# 态叠加原理

## 理論

### 電子自旋範例

$|\psi\rang= c_{\uparrow}|\uparrow \rang + c_{\downarrow}|\downarrow \rang$

$p_{\uparrow}=|c_{\uparrow}|^2$
$p_{\downarrow}=|c_{\downarrow}|^2$

$|\psi\rangle = {3i\over 5} |\uparrow\rang + {4\over 5} |\downarrow\rang$

$p_{\uparrow}=\left|\;\frac{3i}{5}\;\right|^2=\frac{9}{25}$
$p_{\downarrow}=\left|\;\frac{4}{5}\;\right|^2=\frac{16}{25}$

$p=\frac{9}{25}+\frac{16}{25}=1$

### 非相對論性自由粒子案例

$- \frac{\hbar^2}{2m} \nabla^2 \ \Psi(\mathbf{r},t) = i\hbar\frac{\partial}{\partial t} \Psi (\mathbf{r},t)$

$\Psi(\mathbf{r},t) = e^{i(\mathbf{k}\cdot\mathbf{r} - \omega t)}$

$\frac{\hbar^2 k^2}{2m}=\hbar \omega$

$\Psi(\mathbf{r},t)=\frac{1}{(2\pi)^{3/2}} \int_{\mathbb{K}} A(\mathbf{k})e^{i(\mathbf{k}\cdot\mathbf{r} - \omega t)}\mathrm{d}\mathbf{k}$

$\Psi(x,t) = \frac{1}{\sqrt{2\pi}} \int^{ \infty}_{ - \infty} A(k) ~ e^{i(kx - \omega(k)t)} \ \mathrm{d}k$

$A(k) = \frac{1}{\sqrt{2\pi}} \int^{\,\infty}_{ - \infty} \Psi(x,0) ~ e^{ - ikx}\,\mathrm{d}x$

## 參考文獻

1. ^ 1.0 1.1 1.2 French, Anthony, An Introduction to Quantum Physics, W. W. Norton, Inc., 1978, ISBN 0-393-09106-0 请检查|isbn=值 (帮助)