# 全同粒子

## 量子力學的描述

### 對稱與反對稱量子態

${\displaystyle |n_{1}\rangle |n_{2}\rangle \,\!}$

${\displaystyle |n_{1},n_{2};S\rangle \equiv {\frac {1}{\sqrt {2}}}\times {\bigg (}|n_{1}\rangle |n_{2}\rangle +|n_{2}\rangle |n_{1}\rangle {\bigg )}\,\!}$

${\displaystyle |n_{1},n_{2};A\rangle \equiv {\frac {1}{\sqrt {2}}}\times {\bigg (}|n_{1}\rangle |n_{2}\rangle -|n_{2}\rangle |n_{1}\rangle {\bigg )}\,\!}$

${\displaystyle |n_{1}\rangle |n_{2}\rangle \neq |n_{2}\rangle |n_{1}\rangle \,\!}$

### 交換對稱性

${\displaystyle |n_{1},n_{2};?\rangle ={\mbox{constant}}\times {\bigg (}|n_{1}\rangle |n_{2}\rangle +i|n_{2}\rangle |n_{1}\rangle {\bigg )}\,\!}$

${\displaystyle P(|\psi \rangle |\phi \rangle )\equiv |\phi \rangle |\psi \rangle \,\!}$

${\displaystyle P|n_{1},n_{2};S\rangle =+|n_{1},n_{2};S\rangle \,\!}$
${\displaystyle P|n_{1},n_{2};A\rangle =-|n_{1},n_{2};A\rangle \,\!}$

${\displaystyle H={\frac {p_{1}^{2}}{2m}}+{\frac {p_{2}^{2}}{2m}}+U(|x_{1}-x_{2}|)+V(x_{1})+V(x_{2})\,\!}$

${\displaystyle \left[P,H\right]=0\,\!}$

### N 個粒子

${\displaystyle |n_{1},\,n_{2},\,\cdots ,\,n_{N};S\rangle ={\sqrt {\frac {\prod _{j}N_{j}!}{N!}}}\sum _{p\in \mathrm {permutation} (N)}|n_{p_{1}}\rangle |n_{p_{2}}\rangle \cdots |n_{p_{N}}\rangle \,\!}$  ;

${\displaystyle |n_{1},\,n_{2},\,\cdots ,\,n_{N};A\rangle ={\frac {1}{\sqrt {N!}}}\sum _{p\in \mathrm {permutation} (N)}\mathrm {sign} (p)|n_{p_{1}}\rangle |n_{p_{2}}\rangle \cdots |n_{p_{N}}\rangle \ \,\!}$

${\displaystyle \langle n_{1},\,n_{2},\,\cdots ,\,n_{N};S|n_{1},\,n_{2},\,\cdots ,\,n_{N};S\rangle =1\,\!}$
${\displaystyle \langle n_{1},\,n_{2},\,\cdots ,\,n_{N};A|n_{1},\,n_{2},\,\cdots ,\,n_{N};A\rangle =1\,\!}$

### 斯萊特行列式

${\displaystyle \Psi _{n_{1}\cdots n_{N}}^{(A)}(x_{1},\cdots x_{N})={\frac {1}{\sqrt {N!}}}\left|{\begin{matrix}\psi _{n_{1}}(x_{1})&\psi _{n_{1}}(x_{2})&\cdots &\psi _{n_{1}}(x_{N})\\\psi _{n_{2}}(x_{1})&\psi _{n_{2}}(x_{2})&\cdots &\psi _{n_{2}}(x_{N})\\\cdots &\cdots &\cdots &\cdots \\\psi _{n_{N}}(x_{1})&\psi _{n_{N}}(x_{2})&\cdots &\psi _{n_{N}}(x_{N})\\\end{matrix}}\right|\,\!}$

### 範例

#### 二個全同玻色子

{\displaystyle {\begin{aligned}|1,\,1;S\rangle &={\frac {1}{\sqrt {2!2!}}}(|1\rangle |1\rangle +|1\rangle |1\rangle )\\&=|1\rangle |1\rangle \\\end{aligned}}\,\!}
{\displaystyle {\begin{aligned}|1,\,2;S\rangle &={\frac {1}{\sqrt {2!1!1!}}}(|1\rangle |2\rangle +|2\rangle |1\rangle )\\&={\frac {1}{\sqrt {2}}}(|1\rangle |2\rangle +|2\rangle |1\rangle )\\\end{aligned}}\,\!}

