Introduction to Quantum Mechanics · Problem 7.44

Time-Independent Perturbation Theory — David J. Griffiths

Physics — Quantum Mechanics · kramers_relation_applications

Problem

(a) Plug s=0s = 0, s=1s = 1, s=2s = 2, and s=3s = 3 into Kramers' relation (Equation 7.113) to obtain formulas for ⟨r−1⟩\langle r^{-1} \rangle, ⟨r⟩\langle r \rangle, ⟨r2⟩\langle r^2 \rangle, and ⟨r3⟩\langle r^3 \rangle. Note that you could continue indefinitely, to find any positive power. (b) In the *other* direction, however, you hit a snag. Put in s=−1s = -1, and show that all you get is a relation between ⟨r−2⟩\langle r^{-2} \rangle and ⟨r−3⟩\langle r^{-3} \rangle. (c) But if you can get ⟨r−2⟩\langle r^{-2} \rangle by some *other* means, you can apply the Kramers relation to obtain the rest of the negative powers. Use Equation 7.57 (which is derived in Problem 7.42) to determine ⟨r−3⟩\langle r^{-3} \rangle, and check your answer against Equation 7.66.

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