Particle in a Rectangular Three - Dimensional Infinite Well

PARTICLE IN A RECTANGULAR THREE - DIMENSIONAL INFINITE WELL

Let a particle of mass m be in motion in a rectangular deep potential (Fig. 6.14) with sides of lengths a, b, c, parallel to the x,y and z-axes respectively.

If there is no force acting on the particle inside the box, so that in the region.

0 < x < a

0 < y < b

0 < 2 < c

the potential energy V(x, y, z) = 0

and outside the box V (x, y, z) = ∝

Wave Equation of the Particle

For the motion of the particle inside the box, the Schrodinger time - independent wave equation is:

It is assumed that the wave-function Ψ (x, y, z) is equal to the product of three functions X, Y, and Z each of which is a function of one variable only.

Thus we have

Substituting eqn (3) in eqn. (2), we have,

Note: We used ordinary derivatives instead of partial derivatives because each of the functions X, Y and Z is a function of one variable only.

Dividing eqn. (4) by XYZ, we get

In this equation is a constant for a particular value of the kinetic energy.

Since the velocity of the particle, being a vector quantity, can be resolved into three components along the coordinate axes, the kinetic energy E is expressed as the sum of the corresponding terms Ex, Ey and Ez.

Hence,

Therefore, from equations (5) and (6), we get

This equation gives three independent equations:

The eqn. (7) is the equation for the one-dimensional case. The boundary condition applicable to the solution is:

X (0) = X (a) = 0

So the eigen values of Ex are given by

where nx = 1, 2, 3, ......

and the corresponding normalized eigen functions are given by:

The solution for Y and Z are of the same form, therefore, we have:

Eigen Values of Energy 

Substituting the expressions for Ex, Ey and Ez in eqn. (6), we have:

where

nx = 1, 2, 3, ……..

ny = 1, 2, 3, ……..

nz = 1, 2, 3, ……..

This equation gives the eigen values of the energy of the particle. These values are called the energy-levels of the particle.

Wave function

The total normalized wave-function inside the box for the stationary states is given by:

where nx, ny and nz are integers.

The wave-function is zero outside the box. It is easily proved that the wave-function is normalized, because:

From Eqs. (16) and (17) we get the following conclusions: 

1. Three integers nx, ny and nz, which are called quantum numbers, are required to describe each stationary state. If we change the sign of the quantum numbers, there is no change in the energy and in the wave-function except that the minus will appear on the right hand side of Eq. (17). 

Therefore, all the stationary states are given by the positive integral values of nx, ny and nz.

No quantum number can be zero, because if any one of them is taken zero, then Ψ (x, y, z) = 0, which would mean that the particle does not exist in the box. 

2. The lowest possible energy, i.e., the energy in the ground state, occurs when nx = ny =  nz = 1 and it depends on the values of a, b and c. 

3. If the particle is confined in a cubical box in which a = b = c = a, the eigen-values of energy are given by:

In this case energy of the particle in the ground state is given by:

No other state will have this energy, and this state has only one wave-function. Therefore, the ground state and the energy-level are said to be non-degenerate. 

4. In a cubical box, the energy depends on the sum of the squares of the quantum numbers. Consequently the particle having the same energy in an excited state will have several different stationary states, or different wave-functions. Such states and energy-levels are said to be degenerate.