# 期望值 (量子力學)

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

## 量子力學形式論

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

${\displaystyle \langle e_{i}|e_{j}\rangle =\delta _{ij}}$

${\displaystyle {\hat {O}}|e_{i}\rangle =O_{i}|e_{i}\rangle }$

${\displaystyle |\psi \rangle =\sum _{i}c_{i}|e_{i}\rangle }$

${\displaystyle \sum _{i}|e_{i}\rangle \langle e_{i}|=1}$

{\displaystyle {\begin{aligned}\langle O\rangle &=\langle \psi |{\hat {O}}|\psi \rangle \\&=\sum _{i,j}\langle \psi |e_{i}\rangle \langle e_{i}|{\hat {O}}|e_{j}\rangle \langle e_{j}|\psi \rangle \\&=\sum _{i,j}\langle \psi |e_{i}\rangle \langle e_{i}|e_{j}\rangle \langle e_{j}|\psi \rangle O_{i}\\&=\sum _{i}|\langle e_{i}|\psi \rangle |^{2}O_{i}\\\end{aligned}}}

## 系综平均值

${\displaystyle {\hat {\rho }}=\sum _{i}w_{i}|\psi ^{(i)}\rangle \langle \psi ^{(i)}|}$

• 純系綜 ${\displaystyle {\mathcal {E}}_{up}}$ 的密度算符為 ${\displaystyle |\uparrow \rangle \langle \uparrow |}$
• 純系綜 ${\displaystyle {\mathcal {E}}_{down}}$ 的密度算符為 ${\displaystyle |\downarrow \rangle \langle \downarrow |}$
• 混系綜 ${\displaystyle {\mathcal {E}}_{mix}}$ 的密度算符為 ${\displaystyle {\frac {1}{2}}(|\uparrow \rangle \langle \uparrow |+|\downarrow \rangle \langle \downarrow |)}$

${\displaystyle \rho _{jk}=\langle e_{j}|\rho |e_{k}\rangle =\sum _{i}w_{i}\langle e_{j}|\psi ^{(i)}\rangle \langle \psi ^{(i)}|e_{k}\rangle }$

{\displaystyle {\begin{aligned}\langle O\rangle &=\sum _{i}w_{i}\langle \psi ^{(i)}|{\hat {O}}|\psi ^{(i)}\rangle \\&=\sum _{i}\sum _{j}w_{i}\langle \psi ^{(i)}|e_{j}\rangle \langle e_{j}|{\hat {O}}|\psi ^{(i)}\rangle \\&=\sum _{i}\sum _{j}w_{i}\langle \psi ^{(i)}|e_{j}\rangle \langle e_{j}|\psi ^{(i)}\rangle O_{j}\\&=\sum _{i}\sum _{j}w_{i}|\langle \psi ^{(i)}|e_{j}\rangle |^{2}O_{j}\\\end{aligned}}}

{\displaystyle {\begin{aligned}\langle A\rangle &=\sum _{i}w_{i}\langle \psi ^{(i)}|{\hat {A}}|\psi ^{(i)}\rangle \\&=\sum _{i}\sum _{j,k}w_{i}\langle \psi ^{(i)}|e_{j}\rangle \langle e_{j}|{\hat {A}}|e_{k}\rangle \langle e_{k}|\psi ^{(i)}\rangle \\&=\sum _{i}\sum _{j,k}w_{i}\langle e_{k}|\psi ^{(i)}\rangle \langle \psi ^{(i)}|e_{j}\rangle \langle e_{j}|{\hat {A}}|e_{k}\rangle \\&=\sum _{j,k}\rho _{kj}\langle e_{j}|{\hat {A}}|e_{k}\rangle \\&=Trace(\rho A)\\\end{aligned}}}

${\displaystyle \langle A\rangle =\mathrm {Trace} (\rho A)=\sum _{i}w_{i}\langle \psi _{i}|A|\psi _{i}\rangle =\sum _{i}w_{i}\langle A\rangle _{\psi _{i}}}$

## 簡例

${\displaystyle \langle \Lambda \rangle =|{\hat {\Lambda }}\psi |^{2}}$

## 位置空間案例

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

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

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

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

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

${\displaystyle {\hat {\mathfrak {P}}}={\frac {\hbar }{i}}{\frac {\mathrm {d} }{\mathrm {d} x}}}$

${\displaystyle \langle P\rangle ={\frac {\hbar }{i}}\int _{-\infty }^{\infty }\psi ^{*}(x)\,{\frac {d\psi (x)}{dx}}\,\mathrm {d} x}$

## 參考文獻

1. Sakurai, J. J.; Napolitano, Jim, Modern Quantum Mechanics 2nd, Addison-Wesley, 2010, ISBN 978-0805382914
2. ^ Gottfried, Kurt; Yan, Tung-Mow. Quantum Mechanics: Fundamentals 2nd, illustrated. Springer. 2003: pp. 65. ISBN 9780387955766.
• Isham, Chris J. Lectures on Quantum Theory: Mathematical and Structural Foundations. Imperial College Press. 1995. ISBN 978-1860940019.