A Modern Approach to Quantum Mechanics · Problem 2.8

Angular Momentum — John S. Townsend

Physics — Quantum Mechanics · normalization-expectation-value-matrix-mechanics

Problem

The column vector representing the state ∣ψ⟩|\psi\rangle is given by ∣ψ⟩→Sz basis15(i2)|\psi\rangle \xrightarrow{S_z\text{ basis}} \frac{1}{\sqrt{5}}\begin{pmatrix} i \\ 2 \end{pmatrix} Using matrix mechanics, show that ∣ψ⟩|\psi\rangle is properly normalized and calculate the probability that a measurement of SxS_x yields ℏ/2\hbar/2. Also determine the probability that a measurement of SyS_y yields ℏ/2\hbar/2.
Intuitive explanation

Normalization is just checking that the inner product ⟨ψ∣ψ⟩=1\langle\psi|\psi\rangle = 1—sum the squared magnitudes of all components. For spin measurements, you'll need to express ∣ψ⟩|\psi\rangle in the eigenbasis of whichever operator you're measuring (SxS_x or SyS_y), then read off the probability as the squared magnitude of the corresponding eigenstate's amplitude. The key insight is that SxS_x and SyS_y have known matrix forms and eigenstates in the SzS_z basis, so you can change bases using the eigenvector matrices, or equivalently compute ∣⟨eigenstate∣ψ⟩∣2|\langle\text{eigenstate}|\psi\rangle|^2 directly.

Intuitive explanation

Normalization is just checking that the inner product ⟨ψ∣ψ⟩=1\langle\psi|\psi\rangle = 1—sum the squared magnitudes of all components. For spin measurements, you'll need to express ∣ψ⟩|\psi\rangle in the eigenbasis of whichever operator you're measuring (SxS_x or SyS_y), then read off the probability as the squared magnitude of the corresponding eigenstate's amplitude. The key insight is that SxS_x and SyS_y have known matrix forms and eigenstates in the SzS_z basis, so you can change bases using the eigenvector matrices, or equivalently compute ∣⟨eigenstate∣ψ⟩∣2|\langle\text{eigenstate}|\psi\rangle|^2 directly.

Related theory

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