References: Shankar, R. (1994), Principles of Quantum Mechanics, Plenum Press. Section 5.1, Exercise 5.1.1.
Having reviewed the background mathematics and postulates of quantum mechanics as set out by Shankar, we can now revisit some of the classic problems in non-relativistic quantum mechanics using Shankar’s approach, as opposed to that of Griffiths that we’ve already studied.
where is the momentum (we’re working in one dimension) and is the mass. To get the quantum version, we replace by the momentum operator and insert the result into the Schrödinger equation:
Since is time-independent, the solution can be written using a propagator:
To find , we need to solve the eigenvalue equation for the stationary states
where is an eigenvalue representing the allowable energies. Since the Hamiltonian is , and an eigenstate of with eigenvalue is also an eigenstate of with eigenvalue , we can write this equation in terms of the momentum eigenstates :
Using this gives
Assuming that is not a null vector gives the relation between momentum and energy:
Thus each allowable energy has two possible momenta. Once we specify the momentum, we also specify the energy and since each energy state is two-fold degenerate, we can eliminate the ambiguity by specifying only the momentum. Therefore the propagator can be written as
We can convert this to an integral over the energy by using 8 to change variables, and by splittling the integral into two parts. For we have
and for we have
Therefore, we get
Here, is the state with energy and momentum and similarly for . In the first line, the first integral is for and corresponds to the part of 9. The second integral is for and corresponds to the part of 9, which is why the limits on the second integral have at the bottom and 0 at the top. Reversing the order of integration cancels out the minus sign in , which allows us to add the two integrals together to get the final answer.