Finite Square Well

A particle confined in a finite well of depth V0V_0 and width LL has a finite number of bound states whose wavefunctions leak into the classically forbidden region.

10.00
2.00
0.00
4
V(x)ψₙ(x) shifted by EₙEₙ

Bound states

The bound-state energies satisfy the transcendental equations

tan⁡ ⁣(kL2)=κk(even),cot⁡ ⁣(kL2)=−κk(odd)\tan\!\left(\tfrac{kL}{2}\right) = \frac{\kappa}{k} \quad (\text{even}), \qquad \cot\!\left(\tfrac{kL}{2}\right) = -\frac{\kappa}{k} \quad (\text{odd})

with k=2mE/ℏk = \sqrt{2mE}/\hbar and κ=2m(V0−E)/ℏ\kappa = \sqrt{2m(V_0 - E)}/\hbar.

t = 0.00 / 8.00·1/–

Potential

10.00
2.00

Wavepacket

0.00
1.00
0.50

Simulation

8.00
120