Particles in a Box - Civil Engineering (CE) PDF Download

Motion of an electron in an one dimensional potential well of infinite height (Particle in a box)
Consider an electron of mass m, moving along positive x-axis between two walls of infinite height, one located at x = 0 and another at x = a. Let potential energy of the electron is assumed to be zero in the region in-between the two walls and infinity in the region beyond the walls

Region beyond the walls:
The Schrodinger wave equation representing the motion of the particle in the region beyond the two walls is given by

The only possible solution for the above equation is Ψ = 0.
If Ψ = 0, the probability of finding the particle in the region x = 0 and x > a is zero. i.e., particle cannot be found in the region beyond the walls.

Region between two walls:

The Schrodinger’s wave equation representing the motion of the particle in the region between the two walls is given by

Solution of the equation (2.8) is of the form
Ψ = A sin αx + B cos αx

Where A and B are unknown constants to be determined. Since particle cannot be found inside the walls
at x = 0 , Ψ = 0               I
and x = a , Ψ = 0            II

The equations are called boundary conditions. Using the I boundary condition in equation (2.10), we get

0 = A sin 0 + B cos 0
∴ B = 0

Therefore equation (2.10) becomes
Ψ = A sin αx

Using the boundary condition II in equation (2.11) we get

Therefore correct solution of the equation (2.8) can be written as

The above equation represents Eigenfunction, where n=1 , 2 , 3 . . .
(n=0 is not acceptable because, for n=0 the wavefunction Ψ becomes zero for all values of x. Then partial cannot be found anywhere)
Substituting for α in equation (2.9) we get

Therefore energy Eigenvalues are represented by the equation

Where n = 1 , 2 , 3 . . .
It is clear from the above equation that particle can have only discrete values of energies. The lowest energy that particle can have corresponds to n = 1 and is called zero-point energy denoted by E₀. It is given by

It is also called groud state energy. Therefore expression for energy can be written as
En = n²E₀

When n = 2 electron is said to be in first excited state, when n = 3 electron is said to be in second excited state, and so on.

Normalization of wavefunction: We know that particle is definitely found somewhere in space between the walls at x = 0 to a

Therefore Normalised wavefunction is given by

Wavefunction, Probability density and energy of the particle for different energy levels:
For n=1 energy is given by E_n = h²/8ma² = K
Wavefunction and probability density for different values of x is given by

similarly for n=2

Wavefunctions, probability density and energies are as shown in the figure.

Figure 2.2: Graphical Representation of Wavefunctions and probability density at different energies

 2.6 Energy eigenvalues of a Free particle
Consider a particle of mass m moving along positive x-axis. Particle is said to be free if it is not under the influence of any field or force. Therefore for a free particle potential energy can be considered to be constant or zero. The Schrodinger wave equation for a free particle is given by.

(2.14)

The solution of the equation (2.14) is of the form
Ψ = A sin αx + B cos αx

where A and B are unknown constants to be determined. Since there are no boundary conditions A, B and α can have any values. Energy of a particle is given by

E = α²h²/8π²m 

Since there is no restriction on α there is no restriction on E.
Therefore energy of the free particle is not quantized. i.e., free particle can have any value of energy.