# 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 (x,t)=, with a continuum of momentum eigenvalues =kℏ, and energy eigenvalues =/2m. 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)

2

ℏ

2m

2

∂

∂

2

x

∂

∂t

Ψ

k

1

2π

ikℏx

e

-i2mt

2

k

2

ℏ

e

p

k

-∞<k<∞

E

k

2

k

2

ℏ

k=0

k>0

k<0

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

ℏ=m=1

Ψ

k

1

2π

ikx-t2

2

k

e