. Consider an operator A = 2me (p L-Lxp)-k, where p, r, and I denote the...

 

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. Consider an operator A = 2me (p× L-Lxp)-k, where p, r, and I denote the Hermitian operators for momentum, position, and angular momentum respectively. One can think of them as 11+22 + 3e3 p=pie1 +22 +13e3 L = L₁₁+ L22 + L3e3 (1) Here .ej = Sij. The components of r, p, and I are operators. A key property of the angular momentum operators is their commutation relation with the ; and pi operators. Let H = p2 be the Hamiltonian for the hydrogen atom, where k = 2me electron and r = r.ŕ. (1) Show that [H, L₁] = 0, [H, A₁] = 0, i = x, y, z. (2) Find out A.Ĺ and Ã. (3)Define B = -m e2 ATE me is the mass of an A and T = (L+ B). Show that B.B + L.Ĺ = 4T.T = h² - k²me (4) Assuming T2) = t(t + 1)ħ²|v), then using (3) show that E = -mek2 2h2 (2t+1)2 . Consider an operator A = 2me (p× L-Lxp)-k, where p, r, and I denote the Hermitian operators for momentum, position, and angular momentum respectively. One can think of them as 11+22 + 3e3 p=pie1 +22 +13e3 L = L₁₁+ L22 + L3e3 (1) Here .ej = Sij. The components of r, p, and I are operators. A key property of the angular momentum operators is their commutation relation with the ; and pi operators. Let H = p2 be the Hamiltonian for the hydrogen atom, where k = 2me electron and r = r.ŕ. (1) Show that [H, L₁] = 0, [H, A₁] = 0, i = x, y, z. (2) Find out A.Ĺ and Ã. (3)Define B = -m e2 ATE me is the mass of an A and T = (L+ B). Show that B.B + L.Ĺ = 4T.T = h² - k²me (4) Assuming T2) = t(t + 1)ħ²|v), then using (3) show that E = -mek2 2h2 (2t+1)2

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