Momentum Eigenstates

k

2

animate

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

-

2

ℏ

2m

2

∂

2

x

Ψ(x,t)=-iℏ

∂

∂t

Ψ(x,t)

. The eigenfunctions are

Ψ

k

(x,t)=

1

2π

ikℏx

e

-i

2

k

2

ℏ

2mt

e

, with a continuum of momentum eigenvalues

p

k

=kℏ

,

-∞<k<∞

and energy eigenvalues

E

k

=

2

k

2

ℏ

/2m

. The energy eigenvalues are two-fold degenerate except for

k=0

. The degeneracy corresponds to left-to-right motion when

k>0

and right-to-left motion when

k<0

.

For simplicity, we set

ℏ=m=1

. Plots are shown for the real and imaginary parts of

Ψ

k

(x,t)=

1

2π

ikx-

2

k

t2

e

as blue and purple sinusoidal curves, respectively. A three-dimensional representation is also shown.