This Quantum World/Implications and applications/Why energy is quantized

Why energy is quantizedEdit

Limiting ourselves again to one spatial dimension, we write the time independent Schrödinger equation in this form:

 

Since this equation contains no complex numbers except possibly   itself, it has real solutions, and these are the ones in which we are interested. You will notice that if   then   is positive and   has the same sign as its second derivative. This means that the graph of   curves upward above the   axis and downward below it. Thus it cannot cross the axis. On the other hand, if   then   is negative and   and its second derivative have opposite signs. In this case the graph of   curves downward above the   axis and upward below it. As a result, the graph of   keeps crossing the axis — it is a wave. Moreover, the larger the difference   the larger the curvature of the graph; and the larger the curvature, the smaller the wavelength. In particle terms, the higher the kinetic energy, the higher the momentum.

Let us now find the solutions that describe a particle "trapped" in a potential well — a bound state. Consider this potential:


 


Observe, to begin with, that at   and   where   the slope of   does not change since   at these points. This tells us that the probability of finding the particle cannot suddenly drop to zero at these points. It will therefore be possible to find the particle to the left of   or to the right of   where classically it could not be. (A classical particle would oscillates back and forth between these points.)

Next, take into account that the probability distributions defined by   must be normalizable. For the graph of   this means that it must approach the   axis asymptotically as  

Suppose that we have a normalized solution for a particular value   If we increase or decrease the value of   the curvature of the graph of   between   and   increases or decreases. A small increase or decrease won't give us another solution:   won't vanish asymptotically for both positive and negative   To obtain another solution, we must increase   by just the right amount to increase or decrease by one the number of wave nodes between the "classical" turning points   and   and to make   again vanish asymptotically in both directions.

The bottom line is that the energy of a bound particle — a particle "trapped" in a potential well — is quantized: only certain values   yield solutions   of the time-independent Schrödinger equation: