Suppose that a system of two masses as described in has a traditional Hamiltonian in the q, p...

Suppose that a
system of two masses as described in has a traditional Hamiltonian

in the q, p system,
where r = r2 − r1

(a) Write the extended Hamiltonian K(q, p) in the q, p system, and use
the transformation equations to express the same extended Hamiltonian in the Q,
P system as K(Q, P).

(b) Show that
K(Q, P) can be written as K = P0 + HR(R, P) + Hr(r, p) where HR depends only on
the variables R, P and the Hr depends only on the variables r, p. Such systems
are called separable. (b) Use the extended Hamilton equations to show that P0
and P are conserved. Is Hr a conserved quantity?

(c) Use K(Q, P) to
write the Schroedinger equation in the Q, P system, making use of the quantum
substitutions in eqn (17.94) and P0 → −ih¯(∂/∂ Q0) =
−ih¯(∂/∂t).