Schrodinger and Wave Mechanics

      In mathematical physics, the Schrodinger equation, are the most fundamental equations in non-relativistic quantum mechanics, playing the same role as Hamilton’s laws of motion in non-relativistic classical mechanics. The Schrodinger equation has two ‘forms’, one in which time explicitly appears, and so describes how the wave function of a particle will evolve in time. In general, the wave function behaves like a wave, and so the equation is often referred to as the time dependent Schrodinger wave equation. The other is the equation in which the time dependence has been ‘removed’ and hence is known as the time independent Schrodinger equation and is found to describe, amongst other things, what the allowed energies are of the particle. These are not two separate, independent equations – the time independent equation can be derived readily from the time dependent equation.

Time dependent Schrodinger wave equation:

Time independent Schrodinger equation:

The Finite Potential Well

         The infinite potential well is a valuable model since, with the minimum amount of fuss, it shows immediately the way that energy quantization as potentials do not occur in nature. However, for electrons trapped in a block of metal, or gas molecules contained in a bottle, this model serves to describe very accurately the quantum character of such systems. In such cases the potential experienced by an electron as it approaches the edges of a block of metal, or as experienced by a gas molecule as it approaches the walls of its container are effectively infinite as far as these particles are concerned, at least if the particles have sufficiently low kinetic energy compared to the height of these potential barriers.

        But, of course, any potential well is of finite depth, and if a particle in such a well has 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, the behavior in the quantum mechanical case is not necessarily consistent with our classical physics based expectations. Thus we now proceed to look at the quantum properties of a particle in a finite potential well. To find the wave function for a particle of energy E, we have to solve three equations, one for each of the regions:

The Potential Barrier

         In quantum mechanics, it is possible for the particle to be confined to a limited region in space, in which boundary condition is enough to yield the quantization of energy. Although a particle hypothetically behaving as a point mass would be reflected, a particle actually behaving as a matter wave has a finite probability that it will penetrate the barrier and continue its travel as a wave on the other side in case of rectangular potential barrier.

       From the ilustration result, we note that the incident and reflected waves have the same ‘intensity’ and hence they have the same amplitude. This suggests that the incident de Broglie wave is totally reflected, i.e. that the particle merely travels towards the barrier where it ‘bounces off’, as would be expected classically. There is a non-zero probability of finding the particle in the region x > 0 where, classically, the particle has no chance of ever reaching. The distance that the particle can penetrate into this ‘forbidden’ region is given roughly by 1/2% which, for a subatomic particle can be a few nanometers, while for a macroscopic particle, this distance is immeasurably small.