References: Shankar, R. (1994), Principles of Quantum Mechanics, Plenum Press. Section 5.3, Exercise 5.3.1.
Shakar’s derivation of the probability current in 3-d is similar to the one we reviewed earlier, so we don’t need to repeat it here. We can, however, look at a slight variant where the potential has a constant imaginary part, so that
where is a real function of position and is a real constant. A Hamiltonian containing such a complex potential is not Hermitian.
To see what effect this has on the total probability of finding a particle in all space, we can repeat the derivation of the probability current. From the Schrödinger equation and its complex conjugate, we have
Multiply the first equation by and the second by and subtract to get
As in the case with a real potential, the first term on the RHS can be written as the divergence of a vector:
If we define the total probability of finding the particle anywhere in space as
then we can integrate 4 over all space and use Gauss’s theorem to convert the volume integral of a divergence into a surface integral:
We make the usual assumption that the probability current tends to zero at infinity fast enough for the first integral on the RHS to be zero, and we get
This has the solution
That is, the probability of the particle existing decays exponentially. Although Shankar says that such a potential can be used to model a system where particles are absorbed, it’s not clear how realistic it is since the Hamiltonian isn’t hermitian, so technically the energies in such a system are not observables.