I3 In the absence of any spin-orbit coupling the bound eigenstatesof the single-electron hydrogen atom are denoted by In, l, m, m.). They haveenergy En-R/n² where R is the Rydberg constant. You will consider theeffect of including a spin-orbit interactionwhich we will treat as a perturbation. Here is a real parameter, Ĺ is theoperator corresponding to orbital angular momentum of the electron, and Sis its spin operator.a) Before the spin-orbit interaction is switched on, what is the degeneracyof the n = 2 energy level? List the possible states using the notationIn, l, mi, m.).Ĥspspin-orbit = BL-S,b) Once the spin-orbit interaction is switched on, the degeneracy of theEn level is partially lifted and I, my, and m, are no longer "good" quan-tum numbers (i.e. they are no longer the eigenvalues of operators thatcommute with the Hamiltonian). What is a new set of good quantumnumbers?c) Consider the perturbed eigenstates that arise from the n = 2, 1 = 1states. Compute, to first order in 3, the shift in the energies from En-2.How many distinct energies are there after the perturbation, and whatis the degeneracy of each?d) For the energy level with the highest degeneracy, write down all theperturbed eigenstates as explicit linear combinations of the originaln = 2, 1 = 1 basis states [2, 1, m, m.).Hint:Ĵ² = (Î + Ŝ)² = Ĺ² + Ŝ² + 2 Î· Ŝ.Some possibly useful Clebsch-Gordan coefficients (ji, m₁; j2, m2|jtot, mtot):(1,m₁; ,, mtot) =± Mtot3(1, m,, mtot) = ±√FMtot

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In the absence of any spin-orbit coupling the bound eigenstates of the single-electron hydrogen atom are denoted by In, 1, m, m.). They have energy En = -R/n² where R is the Rydberg constant. You will consider the effect of including a spin-orbit interaction Ĥspin-orbit = BL-S, which we will treat as a perturbation. Here 3 is a real parameter, Ĺ is the operator corresponding to orbital angular momentum of the electron, and S is its spin operator. a) Before the spin-orbit interaction is switched on, what is the degeneracy of the n = 2 energy level? List the possible states using the notation In, l, mi, m.). b) Once the spin-orbit interaction is switched on, the degeneracy of the En level is partially lifted and 1, my, and m, are no longer "good" quan- tum numbers (i.e. they are no longer the eigenvalues of operators that commute with the Hamiltonian). What is a new set of good quantum numbers? c) Consider the perturbed eigenstates that arise from the n = 2, 1 = 1 states. Compute, to first order in 3, the shift in the energies from En=2. How many distinct energies are there after the perturbation, and what is the degeneracy of each? d) For the energy level with the highest degeneracy, write down all the perturbed eigenstates as explicit linear combinations of the original n = 2, 1 = 1 basis states (2, 1, m, m.). Hint: Ĵ² = (L+S)² = 1² + S² + 2L.Ŝ. Some possibly useful Clebsch-Gordan coefficients (ji, m₁; j2, m2|jtot, mtot): (1, m,, mot) = mot 3 (1, m,,mtot) = ± #V mtot 3

In the absence of any spin-orbit coupling the bound eigenstates of the single-electron hydrogen atom are denoted by In, 1, m, m.). They have energy En = -R/n² where R is the Rydberg constant. You will consider the effect of including a spin-orbit interaction Ĥspin-orbit = BL-S, which we will treat as a perturbation. Here 3 is a real parameter, Ĺ is the operator corresponding to orbital angular momentum of the electron, and S is its spin operator. a) Before the spin-orbit interaction is switched on, what is the degeneracy of the n = 2 energy level? List the possible states using the notation In, l, mi, m.). b) Once the spin-orbit interaction is switched on, the degeneracy of the En level is partially lifted and 1, my, and m, are no longer "good" quan- tum numbers (i.e. they are no longer the eigenvalues of operators that commute with the Hamiltonian). What is a new set of good quantum numbers? c) Consider the perturbed eigenstates that arise from the n = 2, 1 = 1 states. Compute, to first order in 3, the shift in the energies from En=2. How many distinct energies are there after the perturbation, and what is the degeneracy of each? d) For the energy level with the highest degeneracy, write down all the perturbed eigenstates as explicit linear combinations of the original n = 2, 1 = 1 basis states (2, 1, m, m.). Hint: Ĵ² = (L+S)² = 1² + S² + 2L.Ŝ. Some possibly useful Clebsch-Gordan coefficients (ji, m₁; j2, m2|jtot, mtot): (1, m,, mot) = mot 3 (1, m,,mtot) = ± #V mtot 3