StatMechGuy said:What does the "N" operator look like? Is it just [tex]a^\dagger a[/tex] or something else?
gulsen said:[itex]n[/itex] is the sum of 3 quantum numbers, [itex]n_x, n_y, n_z[/itex]; for the lowest rung, which is [itex]n=0[/itex], there's no degeneracy. For [itex]n=1[/itex], there's 3 fold degeneracy (one of the quantum numbers is 1, and rest is 0). For [itex]n=2[/itex], there's 6 fold degenertacy (one of them may be 2 and rest 0, or two of them may be 1) and so on. The result is, for nth energy level, there's 3n degeneracy.
014137 said:No that's not right.
You can't work it out for the first couple of cases and then presume the trend continues like that.
In fact, the degeneracy g(n) is:
g(n) = (1/2)(n+1)(n+2)
I was thinking about the cubic infinite square well which is more tricky since it involves the sum of squares, so never mind my comment about accidental degeneracy.014137 said:Really? Which ones?
The formula can be derived like this:
n = n1 + n2 + n3
where 1,2,3 are three orthogonal directions.
Choose n1 then
n2 + n3 = n - n1
Can always pick n - n1 + 1 different pairs of n2, n3.
Sum over n1 from 0 to n:
Sum(n - n1 +1) = Sum(n - 1) - Sum(n1)
=(n - 1)*(n - 1) - (1/2)n(n + 1)
=(1/2)(n+1)(n+2)
Maybe there's a mistake somewhere?
014137 said:Does work for n = 10, look:
State
3 x 10, 0, 0
3 x 8, 1, 1
3 x 6, 2, 2
3 x 4, 3, 3
3 x 2, 4, 4
3 x 0, 5, 5
6 x 9, 1, 0
6 x 8, 2, 0
6 x 7, 3, 0
6 x 7, 2, 1
6 x 6, 4, 0
6 x 6, 3, 1
6 x 5, 4, 1
6 x 5, 3, 2
Degeneracy = 6 x 8 + 3 x 6 = 6 x 11 = 66
The formula for g(n) gives (1/2)*(11)*(12) = 66
So it works.
Degeneracy refers to the phenomenon in which two or more states of a system have the same energy. In the case of the 3d harmonic oscillator, degeneracy occurs when multiple energy levels have the same energy value.
Degeneracy causes the energy spectrum of the 3d harmonic oscillator to have multiple energy levels with the same energy value. This results in a more complex energy spectrum compared to non-degenerate systems.
The degeneracy of the 3d harmonic oscillator is primarily influenced by the symmetry of the system. The more symmetrical the system, the higher the degeneracy is likely to be. Other factors such as the potential energy and angular momentum also play a role in determining the degeneracy.
Degeneracy can impact the dynamics of the 3d harmonic oscillator by affecting the probability of a system transitioning to a different energy level. In some cases, degeneracy may result in a system remaining in a certain energy level for a longer period of time.
Yes, degeneracy is a common phenomenon in many physical systems and is not limited to the 3d harmonic oscillator. Examples include atoms, molecules, and other quantum mechanical systems.