Degeneracy of a Quantum Particle in a Cubic Box

Degeneracy refers to the number of different quantum states that have the same energy value. In the case of a quantum particle in a cubic box, the energy levels are given by:

E = (n_x^2 + n_y^2 + n_z^2) * h^2 / 8mL^2

where n_x, n_y, and n_z are integers that define the quantum state, h is Planck's constant, m is the mass of the particle, and L is the length of the side of the box.

Calculating the Degeneracy

To calculate the degeneracy of a quantum particle in a cubic box with energy three times that of the lowest energy, we need to find the number of different quantum states that have an energy value of E = 3 * h^2 / 8mL^2.

Since the energy levels depend on the values of n_x, n_y, and n_z, we can start by finding all the possible combinations of integers that satisfy the equation:

(n_x^2 + n_y^2 + n_z^2) = 24

To do this, we can use a method known as the "sphere-packing" algorithm, which involves finding all the ways to pack 24 identical spheres into three dimensions.

Using this method, we can find that there are three different combinations of integers that satisfy the equation:

n_x = 2, n_y = 2, n_z = 2
n_x = 3, n_y = 1, n_z = 2
n_x = 4, n_y = 0, n_z = 2

Since each combination of integers corresponds to a different quantum state with the same energy value, the degeneracy of the particle is 3.

Therefore, the correct answer is '3'.