Quantum Harmonic Oscillator in 3 Dimensions

The quantum harmonic oscillator is a fundamental concept in quantum mechanics that describes the behavior of a particle trapped in a potential well. In three dimensions, the potential well takes the form of a three-dimensional harmonic oscillator potential.

a) The fourth excited state is degenerate with a level of degeneracy 3

In a three-dimensional quantum harmonic oscillator, the energy levels are given by the formula:

E(n1, n2, n3) = (n1 + n2 + n3 + 3/2)ħω

where n1, n2, and n3 are the quantum numbers corresponding to the energy levels in each dimension, and ω is the angular frequency of the oscillator.

To determine the degeneracy of a particular energy level, we need to find the number of ways in which the quantum numbers (n1, n2, n3) can be arranged to give the same energy.

For the fourth excited state (n1 = n2 = n3 = 2), there are three possible arrangements: (2, 2, 2), (2, 2, 0), and (2, 0, 2). Therefore, the level is degenerate with a degeneracy of 3.

b) The ground state is non-degenerate

The ground state of a quantum harmonic oscillator corresponds to the lowest energy level, where all the quantum numbers are zero (n1 = n2 = n3 = 0). In this case, there is only one possible arrangement, and therefore the ground state is non-degenerate.

c) The third excited state is 10-fold degenerate

To determine the degeneracy of the third excited state (n1 = n2 = n3 = 3), we need to find the number of ways in which the quantum numbers can be arranged. There are 10 different arrangements: (3, 3, 3), (3, 3, 2), (3, 2, 3), (2, 3, 3), (3, 2, 2), (2, 3, 2), (2, 2, 3), (3, 1, 3), (1, 3, 3), and (3, 1, 2). Therefore, the third excited state is 10-fold degenerate.

d) The ground state is non-degenerate

As mentioned earlier, the ground state corresponds to the lowest energy level where all the quantum numbers are zero. Since there is only one possible arrangement for the ground state, it is non-degenerate.

In conclusion, the correct options are C and D. The fourth excited state is degenerate with a level of degeneracy 3, and the ground state is non-degenerate. The third excited state is 10-fold degenerate.