A Modern Approach to Quantum Mechanics · Problem 3.2

Time Evolution — John S. Townsend

Physics — Quantum Mechanics · spin-half-eigenvalue-problem

Problem

Using the ∣+z⟩|{+}\mathbf{z}\rangle and ∣−z⟩|{-}\mathbf{z}\rangle states of a spin-12\frac{1}{2} particle as a basis, set up and solve as a problem in matrix mechanics the eigenvalue problem for S^n=S^⋅n\hat{S}_n = \hat{\mathbf{S}} \cdot \mathbf{n}, where the spin operator S^=S^xi+S^yj+S^zk\hat{\mathbf{S}} = \hat{S}_x\mathbf{i} + \hat{S}_y\mathbf{j} + \hat{S}_z\mathbf{k} and n=sin⁡θcos⁡ϕ i+sin⁡θsin⁡ϕ j+cos⁡θ k\mathbf{n} = \sin\theta\cos\phi\,\mathbf{i} + \sin\theta\sin\phi\,\mathbf{j} + \cos\theta\,\mathbf{k}. Show that the eigenstates may be written as ∣+n⟩=cos⁡θ2∣+z⟩+eiϕsin⁡θ2∣−z⟩∣−n⟩=sin⁡θ2∣+z⟩−eiϕcos⁡θ2∣−z⟩\begin{aligned} |{+}\mathbf{n}\rangle &= \cos\frac{\theta}{2}|{+}\mathbf{z}\rangle + e^{i\phi}\sin\frac{\theta}{2}|{-}\mathbf{z}\rangle\\ |{-}\mathbf{n}\rangle &= \sin\frac{\theta}{2}|{+}\mathbf{z}\rangle - e^{i\phi}\cos\frac{\theta}{2}|{-}\mathbf{z}\rangle \end{aligned} Rather than simply verifying that these are eigenstates by substituting into the eigenvalue equation, obtain these states by directly solving the eigenvalue problem, as in Section 3.6.

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