Simple Systems

The total energy of one mechanical particle is made up of the kinetic and the potential energy. Technically we move to the space-representation

$$(\!x\vert \underset{\sim}{H} \vert\psi\!> = \int \mathrm{d} x... ...\vert\psi\!> = \int \mathrm{d} xx \, H(x,x') \psi(x') \quad .$$ (6)

Nun berechnen wir die gewöhnliche Funktion von zwei Variablen H(x,x') des Operators $\underset{\sim}{H} =\underset{\sim}{H_0} +\underset{\sim}{V}$, der kinetischen und potentiellen Energie. For the kinetic energy operator we know the relation to the momentum operator, $\underset{\sim}{H_0} =\underset{\sim}{p} ^2/2m$.

$$H(x,x')= (\!x\vert \underset{\sim}{H_0} \vert x'\!) + (\!x\vert \underset{\sim}{V} \vert x'\!)$$ (7)