How Are Quantum Numbers Sequenced in a Three-Dimensional Cubic Well?

[TheClincher]

This is a question relating to Problem 7-1 in Modern Physics (5th ed.) by Tipler and Llewellyn. Find the energies E311, E222, E321 and construct an energy-level diagram for the three-dimensional cubic well that includes the third, fourth, and fifth excited states.

How are these states counted? I don't understand the sequencing for successive excited states. E111 would be the ground state, E211 would be the first excited state - I think I can guess that part. I don't know, however, how the next states are numbered E311, E222, and E321, successively.

 

[Jhero]

Hello,

Thank you for your question. The states are counted based on the quantum numbers associated with each state. In a three-dimensional cubic well, the quantum numbers are n1, n2, and n3. The ground state, E111, has all three quantum numbers equal to 1. The first excited state, E211, has the first quantum number, n1, equal to 2 while the other two quantum numbers, n2 and n3, remain at 1. Similarly, the next excited state, E311, has n1 equal to 3 while n2 and n3 remain at 1. This pattern continues, with the fourth excited state being E222, where all three quantum numbers are equal to 2, and the fifth excited state being E321, with n1 equal to 3 and n2 and n3 both equal to 2.

To construct an energy-level diagram for these states, you can draw three axes representing the three quantum numbers, with the energy increasing as you move up the axes. The ground state, E111, will be at the origin (0,0,0) with the next states branching off from it in different directions. For example, E211 would be at (2,1,1) while E311 would be at (3,1,1). You can continue this pattern to plot the other states and connect them with lines to show the energy levels.

I hope this helps to clarify the sequencing for you. Let me know if you have any other questions.