Energy Eigen Values Of A Particle In A Potential Well Of Infinite Depth

If A = 0, the wavefunction will become zero irrespective of the value of x. Hence, A cannot be taken as zero. Therefore,

Here, n cannot be zero as it leads to trivial solution. Hence, the energy eigenvalues may be written as

From this equation, we infer that the energy of the particle is discrete as n can have integer values. In other words, the energy is quantized. We also note that n cannot be zero because in that case, the wave function as well as the probability of finding the particle becomes zero for all values of x. The lowest energy of the particle can possess corresponds to n = 1 is given by

This is called the ground state energy or zero point energy. The first and second excited energies are given by,

The energy levels are like E₁, 4E₁, 9E₁, 16E₁ ....which indicates that the energy levels are not equally spaced.

The eigenfunctions corresponding to the above eigenvalues are given by,

Substituting in the above equation, we get

We apply the normalization condition to fix the value of A, that is