=-r\sin\theta\sin\varphi\cos\varphi\frac{\partial}{\partial x}
+r\sin\theta\cos^2\varphi\frac{\partial}{\partial y}
```

[(5.15.7)](#tag{5.15.7})×$\sin\theta\sin\varphi$より

```math
r\sin^2\sin\varphi\frac{\partial}{\partial r}
+\sin\theta\cos\theta\sin\varphi\frac{\partial}{\partial \theta}
=r\sin\theta\sin\varphi\cos\varphi\frac{\partial}{\partial x}
+r\sin\theta\sin^2\varphi\frac{\partial}{\partial y}
```

両式を加えると，

```math
r\sin^2\sin\varphi\frac{\partial}{\partial r}
+\sin\theta\cos\theta\sin\varphi\frac{\partial}{\partial \theta}
+\cos\varphi\frac{\partial}{\partial \varphi}
=
r\sin\theta\frac{\partial}{\partial y}
```

すなわち

<a name ="tag{5.15.9}">

```math
\frac{\partial}{\partial y}
=
\sin\theta\sin\varphi\frac{\partial}{\partial r}
+\frac{\cos\theta\sin\varphi}{r}\frac{\partial}{\partial \theta}
+\frac{\cos\varphi}{r\sin\theta}\frac{\partial}{\partial \varphi}
\tag{5.15.9}
```

[(5.15.8)](#tag{5.15.8})，[(5.15.9)](#tag{5.15.9})を[(5.15.4)](#tag{5.15.4})に代入すると，

```math
\frac{\partial}{\partial r}
=
\sin^2\theta\cos^2\varphi\frac{\partial}{\partial r}
+\frac{\sin\theta\cos\theta\cos^2\varphi}{r}\frac{\partial}{\partial \theta}
-\frac{\sin\varphi\cos\varphi}{r}\frac{\partial}{\partial \varphi}
+\sin^2\theta\sin^2\varphi\frac{\partial}{\partial r}
+\frac{\sin\theta\cos\theta\sin^2\varphi}{r}\frac{\partial}{\partial \theta}
+\frac{\sin\varphi\cos\varphi}{r}\frac{\partial}{\partial \varphi}
+\cos\theta\frac{\partial}{\partial z},
```

```math
\cos^2\theta\frac{\partial}{\partial r}=
\frac{\sin\theta\cos\theta}{r}\frac{\partial}{\partial \theta}
+\cos\theta\frac{\partial}{\partial z}
```

すなわち

<a name ="tag{5.15.10}">

```math
\frac{\partial}{\partial z}
=
\cos\theta\frac{\partial}{\partial r}
-\frac{\sin\theta}{r}\frac{\partial}{\partial \theta}
\tag{5.15.10}
```

[(5.8)](#tag{5.8}),[(5.15.9)](#tag{5.15.9}),[(5.15.10)](#tag{5.15.10})より,

```math
\begin{align}
y\frac{\partial}{\partial z}-z\frac{\partial}{\partial y}
&=
r\sin\theta\cos\theta\sin\varphi\frac{\partial}{\partial r}
-\sin^2\theta\sin\varphi\frac{\partial}{\partial \theta}
-r\sin\theta\cos\theta\sin\varphi\frac{\partial}{\partial r}
-\cos^2\theta\sin\varphi\frac{\partial}{\partial \theta}
-\cot\theta\cos\varphi\frac{\partial}{\partial \varphi},
\\
&=
-\left(
\sin\varphi\frac{\partial}{\partial \theta}
+\cot\theta\cos\varphi\frac{\partial}{\partial \varphi}
\right)
\end{align}
```

[(5.8)](#tag{5.8}),[(5.15.8)](#tag{5.15.8}),[(5.15.10)](#tag{5.15.10})より,

```math
\begin{align}
z\frac{\partial}{\partial x}-x\frac{\partial}{\partial z}
&=
r\sin\theta\cos\theta\cos\varphi\frac{\partial}{\partial r}
+\cos^2\theta\cos\varphi\frac{\partial}{\partial \theta}
-\cot\theta\sin\varphi\frac{\partial}{\partial \varphi}
-r\sin\theta\cos\theta\cos\varphi\frac{\partial}{\partial r}
+\sin^2\theta\cos\varphi\frac{\partial}{\partial \theta}
\\
&=
\cos\varphi\frac{\partial}{\partial \theta}
-\cot\theta\sin\varphi\frac{\partial}{\partial \varphi}
\end{align}
```

[(5.8)](#tag{5.8}),[(5.15.8)](#tag{5.15.8}),[(5.15.9)](#tag{5.15.9})より,

