Model

Wave function

Energy

Particle in a 1D-box

$\Psi \left(x\right)=\sqrt{2/a}\sin \left(\frac{N\text{π}x}{a}\right)$

$E=\frac{N^2{h}^2}{8m{a}^2},N=1,2,3,\cdots$

1D-Rigid Rotor

$\Psi \left(x\right)={\left(1/2\text{π}\right)}^{1/2}{\text{e}}^{iN\varphi }$

$E=2B\left(N+1\right),N=1,2,3$

Harmonic oscillator

$\Psi \left(x\right)={\left(2^{v}v!\right)}^{-1/2}{\left(\alpha /\text{π}\right)}^{1/4}{\text{e}}^{-\alpha x^2/2}{H}_v\left({\alpha }^{1/2}x\right)$

$E=h{\nu }_e\left(N+1/2\right),N=0,1,2,3,\cdots$