Quantum Mechanics: Concepts and Applications · Problem 5.24

Three-Dimensional Problems — Nouredine Zettili

Physics — Quantum Mechanics · wave-function-lz-measurement

Problem

Consider a system which is in the state ψ(x,y,z)=14πzr+13πxr.\psi(x, y, z) = \frac{1}{4\sqrt{\pi}}\frac{z}{r} + \frac{1}{\sqrt{3\pi}}\frac{x}{r}. (a) Express ψ(x,y,z)\psi(x, y, z) in terms of the spherical harmonics then calculate L^2ψ(x,y,z)\hat{L}^2\psi(x, y, z) and L^zψ(x,y,z)\hat{L}_z\psi(x, y, z). Is ψ(x,y,z)\psi(x, y, z) an eigenstate of L^2\hat{L}^2 or L^z\hat{L}_z? (b) Calculate L^±ψ(x,y,z)\hat{L}_\pm\psi(x, y, z) and ⟨ψ∣L^±∣ψ⟩\langle \psi | \hat{L}_\pm | \psi \rangle. (c) If a measurement of the zz-component of the orbital angular momentum is carried out, find the probabilities corresponding to finding the results 00, ℏ\hbar, and −ℏ-\hbar.

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