パウリ行列 $\sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ が与えられたとき、以下の問題を解く。 I. 次の行列を求めよ。 (1) $\sigma_+ = \sigma_x + i\sigma_y$, (2) $\sigma_- = \sigma_x - i\sigma_y$, (3) $\sigma_x \sigma_x$, (4) $\sigma_x \sigma_y$, (5) $\sigma_y \sigma_z$, (6) $\sigma_x \sigma_y \sigma_z$, (7) $\sigma_y \sigma_z + \sigma_z \sigma_y$, (8) $\sigma_z \sigma_x + \sigma_x \sigma_z$, (9) $\sigma_x \sigma_y \sigma_x$ II. 次の行列式を求めよ。 (1) $|\sigma_x|$, (2) $|\sigma_y|$, (3) $|\sigma_z|$, (4) $|\sigma_x \sigma_x|$, (5) $|\sigma_y \sigma_z|$, (6) $|\sigma_z \sigma_x|$

1. 問題の内容

パウリ行列 $\sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ が与えられたとき、以下の問題を解く。

I. 次の行列を求めよ。

(1) $\sigma_+ = \sigma_x + i\sigma_y$, (2) $\sigma_- = \sigma_x - i\sigma_y$, (3) $\sigma_x \sigma_x$, (4) $\sigma_x \sigma_y$, (5) $\sigma_y \sigma_z$, (6) $\sigma_x \sigma_y \sigma_z$, (7) $\sigma_y \sigma_z + \sigma_z \sigma_y$, (8) $\sigma_z \sigma_x + \sigma_x \sigma_z$, (9) $\sigma_x \sigma_y \sigma_x$

II. 次の行列式を求めよ。

(1) $|\sigma_x|$, (2) $|\sigma_y|$, (3) $|\sigma_z|$, (4) $|\sigma_x \sigma_x|$, (5) $|\sigma_y \sigma_z|$, (6) $|\sigma_z \sigma_x|$

2. 解き方の手順

I. 行列の計算

(1) $\sigma_+ = \sigma_x + i\sigma_y = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} + i \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} + \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 2 \\ 0 & 0 \end{pmatrix}$

(2) $\sigma_- = \sigma_x - i\sigma_y = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} - i \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} - \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 2 & 0 \end{pmatrix}$

(3) $\sigma_x \sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = I$

(4) $\sigma_x \sigma_y = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} = \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix}$

(5) $\sigma_y \sigma_z = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} = \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}$

(6) $\sigma_x \sigma_y \sigma_z = (\sigma_x \sigma_y) \sigma_z = \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} = \begin{pmatrix} i & 0 \\ 0 & i \end{pmatrix} = iI$

(7) $\sigma_y \sigma_z + \sigma_z \sigma_y = \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix} + \begin{pmatrix} 0 & -i \\ -i & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} = 0$

(8) $\sigma_z \sigma_x + \sigma_x \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} + \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} + \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} = 0$

(9) $\sigma_x \sigma_y \sigma_x = (\sigma_x \sigma_y) \sigma_x = \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} 0 & i \\ -i & 0 \end{pmatrix}$

II. 行列式の計算

(1) $|\sigma_x| = (0)(0) - (1)(1) = -1$

(2) $|\sigma_y| = (0)(0) - (-i)(i) = -1$

(3) $|\sigma_z| = (1)(-1) - (0)(0) = -1$

(4) $|\sigma_x \sigma_x| = |I| = 1$

(5) $|\sigma_y \sigma_z| = |\begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}| = (0)(0) - (i)(i) = 1$

(6) $|\sigma_z \sigma_x| = |\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}| = (0)(0) - (1)(-1) = 1$

3. 最終的な答え

I.

(1) $\begin{pmatrix} 0 & 2 \\ 0 & 0 \end{pmatrix}$

(2) $\begin{pmatrix} 0 & 0 \\ 2 & 0 \end{pmatrix}$

(3) $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$

(4) $\begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix}$

(5) $\begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}$

(6) $\begin{pmatrix} i & 0 \\ 0 & i \end{pmatrix}$

(7) $\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$

(8) $\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$

(9) $\begin{pmatrix} 0 & i \\ -i & 0 \end{pmatrix}$

II.

(1) $-1$

(2) $-1$

(3) $-1$

(4) $1$

(5) $1$

(6) $1$