# Momentum Eigenstates

Momentum Eigenstates

This Demonstration shows the motions of nonrelativistic one-dimensional momentum eigenfunctions. The time-dependent Schrödinger equation for a free particle in one dimension is given by . The eigenfunctions are , with a continuum of momentum eigenvalues , and energy eigenvalues . The energy eigenvalues are two-fold degenerate except for . The degeneracy corresponds to left-to-right motion when and right-to-left motion when .

-Ψ(x,t)=-iℏΨ(x,t)

ℏ

2

2m

∂

2

∂x

2

∂

∂t

Ψ(x,t)=ee

k

1

2π

ikℏx

-ikℏ/2mt

2

2

p=kℏ

k

-∞<k<∞

E=kℏ/2m

k

2

2

k=0

k>0

k<0

For simplicity, we set . Plots are shown for the real and imaginary parts of as blue and purple sinusoidal curves, respectively. A three-dimensional representation is also shown.

ℏ=m=1

Ψ(x,t)=e

k

1

2π

ikx-kt/2

2