Time Evolution of a Quantum Free Particle in 2D

wavenumbers

k

x

0

k

y

-0.92

time

0

This Demonstration shows the time evolution and dispersion relation for a 2D quantum free particle represented by

ψ(x,y,t)=

ϕ

k

x

,

k

y

(x,y)

-

E

k

x

,

k

y

t/ℏ

e

, where

ϕ

k

x

,

k

y

(x,y)=

1

2π



k

x

e

1

2π



k

y

e

and

E

k

x

,

k

y

=

2

ℏ



2

k

x

+

2

k

y



2m

are the eigenfunctions and energy levels, respectively, for

-∞<

k

x

,

k

y

<∞

. The left graphic shows the complex wavefunction

ψ(x,y,t)

, where the shape is its modulus and the coloring is according to its argument, the range

0

to

2π

corresponding to colors from red to magenta. The right panel shows a surface plot of the 2D quadratic dispersion relation,

ω(

k

x

,

k

y

)=

E

k

x

,

k

y

ℏ

.