海森堡繪景

數學表述

$\frac{d}{dt}A={i \over \hbar}[H,\,A]+\left(\frac{\partial A}{\partial t}\right)_\mathrm{classical}$

導引

$\lang A \rang _{t} = \lang \psi (t) | A | \psi(t) \rang$

$| \psi (t) \rang = e^{ - iHt / \hbar} | \psi (0) \rang$

$A$的期望值為

$\lang A \rang _{t} = \lang \psi (0) | e^{iHt / \hbar} A e^{ - iHt / \hbar} | \psi(0) \rang$

$A(t)\ \stackrel{def}{=}\ e^{iHt / \hbar} A e^{ - iHt / \hbar}$

$A(t)$對於時間的導數是

\begin{align} {d \over dt} A(t) & = {i \over \hbar} H e^{iHt / \hbar} A e^{ - iHt / \hbar} + \left(\frac{\partial A}{\partial t}\right)_\mathrm{classical} + {i \over \hbar}e^{iHt / \hbar} A \cdot ( - H) e^{ - iHt / \hbar} \\ & = {i \over \hbar } e^{iHt / \hbar} \left( H A - A H \right) e^{-iHt / \hbar} + \left(\frac{\partial A}{\partial t}\right)_\mathrm{classical} \\ & = {i \over \hbar } \left( H A(t) - A(t) H \right) + \left(\frac{\partial A}{\partial t}\right)_\mathrm{classical} \\ \end{align}

${d \over dt} A(t) = {i \over \hbar } [ H , A(t) ] + \left(\frac{\partial A}{\partial t}\right)_\mathrm{classical}$

${e^B A e^{-B}} = A + [B,A] + \frac{1}{2!} [B,[B,A]] + \frac{1}{3!}[B,[B,[B,A]]]+\cdots$

$A(t)=A+\frac{it}{\hbar}[H,A] - \frac{t^{2}}{2!\hbar^{2}}[H,[H,A]] - \frac{it^3}{3!\hbar^3}[H,[H,[H,A]]] + \cdots$

對易關係

$H=\frac{p^{2}(t)}{2m}+\frac{m\omega^{2}x^{2}(t)}{2}$

${d \over dt} x(t)={i \over \hbar } [H,x(t)]=\frac {p(t)}{m}$
${d \over dt} p(t)={i \over \hbar } [H,p(t)]= - m \omega^{2} x(t)$

${d^2 \over dt^2} x(t) = {i \over m\hbar } [H,p(t)]= - \omega^{2} x(t)$
${d^2 \over dt^2} p(t) = { - im \omega^{2} \over \hbar } [H,x(t)]= - \omega^{2} p(t)$

$\dot{p}(0)= - m\omega^{2} x_0$
$\dot{x}(0)=\frac{p_0}{m}$

$x(t)=x_{0}\cos(\omega t)+\frac{p_{0}}{ m\omega}\sin(\omega t)$
$p(t)=p_{0}\cos(\omega t) - m\omega\!x_{0}\sin(\omega t)$

$[x(t_{1}), x(t_{2})]=\frac{i\hbar}{m\omega}\sin(\omega t_{2} - \omega t_{1})$
$[p(t_{1}), p(t_{2})]=i\hbar m\omega\sin(\omega t_{2} - \omega t_{1})$
$[x(t_{1}), p(t_{2})]=i\hbar \cos(\omega t_{2} - \omega t_{1})$

參考文獻

• Cohen-Tannoudji, Claude; Bernard Diu, Frank Laloe. Quantum Mechanics (Volume One). Paris: Wiley. 1977: 312–314. ISBN 047116433X.
1. ^ Sakurai, J. J.; Napolitano, Jim, Modern Quantum Mechanics 2nd, Addison-Wesley, 2010, ISBN 978-0805382914
2. ^ Parker, C.B. McGraw Hill Encyclopaedia of Physics 2nd. Mc Graw Hill. 1994: 786, 1261. ISBN 0-07-051400-3.
3. ^ Y. Peleg, R. Pnini, E. Zaarur, E. Hecht. Quantum mechanics. Schuam's outline series 2nd. McGraw Hill. 2010: 70. ISBN 9-780071-623582.