Non-relativistic Single Particle Quantum Mechanics

The Hamiltonian for two particles of mass $m_1$ and $m_2$ interacting via a potential $V(r)$, $r=\vert{\bf r}_1 -{\bf r}_2\vert$, is given by

$$\hat{H}_2=-\frac{\hbar^2}{2m_1}\Delta_1- \frac{\hbar^2}{2m_2}\Delta_2 + V(r),$$ (1.1)

where $r$ is the distance between the two particles with positions ${\bf r}_1$ and ${\bf r}_2$, and $\Delta_i$is the Laplace operator with respect to coordinate ${\bf r}_i$, cf. the textbook Landau-Lifshitz III [1]. This is reduced to a single particle problem by introducing center-of-mass and relative coordinates,

$${\bf r} \equiv {\bf r}_1 -{\bf r}_2,\quad {\bf R}\equiv \frac{m_1{\bf r}_1+m_2{\bf r}_2}{m_1+m_2},$$ (1.2)

which as in Classical Mechanics leads to a separation

$$\hat{H}_2=-\frac{\hbar^2}{2(m_1+m_2)}\Delta_{\bf R}- \frac{\hbar^2}{2m}\Delta + V(r)\equiv \hat{H}_{\bf R} + \hat{H},$$ (1.3)

where

$$m\equiv \frac{m_1m_2}{m_1+m_2}$$ (1.4)

is called reduced mass and $\Delta_{\bf R}$ and $\Delta$ are the Laplacians with respect to ${\bf R}$ and ${\bf r}$. If we write ${\bf r}=(x,y,z)$ we have

$$\Delta=\frac{\partial^2}{\partial x^2}+ \frac{\partial^2}{\partial y^2}+ \frac{\partial^2}{\partial z^2}.$$ (1.5)

The Hamiltonian $\hat{H}_2$ is now a sum of two independent Hamiltonians.

Exercise: Check Eq. (II.1.3).

Exercise: Prove that the stationary solutions of $\hat{H}_2$ can be written in product form $\Psi({\bf r}_1,{\bf r}_2)=\phi({\bf R})\Psi({\bf r})$.