Finding wave function and energy of a particle inside a 3D-box using Schrodinger wave equation

Question

Find the energy and wave function, ψ of a particle inside a 3D-box
Z




Inside; v=0 LX

LZ



Y




Outside; v= ∞

LY



X




- Consider a particle confined to a 3-Dimensional box by infinitely high potential barriers.

- We assume that inside the box, the potential is constant and this constant is equal to zero i.e.
V=0

- The barriers have infinite potential meaning that the particle cannot escape.

- Therefore the boundary conditions imposed are;




V (x, y, z) = 0 if
{ 0<x<Lx
0<y<Ly
0<z<Lz
…………………………………. 1

, - Otherwise V=Infinite, when the particle is outside the box

- Schrodinger wave equation in 3-D is given by;


2 2 2
∂ψ ∂ψ ∂ψ 8π²m
 2 + 2 + 2 + (E-V) ψ =0
∂x ∂y ∂z h²


1. When particle is outside the box; V=∞

2 2 2
∂ψ ∂ψ ∂ψ 8π²m
 2 + 2 + 2 + (E-∞) ψ =0 ……………….. 2
∂x ∂y ∂z h²
 This is possible only when ψ=0 i.e. particle is not outside the box



2. When the particle is inside the box; V=0

2 2 2
∂ψ ∂ψ ∂ψ 8π²m
 2 + 2 + 2 + (E-0) ψ =0
∂x ∂y ∂z h²

2 2 2
∂ψ ∂ψ ∂ψ 8π²m
 2 + 2 + 2 + E ψ =0 ………………………. 3
∂x ∂y ∂z h²


 ∇.∇ ψ + K2 ψ = 0 {2nd order differential equation}
According to 1D box:
1. X- component


2
∂ψ 2 nxπ
2 + {K ψ}x = 0 Kx = ……………………….. 4
∂x Lx