古典力学から量子力学へ

Newton's 2nd law

$\displaystyle m \frac{d ^2 x}{dt ^2}=F$

Lagrange equation

$\displaystyle \frac{d}{dt} \frac{\partial L}{\partial \dot{q}} - \frac{\partial L}{\partial q} = 0$

Hamilton equations

$\displaystyle \frac{dq}{dt} = + \frac{\partial H}{\partial p}$

$\displaystyle \frac{dp}{dt} = - \frac{\partial H}{\partial q}$

ポアソン括弧を用いると

Hamilton equations with poisson bracket

$\dot{q} = \{ q, H \}$

$\dot{p} = \{ p, H \}$

ハミルトン－ヤコビ方程式

Hamilton-Jacobi equation

$\displaystyle H(q, t) + \frac{\partial S}{\partial t} = 0$

Schrödinger equation (time-independent)

$\displaystyle -\frac{\hbar ^2}{2m} \left( \frac{\partial ^2 \Psi}{\partial q ^2} \right) + V \Psi = E \Psi$

ぶちノート