An electron is in the top rung of the ladder

PHYSICS · Multi-part · 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 Orbital angular momentum, the spherical harmonics and rotations

Problem

An electron is in the top rung of the l=3l = 3 ladder, so that L^2\hat L^2 has the eigenvalue l(l+1)ℏ2l(l+1)\hbar^2 and L^z\hat L_z the eigenvalue mlℏm_l\hbar with ml=3m_l = 3. In this state ⟨Lx⟩=⟨Ly⟩=0\langle L_x\rangle = \langle L_y\rangle = 0, because L^x=12(L^++L^−)\hat L_x = \tfrac12(\hat L_+ + \hat L_-) carries the state to rungs with different L^z\hat L_z eigenvalues, and eigenfunctions belonging to different eigenvalues of a Hermitian operator are orthogonal (Chapter 8); the same holds for L^y\hat L_y. Use the commutator [L^x,L^y]=iℏL^z[\hat L_x, \hat L_y] = i\hbar\hat L_z, which follows from the canonical commutator [x^,p^x]=iℏ[\hat x,\hat p_x] = i\hbar of Chapter 7, and the generalised uncertainty principle ΔA ΔB≥12∣⟨[A^,B^]⟩∣\Delta A\,\Delta B \ge \tfrac12\left|\langle[\hat A,\hat B]\rangle\right| of Chapter 8. ((a)) What is the least value the product ΔLx ΔLy\Delta L_x\,\Delta L_y can take in this state, as a multiple of ℏ2\hbar^2? ((b)) Evaluate ⟨Lx2⟩+⟨Ly2⟩\langle L_x^2\rangle + \langle L_y^2\rangle in this state, as a multiple of ℏ2\hbar^2. ((c)) Given that ⟨Lx2⟩=⟨Ly2⟩\langle L_x^2\rangle = \langle L_y^2\rangle by the symmetry of the state about the zz axis, find ΔLx\Delta L_x, as a multiple of ℏ\hbar to three significant figures. Does the state reach the bound of part (a)? ((d)) Repeat part (c) for the top rung of the l=32l = \tfrac32 ladder, which the algebra permits but orbital motion does not, giving ΔLx\Delta L_x as a multiple of ℏ\hbar to three significant figures.

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