Introduction to Quantum Mechanics · Problem 4.1

Quantum Mechanics in Three Dimensions — David J. Griffiths

Physics — Quantum Mechanics · canonical_commutation_3d

Problem

(a) Work out all of the canonical commutation relations for components of the operators r\mathbf{r} and p\mathbf{p}: [x,y][x, y], [x,py][x, p_y], [x,px][x, p_x], [py,pz][p_y, p_z], and so on. Answer: [ri,pj]=−[pi,rj]=iℏδij,[ri,rj]=[pi,pj]=0,(4.10)[r_i, p_j] = -[p_i, r_j] = i\hbar\delta_{ij}, \quad [r_i, r_j] = [p_i, p_j] = 0, \qquad (4.10) where the indices stand for xx, yy, or zz, and rx=xr_x = x, ry=yr_y = y, and rz=zr_z = z. (b) Confirm the three-dimensional version of Ehrenfest's theorem, ddt⟨r⟩=1m⟨p⟩,andddt⟨p⟩=⟨−∇V⟩.(4.11)\frac{d}{dt}\langle\mathbf{r}\rangle = \frac{1}{m}\langle\mathbf{p}\rangle, \quad \text{and} \quad \frac{d}{dt}\langle\mathbf{p}\rangle = \langle -\nabla V\rangle. \qquad (4.11) (Each of these, of course, stands for three equations—one for each component.) Hint: First check that the "generalized" Ehrenfest theorem, Equation 3.73, is valid in three dimensions. (c) Formulate Heisenberg's uncertainty principle in three dimensions. Answer: σxσpx≥ℏ/2,σyσpy≥ℏ/2,σzσpz≥ℏ/2,(4.12)\sigma_x\sigma_{p_x} \geq \hbar/2, \quad \sigma_y\sigma_{p_y} \geq \hbar/2, \quad \sigma_z\sigma_{p_z} \geq \hbar/2, \qquad (4.12) but there is no restriction on, say, σxσpy\sigma_x\sigma_{p_y}.

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