# 位置算符

${\displaystyle {\hat {x}}|x\rangle =x|x\rangle }$

## 位置空間表現

${\displaystyle \Psi (x)\ {\stackrel {def}{=}}\ \langle x|\Psi \rangle }$
${\displaystyle \psi (x)\ {\stackrel {def}{=}}\ \langle x|\psi \rangle }$

${\displaystyle {\hat {\mathfrak {x}}}\psi (x)\ {\stackrel {def}{=}}\ x\psi (x)}$

${\displaystyle |\psi \rangle =\int _{-\infty }^{\infty }\mathrm {d} x'\ |x'\rangle \langle x'|\psi \rangle }$

${\displaystyle {\hat {x}}|\psi \rangle ={\hat {x}}\int _{-\infty }^{\infty }\mathrm {d} x'\ |x'\rangle \langle x'|\psi \rangle =\int _{-\infty }^{\infty }\mathrm {d} x'\ x'|x'\rangle \langle x'|\psi \rangle =\int _{-\infty }^{\infty }\mathrm {d} x'\ x'\psi (x')|x'\rangle =\int _{-\infty }^{\infty }\mathrm {d} x'\ {\hat {\mathfrak {x}}}\psi (x')|x'\rangle }$

${\displaystyle \langle x|{\hat {x}}|\psi \rangle =\int _{-\infty }^{\infty }\mathrm {d} x'\ {\hat {\mathfrak {x}}}\psi (x')\langle x|x'\rangle =\int _{-\infty }^{\infty }\mathrm {d} x'\ {\hat {\mathfrak {x}}}\psi (x')\delta (x-x')={\hat {\mathfrak {x}}}\psi (x)}$

${\displaystyle \Psi \rangle =\int _{-\infty }^{\infty }\mathrm {d} x'\ |x'\rangle \langle x'|\Psi \rangle =\int _{-\infty }^{\infty }\mathrm {d} x'\ \Psi (x')|x'\rangle }$

${\displaystyle \langle x|\Psi \rangle =\int _{-\infty }^{\infty }\mathrm {d} x'\ \Psi (x')\langle x|x'\rangle =\int _{-\infty }^{\infty }\mathrm {d} x'\ \Psi (x')\delta (x-x')=\Psi (x)}$

${\displaystyle \Psi (x)={\hat {\mathfrak {x}}}\psi (x)}$

## 本徵函數

${\displaystyle {\hat {\mathfrak {x}}}g_{q}(x)=qg_{q}(x)}$

${\displaystyle g_{q}(x)=g_{0}\delta (x-q)}$

${\displaystyle \int _{-\infty }^{\infty }\ g_{q}^{*}(x)g_{q}(x)\ dx=|g_{0}|^{2}\int _{-\infty }^{\infty }\ \delta ^{2}(x-q)\ dx={\mbox{?}}}$

${\displaystyle \int _{-\infty }^{\infty }\ g_{q1}^{*}(x)g_{q2}(x)\ dx=\int _{-\infty }^{\infty }\ \delta (x-q1)\delta (x-q2)\ dx=\delta (q1-q2)}$

${\displaystyle \psi (x)=\int _{-\infty }^{\infty }\ \psi (q)g_{q}(x)\ dq}$

## 期望值

${\displaystyle \langle \psi _{1}|\psi _{2}\rangle =\int _{-\infty }^{\infty }\psi _{1}^{*}(x)\psi _{2}(x)\,\mathrm {d} x}$

${\displaystyle \langle x\rangle \ {\stackrel {def}{=}}\ \langle \psi |{\hat {x}}|\psi \rangle }$

${\displaystyle \langle x\rangle =\int _{-\infty }^{\infty }\psi ^{\ast }(x)\,x\,\psi (x)\,\mathrm {d} x=\int _{-\infty }^{\infty }x\,|\psi (x)|^{2}\,\mathrm {d} x}$

${\displaystyle p(x)\mathrm {d} x=\psi ^{*}(x)\psi (x)\mathrm {d} x}$

## 三維案例

${\displaystyle \langle \mathbf {r} \rangle =\int _{\mathbb {V} }\mathbf {r} |\psi (\mathbf {r} )|^{2}\mathrm {d} ^{3}\mathbf {r} }$

${\displaystyle \mathbf {\hat {\mathfrak {r}}} \psi =\mathbf {r} \psi }$

## 對易關係

${\displaystyle [{\hat {x}},\ {\hat {p}}]\psi =({\hat {x}}{\hat {p}}-{\hat {p}}{\hat {x}})\psi =x{\frac {\hbar }{i}}{\frac {\partial \psi }{\partial x}}-{\frac {\hbar }{i}}{\frac {\partial (x\psi )}{\partial x}}=i\hbar \psi }$

${\displaystyle \Delta A\ \Delta B\geq \left|{\frac {\langle [A,\ B]\rangle }{2i}}\right|}$

## 參考文獻

1. ^ Griffiths, David J. Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. 2004: pp. 17, 104–109. ISBN 0-13-111892-7.
2. Sakurai, J. J.; Napolitano, Jim, Modern Quantum Mechanics 2nd, Addison-Wesley, 2010, ISBN 978-0805382914