# 角動量圖

## 狄拉克符號與朱西斯角動量圖的等價

### 角動量量子態

• 量子數jm常標記在箭頭附近，以表示特定的角動量量子態，
• 箭頭常在線段的中間
• 等號"="置於等價的圖樣之間，如同相應的代數運算。

• 標準表象（standard representation）以一條離開頂點的指向線段表示，
• 反標準表象（contrastandard representation）則是以一條進入頂點的指向線段表示。

${\displaystyle T{\hat {\mathbf {x} }}T^{\dagger }={\hat {\mathbf {x} }}}$

${\displaystyle T{\hat {\mathbf {p} }}T^{\dagger }=-{\hat {\mathbf {p} }}}$

${\displaystyle T{\hat {\mathbf {S} }}T^{\dagger }=-{\hat {\mathbf {S} }}}$

${\displaystyle T{\hat {\mathbf {L} }}T^{\dagger }=-{\hat {\mathbf {L} }}}$

${\displaystyle T{\hat {\mathbf {J} }}T^{\dagger }=-{\hat {\mathbf {J} }}}$

${\displaystyle T\left|j,m\right\rangle \equiv \left|T(j,m)\right\rangle ={(-1)}^{j-m}\left|j,-m\right\rangle }$

### 內積

${\displaystyle \langle j_{2},m_{2}|j_{1},m_{1}\rangle =\delta _{j_{1}j_{2}}\delta _{m_{1}m_{2}}}$

|j1, m1|j2, m2內積，亦即j2, m2|j1, m1

${\displaystyle \sum _{m}\langle j,m|j,m\rangle =2j+1}$

### 外積

${\displaystyle \left|j_{2},m_{2}\right\rangle \left\langle j_{1},m_{1}\right|}$

|j1, m1|j2, m2外積，亦即|j2, m2j1, m1|

{\displaystyle {\begin{aligned}\sum _{m}|j,m\rangle \langle j,m|&=\sum _{m}|j,-m\rangle \langle j,-m|\\&=\sum _{m}{(-1)}^{2(j-m)}|j,-m\rangle \langle j,-m|\\&=\sum _{m}{(-1)}^{j-m}|j,-m\rangle \langle j,-m|{(-1)}^{j-m}\\&=\sum _{m}T|j,m\rangle \langle j,m|T^{\dagger }\end{aligned}}}

### 張量積

n狀態|j1, m1, |j2, m2, ... |jn, mn的張量積⊗可寫為：

{\displaystyle {\begin{aligned}\left|j_{1},m_{1},j_{2},m_{2},...j_{n},m_{n}\right\rangle &\equiv \left|j_{1},m_{1}\right\rangle \otimes \left|j_{2},m_{2}\right\rangle \otimes \cdots \otimes \left|j_{n},m_{n}\right\rangle \\&\equiv \left|j_{1},m_{1}\right\rangle \left|j_{2},m_{2}\right\rangle \cdots \left|j_{n},m_{n}\right\rangle \end{aligned}}}

• 負號(−)表示順序為順時針走向${\displaystyle \circlearrowright }$
• 正號(+)表示順序為逆時針走向${\displaystyle \circlearrowleft }$ .

|j1, m1, |j2, m2, |j3, m3張量積，亦即|j1, m1|j2, m2|j3, m3 = |j1, m1, j2, m2, j3, m3

{\displaystyle {\begin{aligned}&\left\langle j'_{n},m'_{n},...,j'_{2},m'_{2},j'_{1},m'_{1}|j_{1},m_{1},j_{2},m_{2},...j_{n},m_{n}\right\rangle \\=&\langle j'_{n},m'_{n}|...\langle j'_{2},m'_{2}|\langle j'_{1},m'_{1}||j_{1},m_{1}\rangle |j_{2},m_{2}\rangle ...|j_{n},m_{n}\rangle \\=&\prod _{k=1}^{n}\left\langle j'_{k},m'_{k}|j_{k},m_{k}\right\rangle \end{aligned}}}
|j1, m1, j2, m2, j3, m3|j1, m1, j2, m2, j3, m3內積，亦即j3, m3, j2, m2, j1, m1|j1, m1, j2, m2, j3, m3