Calculate the Zero Point Energy of Electron

Suppose an electron is confined in a cube of length L. Calculate the zero point energy of an electron. Discuss the situation when
a. walls are removed and
b. one of the wall is elongated

We know that the total energy of an electron in a cubical box of side length 'L' is-
E = E_x + E_y + E_z = [h² / 8mL²](n²_x + n²_y + n²_z) -----Equation-1
Although the zero value of n_x, n_y or n_z is possible, but not acceptable because the ψ function then becomes zero but an electron is assumed to be already present inside the box. Therefore, the lowest kinetic energy permissible to electron in a cubical box is one with n_x = n_y = n_z = 1. This lowest kinetic energy is called zero point energy, which is given as-
E_zero = 3h² / 8mL²
It shows that the electron inside the box is not at rest even at 0°K. Therefore the position of the electron cannot be precisely known. Since only the mean value of kinetic energy is known, the momentum of electron is also not precisely known. The occurrence of zero point energy is therefore in accordance with Heisenberg's uncertainty principle.

A. When Walls are Removed
The energy of an electron confined between two infinitely large walls. of a distance 'L' along x-axis and supposed to have zero potentia energy is given by-
E_x = n²_x h² / 8mL²
where n_x is a quantum number which can only be positive integer excluding zero. Therefore, a bound electron has only quantised energy levels with values Ex₁, Ex₂ and Ex₃... with n_x = 1, 2, 3 ... respectively i.e. the energy of a bound electron is not continuous, rather discrete or quantised. If the walls of the box are removed and an electron is free to move without any restriction in a field whose potential energy may be assumed to be zero then Schrödinger equation and its solution are given by-
δ²X/δX² + K²_xX = 0
where K²_x = [8π²m/h²] E_x

A = A cos k_x X + B sin k_x X
The arbitrary constant A, B and k²_x can now have any value one chooses to given them.
E_x = k²_x h² / 8π²m
The energy is therefore not quantised in this case. A free electron has a continuous energy spectrum. It can have any value of energy what so every possible. This quantitatively explains the occurrence of continuum in the atomic or molecular spectra on ionisation because electron lost by an atom or molecule is a free electron which can move without any restriction.

B. One of the Wall is Elongated
The occurrence of the three quantum number (n_x, n_y and n_z) in the energy expression of an electron in equation (1) enclosed in a cube shows that each state is characterised by three quantum numbers and several states of identical energy are possible e.g. there are three different states having quantum number (2, 1, 1), (1, 2, 1) and (1, 1, 2) for (n_x, n_y and n_z) each with the same energy 6h² / 8mL². Such a level is said to be the three fold degenerate or triply degenerate. The wave function of these three triply degenerate states are different.
Let one of the walls of the cube are elongated along x-axis by dL i.e. the cube is distorted slightly. For the state (2, 1, 1) the energy of electron in the undistorted cube-
E = E_x + E_y + E_z


The new energy on distribution along x-axis is given by-
E + dE = E_x + dE_x + E_y + E_z

Whereas the new energy for other states i.e. (1,2,1) and (1,1,2) is given by-

Thus, the initial three fold degnerate levels are split on distortion of the cube into a non-degenerate level and doubly degenerate levels.That electron degeneracy is either reduced or removed on slight distortion of the system is a common phenomenon (John-Teller Distortion).

NOTE: Very Important Question for B.Sc. and M.Sc. Exams.