# 自由粒子

## 古典自由粒子

$\mathbf{p}=m\mathbf{v}\,\!$

$E=\frac{1}{2}mv^2\,\!$

## 非相對論性的自由粒子

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

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

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

$\langle\mathbf{p}\rangle=\langle \Psi | - i\hbar\nabla|\Psi\rangle = \hbar\mathbf{k}\,\!$

$\langle E\rangle=\langle \Psi |i\hbar\frac{\partial}{\partial t}|\Psi\rangle = \hbar\omega\,\!$

$\langle E \rangle =\frac{\langle p \rangle^2}{2m}\,\!$

$v_g= \frac{\mathrm{d}\omega}{\mathrm{d}k} = \frac{\mathrm{d}E}{\mathrm{d}p} = v\,\!$

$v_p=\frac{\omega}{k} = \frac{E}{p} = \frac{p}{2m} = \frac{v}{2}\,\!$

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

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

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