Introduction to Quantum Mechanics · Problem 4.64

Quantum Mechanics in Three Dimensions — David J. Griffiths

Physics — Quantum Mechanics · angular_momentum_measurements

Problem

The electron in a hydrogen atom occupies the combined spin and position state R21(1/3 Y10 χ++2/3 Y11 χ−).R_{21}\left(\sqrt{1/3}\, Y_1^0\, \chi_+ + \sqrt{2/3}\, Y_1^1\, \chi_-\right). (a) If you measured the orbital angular momentum squared (L2L^2), what values might you get, and what is the probability of each? (b) Same for the zz component of orbital angular momentum (LzL_z). (c) Same for the spin angular momentum squared (S2S^2). (d) Same for the zz component of spin angular momentum (SzS_z). Let J≡L+S\mathbf{J} \equiv \mathbf{L} + \mathbf{S} be the total angular momentum. (e) If you measured J2J^2, what values might you get, and what is the probability of each? (f) Same for JzJ_z. (g) If you measured the position of the particle, what is the probability density for finding it at r,θ,ϕr, \theta, \phi? (h) If you measured both the zz component of the spin and the distance from the origin (note that these are compatible observables), what is the probability per unit rr for finding the particle with spin up and at radius rr?
Intuitive explanation

The key insight is that this state is a superposition of eigenstates for each observable—the angular part mixes two different spherical harmonics with different mℓm_\ell values, while the spin part mixes spin-up and spin-down. To find measurement outcomes, identify which eigenstates appear in the superposition, read off their eigenvalues, and square the coefficients to get probabilities. The radial function R21R_{21} is shared by both terms, so it affects position and combined measurements but not the angular/spin observables separately.

Intuitive explanation

The key insight is that this state is a superposition of eigenstates for each observable—the angular part mixes two different spherical harmonics with different mℓm_\ell values, while the spin part mixes spin-up and spin-down. To find measurement outcomes, identify which eigenstates appear in the superposition, read off their eigenvalues, and square the coefficients to get probabilities. The radial function R21R_{21} is shared by both terms, so it affects position and combined measurements but not the angular/spin observables separately.

Related theory

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