# 自由粒子

## 古典自由粒子

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

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

## 非相對論性的自由粒子

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

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

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

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

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

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

${\displaystyle v_{g}={\frac {\mathrm {d} \omega }{\mathrm {d} k}}={\frac {\mathrm {d} E}{\mathrm {d} p}}=v\,\!}$

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

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

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

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