One-Dimensional Barrier Transmission Problems

Abstract

In the previous two chapters we considered the bound states corresponding to one-dimensional potentials. For example, we showed that for a particle in a onedimensional potential well of finite depth (see Sec. 6.6.2 and Fig. 6.3) there exist a number of discrete states (for E < V ₀), the corresponding wave functions vanishing at large distances from the origin. In this chapter we will consider solutions for E > V ₀ and will show that the corresponding wave functions in the region |x| > a/2 do not vanish at large distances from the origin and E can have any arbitrary value (greater than V ₀). Thus, for such a potential well there exists a finite number of bound states (with E < V ₀) and a continuum of states for which E > V ₀. A good example is the neutron-proton problem. For E < 0 we have the deuteron nucleus (see Example 10.6) and for E > 0 we have the neutron-proton scattering problem (see, e.g. Sec. 24.5.3).

It is possible in quantum mechanics to sneak quickly across a region which is illegal energetically.

Richard Feynman¹