Principles of Quantum Mechanics · Problem 12.2.4

Rotational Invariance and Angular Momentum — R. Shankar

Physics — Quantum Mechanics · rotational-invariance-angular-momentum

Problem

Rederive the equivalent of Eq. (12.2.23) keeping terms of order εxεz2\varepsilon_x \varepsilon_z^2. (You may assume εy=0\varepsilon_y = 0.) Use this information to rewrite Eq. (12.2.24) to order εxεz2\varepsilon_x \varepsilon_z^2. By equating coefficients of this term deduce the constraint −2LzPxLz+PxLz2+Lz2Px=ℏ2Px-2L_z P_x L_z + P_x L_z^2 + L_z^2 P_x = \hbar^2 P_x This seems to conflict with statement (1) made above, but not really, in view of the identity −2ΛΩΛ+ΩΛ2+Λ2Ω≡[Λ,[Λ,Ω]]-2\Lambda\Omega\Lambda + \Omega\Lambda^2 + \Lambda^2\Omega \equiv [\Lambda, [\Lambda, \Omega]] Using the identity, verify that the new constraint coming from the εxεz2\varepsilon_x \varepsilon_z^2 term is satisfied given the commutation relations between PxP_x, PyP_y, and LzL_z.

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