期望值 (量子力學)

$\lang O \rang\ \stackrel{def}{=}\ \lang \psi |\hat{O} | \psi \rang$

量子力學形式論

$\lang O \rang\ \stackrel{def}{=}\ \lang \psi |\hat{O} | \psi \rang$

$\lang e_i |e_j\rang=\delta_{ij}$

$\hat{O} |e_i \rang=O_i|e_i \rang$

$|\psi\rang=\sum_i c_i|e_i\rang$

$\sum_{i} |e_i\rang\lang e_i |=1$

\begin{align} \lang O \rang & = \lang \psi |\hat{O} | \psi \rang \\ & = \sum_{i,j} \lang\psi|e_i\rang\lang e_i |\hat{O}|e_j\rang\lang e_j |\psi \rang \\ & = \sum_{i,j} \lang\psi|e_i\rang\lang e_i |e_j\rang\lang e_j |\psi \rang O_i \\ & = \sum_i |\lang e_i |\psi \rang|^2 O_i \\ \end{align}

系综平均值

$\hat{\rho}= \sum_i w_i |\psi^{(i)}\rang\lang\psi^{(i)} |$

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

$\rho_{jk}= \lang e_j|\rho|e_k\rang=\sum_i w_i \lang e_j|\psi^{(i)}\rang\lang \psi^{(i)} |e_k\rang$

\begin{align}\lang O\rang & = \sum_i w_i \lang \psi^{(i)} |\hat{O} | \psi^{(i)} \rang \\ & = \sum_i\sum_j w_i \lang \psi^{(i)} |e_j\rang\lang e_j |\hat{O}| \psi^{(i)} \rang \\ & = \sum_i\sum_j w_i \lang \psi^{(i)} |e_j\rang\lang e_j | \psi^{(i)} \rang O_j \\ & = \sum_i\sum_j w_i |\lang \psi^{(i)} |e_j\rang|^2 O_j \\ \end{align}

\begin{align}\lang A\rang & = \sum_i w_i \lang \psi^{(i)} |\hat{A} | \psi^{(i)} \rang \\ & = \sum_i\sum_{j,k} w_i \lang \psi^{(i)} |e_j\rang\lang e_j |\hat{A}| e_k\rang\lang e_k |\psi^{(i)} \rang \\ & = \sum_i\sum_{j,k} w_i \lang e_k |\psi^{(i)} \rang\lang \psi^{(i)} |e_j\rang\lang e_j |\hat{A}| e_k\rang \\ & = \sum_{j,k} \rho_{kj}\lang e_j |\hat{A}| e_k\rang \\ & = Trace(\rho A) \\ \end{align}

$\lang A \rang= \mathrm{Trace} (\rho A) = \sum_i w_i \lang \psi_i | A| \psi_i \rang=\sum_i w_i \lang A \rang_{\psi_i}$

簡例

$\lang \Lambda\rang= | \hat{\Lambda} \psi |^2$

位置空間案例

$\psi(x)\ \stackrel{def}{=}\ \lang x|\psi\rang$

$\lang \psi_1| \psi_2 \rang = \int \psi_1^*(x)\psi_2(x) \, \mathrm{d}x$

$\lang x \rang\ \stackrel{def}{=}\ \lang \psi | \hat{x} |\psi \rang$

$\lang x \rang =\int_{ - \infty}^{\infty} \psi^\ast (x) \, x \, \psi(x) \, \mathrm{d}x = \int_{ - \infty}^{\infty} x \, |\psi(x)|^2 \, \mathrm{d}x$

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

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

$\lang P \rang=\frac{\hbar}{i} \int_{-\infty}^{\infty} \psi^*(x) \, \frac{d\psi(x)}{dx}\, \mathrm{d}x$

參考文獻

1. ^ 1.0 1.1 1.2 1.3 1.4 1.5 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.