Question

Sketch the potential-well diagram of finite height U and length L, obtain the general solution of the Schrödinger equation for a particle of mass m in it

Answer 1

The Finite Potential Well diagram:

For any potential well is of finite depth, and if a particle in such a well has an energy comparable to the height of the potential barriers that define the well, there is the prospect of the particle escaping from the well. This is true both classically and quantum mechanically, though, as you might expect. Thus we now proceed to look at the quantum properties of a particle in a finite potential well.

As, we know that the Schrodinger equation is given by:

i.e. this solution represents a wave associated with the particle heading towards the barrier and a reflected wave associated with the particle heading away from the barrier. Later we will see that these two waves have the same amplitude, implying that the particle is perfectly reflected at the barrier.

The problem here is that the exp(αx) solution grows exponentially with x, and we do not want wave functions that become infinite: it would essentially mean that the particle is forever to be found at x=infinity, which does not make physical sense. So we must put D = 0.

which is not a travelling wave at all. It is a stationary wave that simply diminishes in amplitude for increasing x.

We still need to determine the constants A, B, and C. To do this we note that for arbitrary choice of these coefficients, the wave function will be discontinuous at x = 0. Let us assume the wave function and its first derivative both be continuous at x = 0.