Quantum chemistry (particle in a box)

1. Particle in a box

2. Particle in 1D
• Consider a particle moving in 1D box of length (l) with height (∞) at two ends
• Inside the box P.E.(v) = 0
• Outside the box P.E.(V) = ∞
• Particle can move along length (l) which can be known as potential well
• Due to discontinuous P.E. there will be two Schrödinger equation
• Outside the box
HΨ=EΨ
as P.E. is ∞ therefore probability of finding the particle is 0
Boundary conditions are Ψ=0 at x=0 and x=L
• Inside the box
Ψ=Asinkx + Bcoskx
At x=0 Ψ=0
Ψ=Asinkx for all value of x
At x=L Ψ=0
0=AsinkL
Therefor the energy quantised in confined space is
E=n²h²/8mL²

3. Particle in 2D
• Particle in 2D means particle is between x=x and x=y
• Ψ(x,y)
• Energy in 2D can be written as
• E=Ex+Ey
• E= h²/8m[nx²+ ny²/Lx²+Ly²]

4. Particle in 3D
• Particle in 3D means particle is between x,y&z
• Ψ(x,y,z)
• Energy in 3D can be written as
• E=Ex+Ey+Ez
• E= h²/8m[nx²+ ny²+nz²/Lx²+Ly²+Lz²]