Question

Assuming that the potential V() is invariant under rotation, let im be a solution of the time-independent Schrödinger equation: Use the commutation relation [L dent of m H 0 to show that the energy Elm is in fact indepen For this problem, you can use the fact that the orbital angular momentum L is a special case of the total angular momentum 1, and so you can use eq 10.15 with the replacementJ-L.j - letc. (10.15)

Answer 1

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[L_+, H] = 0 [L_+, H] psi_Lm = (L_+ H) psi_Lm - (H L_+) psi_Lm = 0 since H psi_Lm = E_Lm psi_Lm L_+ psi_Lm = [l(l + 1) - m(m + 1)]^1/2 Vertical-bar l, m + 1 > therefore [L_+, H]psi_L, m = E_Lm (L_+ psi_Lm) - [l(l + 1) - m(m + 1)]^1/2 H psi_L, m + 1 = E_Lm [l(l + 1) - m(m + 1)]^1/2 psi_L, m + 1 - [l(l + 1) - m(m + 1)]^1/2 E_L, m + 1 psi_L, m + 1 therefore (E_L, m - E_L, m + 1)[l(l + 1) - m(m + 1)]^1/2 psi_L, m + 1 = 0 therefore E_L, m = E_L, m + 1 This is only possible if E_L, m is independent of m. So, E_L, m is independent of m rightarrow Proved.