# Time Evolution of a Quantum Free Particle in 1D

Time Evolution of a Quantum Free Particle in 1D

This Demonstration shows the time evolution and dispersion relation of a 1D quantum free particle represented by , where and are the eigenfunctions and energy levels, respectively, for . The upper graphics show two different visualizations of the complex function . On the left, the shape is, and the coloring is according to the argument, with the range to corresponding to colors from red to magenta. On the right, the real and imaginary parts are shown individually. The lower graphic shows the quadratic dispersion relation, .

ψ(x,t)=ϕ(x)e

k

-Et/ℏ

k

ϕ(x)=e

k

1

2π

kx

E=

k

ℏk

2

2

2m

-∞<k<∞

ψ(x,t)

|ψ(x,t)|

0

2π

ω(k)=E/ℏ

k