Shankar, R. (1994), Principles of Quantum Mechanics, Plenum Press. Chapter 10, Exercise 10.1.3.
Shankar shows that, for a two-particle system, the state vector is an element of the direct product space . Its evolution in time is determined by the Schrödinger equation, as usual, so that
The method by which this equation can be solved (if it can be solved, that is) depends on the form of the potential . If the two particles interact only with some external potential, and not with each other, then is composed of a sum of terms, each of which depends only on or , but not on both. In such cases, we can split into two parts, one of which () depends only on operators pertaining to particle 1 and the other () on operators pertaining to particle 2. If the eigenvalues (allowed energies) of particle are given by , then the stationary states are direct products of the corresponding single particle eigenstates. That is, in general
Thus the two-particle state . Since a stationary state evolves in time according to
the compound two-particle state evolves according to
In this case, the two particles are essentially independent of each other, and the compound state is just the product of the two separate one-particle states.
If is not separable, which will occur if contains terms involving both and in the same term, we cannot, in general, reduce the system to the product of two one-particle systems. There are a couple of instances, however, where such a reduction can be done.
The first instance is if the potential is a function of only, in other words, that the interaction between the particles depends only on the distance between them. Shankar shows that in this case we can transform the system to that of a reduced mass and a centre of mass . We’ve already seen this problem solved by means of separation of variables. The result is that the state vector is the product of a vector for a free particle of mass and of a vector of a particle with reduced mass moving in the potential .
Another case where we can decouple the Hamiltonian is in a system of harmonic oscillators. We’ve already seen this system solved for two masses in classical mechanics using diagonalization of the matrix describing the equations of motion. The classical Hamiltonian is
The earlier solution involved introducing normal coordinates
and corresponding momenta
These normal coordinates are canonical as we can verify by calculating the Poisson brackets. For example
and so on, with the general result
We can invert the transformation to get
Inserting these into 7 we get
We can now subsitute the usual quantum mechanical operators to get the quantum Hamiltonian:
In the coordinate basis, this is
The Hamiltonian is now decoupled and can be solved by separation of variables.
We could have arrived at this result by starting with 7 and promoting and to quantum operators directly, then made the substitution to normal coordinates. We would then start with
The potential term on the right transforms the same way as before, so we get
To transform the two derivatives, we need to use the chain rule a couple of times. To get the first derivatives:
Now the second derivatives:
Combining the two derivatives, we get