#### 三個全同玻色子

{\displaystyle {\begin{aligned}|1,\,1,\,1;S\rangle &={\frac {1}{\sqrt {3!3!}}}(|1\rangle |1\rangle |1\rangle +|1\rangle |1\rangle |1\rangle +|1\rangle |1\rangle |1\rangle \\&\qquad \qquad +|1\rangle |1\rangle |1\rangle +|1\rangle |1\rangle |1\rangle +|1\rangle |1\rangle |1\rangle )\\&=|1\rangle |1\rangle |1\rangle \\\end{aligned}}\,\!}
{\displaystyle {\begin{aligned}|1,\,1,\,2;S\rangle &={\frac {1}{\sqrt {3!2!1!}}}(|1\rangle |1\rangle |2\rangle +|1\rangle |2\rangle |1\rangle +|1\rangle |1\rangle |2\rangle \\&\qquad \qquad +|1\rangle |2\rangle |1\rangle +|2\rangle |1\rangle |1\rangle +|2\rangle |1\rangle |1\rangle )\\&={\frac {1}{\sqrt {3}}}(|1\rangle |1\rangle |2\rangle +|1\rangle |2\rangle |1\rangle +|2\rangle |1\rangle |1\rangle )\\\end{aligned}}\,\!}
{\displaystyle {\begin{aligned}|1,\,2,\,3;S\rangle &={\frac {1}{\sqrt {3!1!1!1!}}}(|1\rangle |2\rangle |3\rangle +|1\rangle |3\rangle |2\rangle +|2\rangle |1\rangle |3\rangle \\&\qquad \qquad +|2\rangle |3\rangle |1\rangle +|3\rangle |1\rangle |2\rangle +|3\rangle |2\rangle |1\rangle )\\&={\frac {1}{\sqrt {6}}}(|1\rangle |2\rangle |3\rangle +|1\rangle |3\rangle |2\rangle +|2\rangle |1\rangle |3\rangle \\&\qquad \qquad +|2\rangle |3\rangle |1\rangle +|3\rangle |1\rangle |2\rangle +|3\rangle |2\rangle |1\rangle )\\\end{aligned}}\,\!}

#### 二個全同費米子

{\displaystyle {\begin{aligned}|1,\,1;A\rangle &={\frac {1}{\sqrt {2!}}}(|1\rangle |1\rangle -|1\rangle |1\rangle )\\&=0\\\end{aligned}}\,\!}
{\displaystyle {\begin{aligned}|1,\,2;A\rangle &={\frac {1}{\sqrt {2!}}}(|1\rangle |2\rangle -|2\rangle |1\rangle )\\&={\frac {1}{\sqrt {2}}}(|1\rangle |2\rangle -|2\rangle |1\rangle )\\\end{aligned}}\,\!}

#### 三個全同費米子

{\displaystyle {\begin{aligned}|1,\,1,\,1;A\rangle &={\frac {1}{\sqrt {3!}}}(|1\rangle |1\rangle |1\rangle -|1\rangle |1\rangle |1\rangle +|1\rangle |1\rangle |1\rangle \\&\qquad \qquad -|1\rangle |1\rangle |1\rangle +|1\rangle |1\rangle |1\rangle -|1\rangle |1\rangle |1\rangle )\\&=0\\\end{aligned}}\,\!}
{\displaystyle {\begin{aligned}|1,\,1,\,2;A\rangle &={\frac {1}{\sqrt {3!}}}(|1\rangle |1\rangle |2\rangle -|1\rangle |2\rangle |1\rangle -|1\rangle |1\rangle |2\rangle \\&\qquad \qquad +|1\rangle |2\rangle |1\rangle +|2\rangle |1\rangle |1\rangle -|2\rangle |1\rangle |1\rangle )\\&=0\\\end{aligned}}\,\!}
{\displaystyle {\begin{aligned}|1,\,2,\,3;A\rangle &={\frac {1}{\sqrt {3!}}}(|1\rangle |2\rangle |3\rangle -|1\rangle |3\rangle |2\rangle -|2\rangle |1\rangle |3\rangle \\&\qquad \qquad +|2\rangle |3\rangle |1\rangle +|3\rangle |1\rangle |2\rangle -|3\rangle |2\rangle |1\rangle )\\&={\frac {1}{\sqrt {6}}}(|1\rangle |2\rangle |3\rangle -|1\rangle |3\rangle |2\rangle -|2\rangle |1\rangle |3\rangle \\&\qquad \qquad +|2\rangle |3\rangle |1\rangle +|3\rangle |1\rangle |2\rangle -|3\rangle |2\rangle |1\rangle )\\\end{aligned}}\,\!}

## 統計性質

• ${\displaystyle |0\rangle |0\rangle \,\!}$
• ${\displaystyle |1\rangle |1\rangle \,\!}$
• ${\displaystyle |0\rangle |1\rangle \,\!}$
• ${\displaystyle |1\rangle |0\rangle \,\!}$

• ${\displaystyle |0\rangle |0\rangle \,\!}$
• ${\displaystyle |1\rangle |1\rangle \,\!}$
• ${\displaystyle {\frac {1}{\sqrt {2}}}(|0\rangle |1\rangle +|1\rangle |0\rangle )\,\!}$

${\displaystyle {\frac {1}{\sqrt {2}}}(|0\rangle |1\rangle -|1\rangle |0\rangle )\,\!}$

## 註釋

1. ^ 反對稱性波函數為 ${\displaystyle [\sin(x)sin(3y)-sin(3x)sin(y)]/{\sqrt {2}},\qquad 0\leq x,y\leq \pi }$  。注意到在 ${\displaystyle x=y}$  附近，機率輻絕對值很微小，兩個費米子趨向於彼此互相遠離對方。
2. ^ 對稱性波函數為 ${\displaystyle -[\sin(x)sin(3y)+sin(3x)sin(y)]/{\sqrt {2}},\qquad 0\leq x,y\leq \pi }$  。注意到在 ${\displaystyle x=y}$  附近，機率輻絕對值較大，兩個費米子趨向於彼此互相接近對方。