[Quantum Mechanics] I need the solution for point (c). Use the hint and the result from the previous points, if necessar

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[Quantum Mechanics] I need the solution for point (c). Use the
hint and the result from the previous points, if necessary.

Quantum Mechanics I Need The Solution For Point C Use The Hint And The Result From The Previous Points If Necessar 1 (105.31 KiB) Viewed 25 times

The deuterium atom is composed of a spin-1 nucleus and a spin-ı electron. We denote the orbital angular momentum operator of the electron with ĩ, the electron's spin operator with , and the nucleus spin operator with ô. Throughout this exercise we assume that the deuterium atom is in the electronic ls state. (a) Let I = $ + 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 Iqma). Write down the respective expectation values of the operators from the set { 32, 33,0?, Q3 } for each possible product eigenstate |jm;) |qma). [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 È = 9 +ộ be the total angular momentum of the deuterium atom. Show that [4P] F? = $_+Q2 + (1+Q_ +Î_@+) +25.0:. O2 + (1) Hint: Use the relations )t = , Eiſ, and Q = 0 +iQy. (c) Use Eq. (1) to show that the states | j = ž, m; = }}®]9 = 1, ma = 1) and \j = 1, m; = – }) ®]9 = 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: Í: \jm;) =ħVG Fm;) ( #m; +1)]; (m; + 1)). An analogous relation holds for @ +|qmg). = +