```math
\begin{align}
x\frac{\partial}{\partial y}-y\frac{\partial}{\partial x}
&=
r\sin^2\theta\sin\varphi\cos\varphi\frac{\partial}{\partial r}
+\sin\theta\cos\theta\sin\varphi\cos\varphi\frac{\partial}{\partial \theta}
+\cos^2\varphi\frac{\partial}{\partial \varphi}
-r\sin^2\theta\sin\varphi\cos\varphi\frac{\partial}{\partial r}
-\sin\theta\cos\theta\sin\varphi\cos\varphi\frac{\partial}{\partial \theta}
+\sin^2\varphi\frac{\partial}{\partial \varphi}
\\
&=
\frac{\partial}{\partial \varphi}
\end{align}
```

これら3式を[(5.15)](#tag{5.15})に代入すると,

<a name ="tag{5.16}">

```math
\begin{align}
\hat{L}_x&=-\frac{\hbar}{i}\left(
\sin\varphi\frac{\partial}{\partial \theta}
+\cot\theta\cos\varphi\frac{\partial}{\partial \varphi}
\right)
\\
\hat{L}_y&=\frac{\hbar}{i}\left(
\cos\varphi\frac{\partial}{\partial \theta}
-\cot\theta\sin\varphi\frac{\partial}{\partial \varphi}
\right)
\\
\hat{L}_z&=\frac{\hbar}{i}
\frac{\partial}{\partial \varphi}
\end{align}
\tag{5.16}
```

となる.

さらに,

```math
\begin{align}
\left(
\sin\varphi\frac{\partial}{\partial \theta}
+\cot\theta\cos\varphi\frac{\partial}{\partial \varphi}
\right)^2
&=
\sin\varphi\frac{\partial}{\partial \theta}
\left(
\sin\varphi\frac{\partial}{\partial \theta}
+\cot\theta\cos\varphi\frac{\partial}{\partial \varphi}
\right)
+
\cot\theta\cos\varphi\frac{\partial}{\partial \varphi}
\left(
\sin\varphi\frac{\partial}{\partial \theta}
+\cot\theta\cos\varphi\frac{\partial}{\partial \varphi}
\right)
\\
&=
\sin^2\varphi\frac{\partial^2}{\partial \theta^2}
+\sin\varphi\cos\varphi\frac{\partial}{\partial \theta}\left(\cot\theta\right)\frac{\partial}{\partial \varphi}
+\cot\theta\cos\varphi\frac{\partial}{\partial \theta}\frac{\partial}{\partial \varphi}\left(\sin\varphi\right)
+\cot^2\theta\cos\varphi\frac{\partial}{\partial \varphi}\left(\cos\varphi\frac{\partial}{\partial \varphi}\right)
\\
&=
\sin^2\varphi\frac{\partial^2}{\partial \theta^2}
+\sin\varphi\cos\varphi
\left(
-\frac{1}{\sin^2\theta}
+\cot\theta\frac{\partial}{\partial \theta}
\right)\frac{\partial}{\partial \varphi}
+\cot\theta\cos\varphi\frac{\partial}{\partial \theta}
\left(
\cos\varphi+\sin\varphi\frac{\partial}{\partial \varphi}
\right)
+\cot^2\theta\cos\varphi
\left(
-\sin\varphi\frac{\partial}{\partial \varphi}
+\cos\varphi\frac{\partial^2}{\partial \varphi^2}
\right)
\end{align}
```

```math
\begin{align}
\left(
\cos\varphi\frac{\partial}{\partial \theta}
-\cot\theta\sin\varphi\frac{\partial}{\partial \varphi}
\right)^2
&=
\cos\varphi\frac{\partial}{\partial \theta}
\left(
\cos\varphi\frac{\partial}{\partial \theta}
-\cot\theta\sin\varphi\frac{\partial}{\partial \varphi}
\right)
-\cot\theta\sin\varphi\frac{\partial}{\partial \varphi}
\left(
\cos\varphi\frac{\partial}{\partial \theta}
-\cot\theta\sin\varphi\frac{\partial}{\partial \varphi}
\right)
\\
&=
\cos^2\varphi\frac{\partial^2}{\partial \theta^2}
-\sin\varphi\cos\varphi\frac{\partial}{\partial \theta}\left(\cot\theta\right)\frac{\partial}{\partial \varphi}
-\cot\theta\sin\varphi\frac{\partial}{\partial \theta}\frac{\partial}{\partial \varphi}\left(\cos\varphi\right)
+\cot^2\theta\sin\varphi\frac{\partial}{\partial \varphi}\left(\sin\varphi\frac{\partial}{\partial \varphi}\right)
\\
&=
\cos^2\varphi\frac{\partial^2}{\partial \theta^2}
-\sin\varphi\cos\varphi
\left(
-\frac{1}{\sin^2\theta}+\cot\theta\frac{\partial}{\partial \theta}
\right)\frac{\partial}{\partial \varphi}
-\cot\theta\sin\varphi\frac{\partial}{\partial \theta}
\left(
-\sin\varphi+\cos\varphi\frac{\partial}{\partial \varphi}
\right)
+\cot^2\theta\sin\varphi
\left(
\cos\varphi\frac{\partial}{\partial \varphi}
+\sin\varphi\frac{\partial^2}{\partial \varphi^2}
\right)
\end{align}
```

