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Three-Dimensional Problems — Nouredine Zettili
Physics — Quantum Mechanics · spin-half-eigenstates-sx-sy
Problem Use the following general relations:
∣ ψ x ⟩ ± = 1 2 [ ∣ 1 2 , 1 2 ⟩ ± ∣ 1 2 , − 1 2 ⟩ ] , ∣ ψ y ⟩ ± = 1 2 [ ∣ 1 2 , 1 2 ⟩ ± i ∣ 1 2 , − 1 2 ⟩ ] |\psi_x\rangle_\pm = \frac{1}{\sqrt{2}}\left[\left|\frac{1}{2},\ \frac{1}{2}\right\rangle \pm \left|\frac{1}{2},\ -\frac{1}{2}\right\rangle\right], \qquad |\psi_y\rangle_\pm = \frac{1}{\sqrt{2}}\left[\left|\frac{1}{2},\ \frac{1}{2}\right\rangle \pm i\left|\frac{1}{2},\ -\frac{1}{2}\right\rangle\right] ∣ ψ x ⟩ ± = 2 1 [ 2 1 , 2 1 ⟩ ± 2 1 , − 2 1 ⟩ ] , ∣ ψ y ⟩ ± = 2 1 [ 2 1 , 2 1 ⟩ ± i 2 1 , − 2 1 ⟩ ] S ^ x ∣ ψ x ⟩ ± = ± ℏ 2 ∣ ψ x ⟩ ± and S ^ y ∣ ψ y ⟩ ± = ± ℏ 2 ∣ ψ y ⟩ ± . \hat{S}_x|\psi_x\rangle_\pm = \pm\frac{\hbar}{2}|\psi_x\rangle_\pm \qquad \text{and} \qquad \hat{S}_y|\psi_y\rangle_\pm = \pm\frac{\hbar}{2}|\psi_y\rangle_\pm. S ^ x ∣ ψ x ⟩ ± = ± 2 ℏ ∣ ψ x ⟩ ± and S ^ y ∣ ψ y ⟩ ± = ± 2 ℏ ∣ ψ y ⟩ ± . Browse the problem library