A Modern Approach to Quantum Mechanics · Problem 9.1

Translational and Rotational Symmetry in the Two-Body Problem — John S. Townsend

Physics — Quantum Mechanics · commuting-translation-generators

Problem

Show that successive translations in different directions commute by substituting the series expansion T^(axi)=1−ip^xaxℏ−p^x2ax22ℏ2+⋯\hat{T}(a_x \mathbf{i}) = 1 - \frac{i\hat{p}_x a_x}{\hbar} - \frac{\hat{p}_x^2 a_x^2}{2\hbar^2} + \cdots and the corresponding expression for T^(ayj)\hat{T}(a_y \mathbf{j}) into (9.11) and retaining terms through second order, thereby showing that the generators of translations along different directions commute: [p^x,p^y]=0[\hat{p}_x, \hat{p}_y] = 0.

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