Quantum Mechanics: Concepts and Applications · Problem 5.12

Three-Dimensional Problems — Nouredine Zettili

Physics — Quantum Mechanics · orbital-angular-momentum-wave-function

Problem

Consider a particle whose wave function is ψ(x,y,z)=14π2z2−x2−y2r2+3πxzr2.\psi(x, y, z) = \frac{1}{4\sqrt{\pi}}\frac{2z^2 - x^2 - y^2}{r^2} + \sqrt{\frac{3}{\pi}}\frac{xz}{r^2}. (a) 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). Find the total angular momentum of this particle. (b) Calculate L^+ψ(x,y,z)\hat{L}_+\psi(x, y, z) and ⟨ψ∣L^+∣ψ⟩\langle \psi | \hat{L}_+ | \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. (d) What is the probability of finding the particle at the position θ=π/3\theta = \pi/3 and φ=π/2\varphi = \pi/2 within dθ=0.03d\theta = 0.03 rad and dφ=0.03d\varphi = 0.03 rad?

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