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[Quantum Mechanics] I need the solution for point (d). Use the
hint and the result from the previous points, if necessary.
The deuterium atom is composed of a spin-1 nucleus and a spin- electron. We denote the orbital angular momentum operator of the electron with ĩ, the electron's spin operator with S, and the nucleus’ spin operator with @. Throughout this exercise we assume that the deuterium atom is in the electronic ls state. (a) Let ) = $ + Î be the total angular momentum operator of the electron. We denote the angular momentum eigenstates of the electron by jm;) and the angular momentum eigenstates of the nucleus by gmg). Write down the respective expectation values of the operators from the set {j2,32,0?, Q3} for each possible product eigenstate |jm;) qmg). [9P] Hint: Motivate and use the fact that, for the ls state, the total angular momentum of the electron is equal to its spin angular momentum. (b) Let Ể = À +be the total angular momentum of the deuterium atom. Show that (AP) F2 = 52 + O2 + +(1+Q_+Ì_Q+) +25.0:1 (1) Hint: Use the relations It = ), tij, and Q + = Q. +iQy. (c) Use Eq. (1) to show that the states |j = 1, m; = 1) 1 = 1, m, = 1) and j = 1, m; = -1)q = 1, m, = -1) from part (a) are eigenstates to the operator F2 and calculate the corresponding eigenvalues. Furthermore, use Eq. (1) to find an example for a product state which is not an eigenstate to F2. [8P] Hint: }\jm;) = ħ/GF m;) (i+m;+1) |j (m;+1)). An analogous relation holds for Q+Igma). (d) For states where 1 = 0, the angular coupling between electron and nuclear spin can be described by the perturbation Hamiltonian Ĥ, (n) = 1$. , where y ER. Show that the energy correction in first-order per- turbation theory for a state with total angular momentum quantum number f is given by the expression AE") = (f (f +1) - 1] - [7P] Hint: For simplicity, you may only consider the angular part of the total wave function and neglect the contri- bution of the radial part to expectation values. Furthermore, you may assume that nondegenerate perturbation theory is applicable. = ญ" =