A Modern Approach to Quantum Mechanics · Problem 9.5

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

Physics — Quantum Mechanics · center-of-mass-relative-hamiltonian

Problem

Show that the two-body Hamiltonian (9.26) can be written in the form

\hat{H} = \frac{\hat{\mathbf{P}}^2}{2M} + \frac{\hat{\mathbf{p}}^2}{2\mu} + V(|\hat{\mathbf{r}}|)

where

M = m_1 + m_2

is the total mass,

\mu = \frac{m_1 m_2}{m_1 + m_2}

is the reduced mass,

\hat{\mathbf{P}} = \hat{\mathbf{p}}_1 + \hat{\mathbf{p}}_2

is the total momentum, and

\hat{\mathbf{p}} = \frac{m_2 \hat{\mathbf{p}}_1 - m_1 \hat{\mathbf{p}}_2}{m_1 + m_2}

is the relative momentum.