Multiple Dimensional Systems | Modern Physics PDF Download

If x, y, z are independent, then ψ(x, y, z) can be written as X(x) Y(y) Z(z) and Energy Eigen value can be written as  E_x,y,z = E_x + E_y + E_z 
Where, H_xX(x) = E_xX(x), H_yY(y) = E_yY(y), H_zZ(z) = E_zZ(z)
and H = H_x + H_y + H_z and Hψ = E(ψ)

Two Dimensional Free Particle

So, two dimensional free particle is infinitely degenerate.
Three Dimensional Free Particle
The wave function is ψ(x, y, z) is defined as

So, three dimensional free particle is infinitely degenerate. 

Particle in Two Dimensional Box

The two dimensional system is defined as,

The wave function is given as

 where n_x = 1, 2, 3.....  and 1, 2, 3,......
Ground state energy (nx = 1, ny = 1) i.e.,

First excited state (n_x = 1, n_y = 2) i.e.,

(Degeneracy of first excited state is doubly degenerate)
Second excited state n_x = 2, n_y = 2,

Third excited state n_x = 1, n_y = 3,

(Third excited state is doubly generated) 

Particle in Three Dimensional Box  

where n_x = 1,2,3, ..... n_y = 1, 2, 3, .....
For cubic box  L_x = L_y = L_z = L

Where, n_x = 1, 2, 3, ...., n_y =  1, 2, 3, ..., n_z = 1, 2, 3, ....
Ground state energy (n_x = 1, n_y = 1, n_z = 1) i.e., 

First excited state energy (n_x = 1, n_y = 1, n_z = 2) i.e.,

Other configurations are, n_x = 1, n_y = 2, n_z = 1 and n_x = 2, n_y = 1, n_z = 1
Second excited state energy (n_x = 2, n_y = 2, n_z = 1) i.e.,

Other configurations are, n_x = 2, n_y = 1, n_z = 2 and n_x = 1, n_y = 2, n_z = 2

Two Dimensional Harmonic Oscillator   

= (nx + ny + 1)ℏω, where n_x 0,1, 2, 3, ..... and n_y = 0,1, 2, 3, ..... 
= (n + 1)ℏω, where n = 0,1, 2, 3, ........

So, degeneracy of two dimensional harmonic oscillators for n^th state is (n +1)

Three Dimensional Harmonic Oscillators

For isotropic harmonic oscillator, ω_x = ω_y = ω_z 

The degeneracy of the isotropic harmonic oscillator for n^th state is, g_n = 1/2 (n + 1)(n + 2)
Where, n = 0 corresponds to ground state. 

Example: If b = 2a , write down ground, first, second and third excited state energy.

For ground state: 

For first excited state:

For second excited state:

For third excited state:

For fourth excited state:  

Example:

If b = 2a and c = 3a, then write down energy eigenvalue for ground state, first excited state and second excited state.

Example: If the potential of two dimensional harmonic oscillator is then find energy Eigen value. 

For ground state:  E_0,0 = 3/2 ℏω
For first excited state: E_1,0 = 5/2 ℏω
For second excited state: E_0,1 = E_2,0 = 7/2 ℏω (Doubly degenerate)

For third excited state: E_3,0 = 9/2 ℏω
For fourth excited state:  E_2,1 = E_4,0 = 11/2 ℏω (Doubly degenerate)

Example: If the potential of three dimensional harmonic oscillator is,  

then write down unnormalised wavefunction  for ground state, first excited state and second excited state. 

Wave function

For ground state: (n_x, n_y, n_z) then wave function

For first excited state: (nx, ny, nz) = (1, 0, 0) then wave function

For second excited state: (n_x, n_y, n_z)  = (1, 1,0), (2,0,0), then wave function