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A more detailed analysis by Quantum Mechanics shows that Planck's hypothesis must be modified slightly. The energy of a simple harmonic oscillator is given by

En = ( n + 1/2 ) h f, n = 0, 1, 2, 3, . . .

Again, n is an integer. An integer like this is usually called a quantum number. The startling point of this result is that the lowest energy is not zero! The lowest possible energy is

Eo = (1/2) h f

This is called the zero point energy. This can be shown graphically as an energy level diagram like that below. Energy level diagrams will be especially useful when we look at atomic and nuclear energies. The important thing such diagrams show is that a system can have only the energies shown. We never find an oscillator with E = 2.7 hf nor with E = 6.3 hf. We only find energies of 0.5 hf or 1.5 hf or 2.5 hf or 3.5 hf and so on. And we never find an oscillator with zero energy! This is not at all what we would expect from our classical example of a simple harmonic oscillator, a mass on a spring!

The energy level diagram for a simple harmonic oscillator shows the quantization of energy and the ground state (or lowest or zero point) energy.

One of the most curious results of Quantum Mechanics is called tunneling. Below is a one-dimensional "potential well" with position along the x-axis plotted horizontally and energy plotted vertically. You might think of a bead sliding along a wire shaped just like this potential energy function. Classically, the bead can slide back and forth, with constant total energy E, until it reaches a position where its total energy is potential energy. At that point its kinetic energy vanishes, it stops, turns around, and continues in the opposite direction. This position is known as a turning point. The object is allowed to move back and forth between the turning points; this region is called the allowed region. It can never get beyond the turning point for that would require potential energy greater than the total energy or a negative kinetic energy. Such regions are called forbidden regions. This is what we would expect from the ideas of energy conservation that we discussed in PHY 1150.

Below is another graph shows a more complicated potential energy function. On the right, the potential energy increases and then decreases. From the basis of class mechanics, there are two allowed regions. The object can oscillate back and forth in the central region just as for the potential energy function above. But for our new potential energy function here, the particle can also exist far to the right. If it is there and moving to the left, it will come to the turning point where E = V and KE = 0 (so v = 0), stop, turn around and go on out to the right forever, never to return. The object can exist in either allowed region. But if we put it in one region it will stay in that region and never be found in the other. There is no way a classical bead on a wire can cross a "forbidden region".

From Classical Mechanics, a particle placed in the central allowed region will stay there; it will never be found in the allowed region on the right. If a particle is placed in the allowed region on the right it will never be found in the central allowed region. Quantum Mechanics makes quite different predictions!

However, if we analyze this same situation with Quantum Mechanics we find that there is some non-zero probability that a particle placed in the central well with the energy shown will later be found off to the right. The particle has, somehow, "tunnelled" through the potential energy barrier! Because of the energies involved we can only look for the particle in the central well or out to the right. We can not look within the forbidden region. But we know its total energy is less than the potential energy in the forbidden region. Yet a particle placed in the central well will later be found off to the right, moving away. It will have "leaked" through the potential barrier! This curious behavior will be quite useful when we study radioactive nuclear decay.

This same curious behavior is observed with electromagnetic radiation (light). The sketch below shows light incident on a glass-air interface at an angle of incidence greater than the critical angle; total internal reflection occurs. No light continues into the air; the air is a "forbidden region". However, if another piece of glass is brought close to the first--so the air gap between the two pieces is small--light will pass on through the gap. Light is not detected in the air in the first case yet it is detected beyond the air gap in the second case. This is known as frustrated total internal reflection.

Frustrated total internal reflection occurs when there is a narrow air gap between two pieces of glass. Light, which should never make it into the air, is observed to pass through the second piece of glass.


Return to Ch 28, Quantum Mechanics

(c) Doug Davis, 2002; all rights reserved