Model

Wave function

Energy

Particle in a 1D-box

Ψ ( x ) = 2 / a sin ( N π x a )

E = N 2 h 2 8 m a 2 , N = 1 , 2 , 3 ,

1D-Rigid Rotor

Ψ ( x ) = ( 1 / 2 π ) 1 / 2 e i N ϕ

E = 2 B ( N + 1 ) , N = 1 , 2 , 3

Harmonic oscillator

Ψ ( x ) = ( 2 v v ! ) 1 / 2 ( α / π ) 1 / 4 e α x 2 / 2 H v ( α 1 / 2 x )

E = h ν e ( N + 1 / 2 ) , N = 0 , 1 , 2 , 3 ,