The chapter starts the precessing spin in

PHYSICS · Subjective · Fermi University Physics · Quantum Mechanics · Symmetry, angular momentum and spin · Symmetry, angular momentum and spin

Quantum Mechanics: From the Photon to the Hydrogen Atom, Ch Spin one-half, the Pauli matrices and two-level systems

Problem

The chapter starts the precessing spin in χ+(n)(θ)=(cos⁡θ2,sin⁡θ2)T\chi_+^{(n)}(\theta) = (\cos\tfrac{\theta}{2}, \sin\tfrac{\theta}{2})^T, the state that reads +ℏ/2+\hbar/2 for certain along the axis at angle θ\theta from zz in the xzxz plane. Using the same method as SxS_x's eigenvectors in this chapter, find χ−(n)(θ)\chi_-^{(n)}(\theta), the normalised eigenspinor of SnS_n with eigenvalue −ℏ/2-\hbar/2, and check it is orthogonal to χ+(n)(θ)\chi_+^{(n)}(\theta). Then, for a spin started in χ−(n)(θ)\chi_-^{(n)}(\theta) in a field B=B0z^\mathbf B = B_0\hat z, derive ⟨Sx⟩(t)\langle S_x\rangle(t), ⟨Sy⟩(t)\langle S_y\rangle(t) and ⟨Sz⟩(t)\langle S_z\rangle(t), and compare with the chapter's result for a spin started in χ+(n)(θ)\chi_+^{(n)}(\theta).

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