Principles of Quantum Mechanics · Problem 12.5.14

Rotational Invariance and Angular Momentum — R. Shankar

Physics — Quantum Mechanics · rotation-operator-angular-momentum

Problem

Consider a rotation θxi^\theta_x \hat{i}. Under this x→xx \to x y→ycos⁡θx−zsin⁡θxy \to y\cos\theta_x - z\sin\theta_x z→zcos⁡θz+ysin⁡θxz \to z\cos\theta_z + y\sin\theta_x Therefore we must have ψ(x,y,z)→U[R(θxi^)]ψR=ψ(x, ycos⁡θx−zsin⁡θx, zcos⁡θx+ysin⁡θx)\psi(x, y, z) \xrightarrow{U[R(\theta_x \hat{i})]} \psi_R = \psi(x,\, y\cos\theta_x - z\sin\theta_x,\, z\cos\theta_x + y\sin\theta_x) Let us verify this prediction for a special case ψ=Az e−r2/a2\psi = Az\, e^{-r^2/a^2}, which must go into ψR=A(zcos⁡θx−ysin⁡θx) e−r2/a2\psi_R = A(z\cos\theta_x - y\sin\theta_x)\, e^{-r^2/a^2} (1) Expand ψ\psi in terms of Y11Y_1^1, Y10Y_1^0, Y1−1Y_1^{-1}. (2) Use the matrix e−iθxLx/ℏe^{-i\theta_x L_x/\hbar} to find the fate of ψ\psi under this rotation. Check your result against that anticipated above. [Hint: (1) ψ∼Y10\psi \sim Y_1^0, which corresponds to [010]\begin{bmatrix}0\\1\\0\end{bmatrix}. (2) Use Eq. (12.5.42).]

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