# Time Evolution of a Quantum Free Particle in 2D

Time Evolution of a Quantum Free Particle in 2D

This Demonstration shows the time evolution and dispersion relation for a 2D quantum free particle represented by , where and are the eigenfunctions and energy levels, respectively, for . The left graphic shows the complex wavefunction , where the shape is its modulus and the coloring is according to its argument, the range to corresponding to colors from red to magenta. The right panel shows a surface plot of the 2D quadratic dispersion relation, .

ψ(x,y,t)=ϕ(x,y)e

k,k

x

y

-Et/ℏ

k,k

x

y

ϕ(x,y)=ee

k,k

x

y

1

2π

kx

x

1

2π

ky

y

E=

k,k

x

y

ℏk+k

2

2

x

2

y

2m

-∞<k,k<∞

x

y

ψ(x,y,t)

0

2π

ω(k,k)=E/ℏ

x

y

k,k

x

y