Question

In a 1D harmonic oscillator, there is an energy degeneracy from solving the angular part of the Schrodinger equation. This energy degeneracy is 2(2$l$ + 1). The energy of the harmonic oscillator is given by $E_{nl} = \hbar\omega \left(2n + l - \frac{3}{2}\right)$ If we let $\Lambda = 2n + l - 2$, we have $E_{nl} = \hbar\omega \left(\Lambda + \frac{1}{2}\right)$, $\Lambda = 0, 1, 2, ...$ (a) Find the number of ways $n$ and $l$ give the same value of $\Lambda$; that is, find the degeneracy in energy corresponding to different combinations of $n$ and $l$ that give the same $\Lambda$. (b) The total degeneracy is therefore the sum of 2(2$l$ + 1) and the degeneracy you computed in part (a). Show that the total degeneracy is $(\Lambda + 1)(\Lambda + 2)$

          In a 1D harmonic oscillator, there is an energy degeneracy from solving the angular part of the
Schrodinger equation. This energy degeneracy is 2(2$l$ + 1). The energy of the harmonic
oscillator is given by
$E_{nl} = \hbar\omega \left(2n + l - \frac{3}{2}\right)$
If we let $\Lambda = 2n + l - 2$, we have
$E_{nl} = \hbar\omega \left(\Lambda + \frac{1}{2}\right)$,  $\Lambda = 0, 1, 2, ...$
(a) Find the number of ways $n$ and $l$ give the same value of $\Lambda$; that is, find the degeneracy in
energy corresponding to different combinations of $n$ and $l$ that give the same $\Lambda$.
(b) The total degeneracy is therefore the sum of 2(2$l$ + 1) and the degeneracy you
computed in part (a). Show that the total degeneracy is
$(\Lambda + 1)(\Lambda + 2)$

In a 1D harmonic oscillator, there is an energy degeneracy from solving the angular part of the
Schrodinger equation. This energy degeneracy is 2(2l + 1). The energy of the harmonic
oscillator is given by
Enl = ħω(2n + l - (3)/(2))
If we let Λ = 2n + l - 2, we have
Enl = ħω(Λ + (1)/(2)), Λ = 0, 1, 2, ...
(a) Find the number of ways n and l give the same value of Λ; that is, find the degeneracy in
energy corresponding to different combinations of n and l that give the same Λ.
(b) The total degeneracy is therefore the sum of 2(2l + 1) and the degeneracy you
computed in part (a). Show that the total degeneracy is
(Λ + 1)(Λ + 2)

Added by Matthew J.

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