```math
\left(\frac{\partial}{\partial \varphi}\right)^2
=
\frac{\partial^2}{\partial \varphi^2}
```

以上3式を,

```math
\frac{1}{\sin\theta}\frac{\partial}{\partial \theta}
\left(\sin\theta\frac{\partial}{\partial \theta}\right)
=
\frac{1}{\sin\theta}
\left(
\cos\theta\frac{\partial}{\partial \theta}+\sin\theta\frac{\partial^2}{\partial \theta^2}
\right)
=\cot\theta\frac{\partial}{\partial \theta}+\frac{\partial^2}{\partial \theta^2}
```

であることを用いて加えると，

```math
\begin{align}
\left(
\sin\varphi\frac{\partial}{\partial \theta}
+\cot\theta\cos\varphi\frac{\partial}{\partial \varphi}
\right)^2
+
\left(
\cos\varphi\frac{\partial}{\partial \theta}
-\cot\theta\sin\varphi\frac{\partial}{\partial \varphi}
\right)^2
+
\left(\frac{\partial}{\partial \varphi}\right)^2
&=
\frac{\partial^2}{\partial \theta^2}
+\cot\theta\frac{\partial}{\partial \theta}
+
\left(\cot^2\theta+1\right)\frac{\partial^2}{\partial \varphi^2}
=
\frac{\partial^2}{\partial \theta^2}
+\cot\theta\frac{\partial}{\partial \theta}
+\frac{1}{\sin^2\theta}\frac{\partial^2}{\partial \varphi^2}
\\
&=
\frac{1}{\sin\theta}\frac{\partial}{\partial \theta}
\left(\sin\theta\frac{\partial}{\partial \theta}\right)
+\frac{1}{\sin^2\theta}\frac{\partial^2}{\partial \varphi^2}
\end{align}
```

したがって,

<a name ="tag{5.17}">

```math
\begin{align}
\left(\hat{L}\right)^2
&=\left(\hat{L}_x\right)^2+\left(\hat{L}_y\right)^2+\left(\hat{L}_z\right)^2
\\
&=
-\hbar^2\left[
\frac{1}{\sin\theta}\frac{\partial}{\partial \theta}
\left(\sin\theta\frac{\partial}{\partial \theta}\right)
+\frac{1}{\sin^2\theta}\frac{\partial^2}{\partial \varphi^2}
\right]
\end{align}
\tag{5.17}
```

また，

```math
\hat{p}_r=\frac{\hbar}{i}\frac{1}{r}\frac{\partial}{\partial r}r
```

を2乗すると，

```math
\begin{align}
\left(\hat{p}_r\right)^2
&=
-\hbar^2
\left(\frac{1}{r}\frac{\partial}{\partial r}r\right)
\left(\frac{1}{r}\frac{\partial}{\partial r}r\right)
\\
&=
-\hbar^2
\left(\frac{1}{r}\frac{\partial}{\partial r}r\right)
\left\{
\frac{1}{r}\left(1+r\frac{\partial}{\partial r}\right)
\right\}
\\
&=
-\hbar^2
\left(\frac{1}{r}\frac{\partial}{\partial r}r\right)
\left(\frac{1}{r}+\frac{\partial}{\partial r}\right)
\\
&=
-\hbar^2
\left(\frac{1}{r}\frac{\partial}{\partial r}\right)
\left(1+r\frac{\partial}{\partial r}\right)
\\
&=
-\hbar^2
\frac{1}{r}
\left(
\frac{\partial}{\partial r}+\frac{\partial}{\partial r}
+r\frac{\partial^2}{\partial r^2}
\right)
\\
&=
-\hbar^2
\left(
\frac{2}{r}\frac{\partial}{\partial r}
+\frac{\partial^2}{\partial r^2}
\right)
\end{align}
```

ここで，

```math
\frac{1}{r}\frac{\partial^2}{\partial r^2}r
=
\frac{1}{r}\frac{\partial}{\partial r}\left(1+r\frac{\partial}{\partial r}\right)
=
\frac{1}{r}
\left(
\frac{\partial}{\partial r}+\frac{\partial}{\partial r}+r\frac{\partial^2}{\partial r^2}
\right)
=
\frac{2}{r}\frac{\partial}{\partial r}
+\frac{\partial^2}{\partial r^2}
```

より

<a name ="tag{5.18}">

```math
\left(\hat{p}_r\right)^2=-\hbar^2\frac{1}{r}\frac{\partial^2}{\partial r^2}r
\tag{5.18}
```

なので，[(5.9)](#tag{5.9})，[(5.17)](#tag{5.17})，[(5.18)](#tag{5.18})より，

<a name ="tag{5.19}">

```math
\frac{\hat{p}_r^2}{2m}
+\frac{\hat{L}_r^2}{2mr^2}
=
-\frac{\hbar^2}{2m}\Delta
\tag{5.19}
```

であることが確かめられた．したがって[(5.14)](#tag{5.14})が極座標のエルミート演算子のみで表された正しいハミルトニアンである．