Quantum Particle in a Multi-step Potential Well

potential heights

u

1

0

u

2

0

u

3

0

u

4

0

u

5

0

well regions

a

0

b

1

c

2

d

3

e

4

quantum number

n

1

This Demonstration shows solutions of the one-dimensional Schrödinger equation in a multi-step potential well, confined between

-3⩽x⩽5

. The controls on the left determine the heights and widths of the segments of the potential well. Select a quantum number

n

(from 1 to 20) to plot the

th

n

eigenfunction of the Schrödinger equation

ψ

n

(x)

. The Schrödinger equation is then solved numerically using Mathematica's built-in function NDEigensystem. Energies are expressed in units of

100

2

ℏ

/2m

. The wavefunctions are plotted in red, in arbitrary units, with the black curves showing the corresponding probability densities.