Changing the Hamiltonian to a new frame of reference

[AxiomOfChoice]

Suppose I'm considering particles of mass $\mu_i$, $1 \leq i \leq 3$, located at positions $r_i$. Suppose I ignore the potential between $\mu_1$ and $\mu_2$. Then the Hamiltonian I'd write down would be

$$H = -\frac{1}{2\mu_1}\Delta_1 -\frac{1}{2\mu_2}\Delta_2 - \frac{1}{2\mu_3}\Delta_3 + V_1(r_3 - r_1) + V_2(r_3 - r_2).$$

But what if I instead want to work in a frame of reference in which $\mu_1$ is at rest? How should I go about changing $H$? I'm never very sure of myself when I do these kinds of calculations, so any help would be appreciated...thanks!

 

[tom.stoer]

The are several options, but first of all no particle will be "at rest" at the level of the Hamiltonian; it's a special solution (in classical mechanics) where one momentum p vanishes, i.e. where one particle is at rest. You must not set p=0 in H.

Coordinate changes can be done at three levels
- on the level of the Lagrangian, as usual in classical mechanics
- on the level of the Hamiltonian function, i.e. using canonical transformations
- on the level of the Hamiltonian operator, i.e. using unitary transformations

There is a nice example using unitary transformations showing what really happens when we introduce the c.o.m. frame for a central potential; the new coordinates can be interpreted as r,R,p,P, where R does not appear in H, so P is conserved. That means tha strictly speaking we should not set P=0, but we have a plane wave in R and P.