Introduction to Quantum Mechanics

We have discussed 1D problems in Chapter 7. In this chapter, we discuss mainly 3D problems, although some of the generalities might apply to spaces whose dimensions are difierent.

Generalities

The stationary state form of the Schrödinger equation for a single particle system was given in Eq. (4.24, p. 86). For non-relativistic systems with velocity-independent potential energies, the general form of the Hamiltonian would be

for a particle of mass M. Accordingly, the Schrӧdinger equation for stationary states has the form

The Laplacian operator, ▿2, involves second-order derivatives with respect to the coordinates. This is the difierential equation we want to solve in this chapter for various choices of the potential.

Particle in a 3D box

The problem of a particle in an infinite rectangular-shaped 3D potential well is just an extension of the same problem in 1D. So, the solution can be borrowed from what we have already done in §7.2, but there will be some new features of the solutions which need some attention.

The potential is infinite except within the box defined by

Inside the box, the potential vanishes. The wavefunctions for the stationary states can be written in the form

and the difierential equation corresponding to each factor will be the same as the corresponding 1D problem.