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Section 5: Introducing Quantum Mechanics

As we saw in the previous section, there is strong evidence that atoms can behave like waves. So, we shall take the wave nature of atoms as a fact and turn to the questions of how matter waves behave and what they mean.

Werner Heisenberg (left) and Erwin Schrödinger (right).

Figure 16: Werner Heisenberg (left) and Erwin Schrödinger (right).

Source: © Left: German Federal Archive, Creative Commons Attribution ShareAlike 3.0 Germany License. Right: Francis Simon, courtesy AIP Emilio Segrè Visual Archives. More info

Mathematically, waves are described by solutions to a differential equation called the "wave equation." In 1925, the Austrian physicist Erwin Schrödinger reasoned that since particles can behave like waves, there must be a wave equation for particles. He traveled to a quiet mountain lodge to discover the equation; and after a few weeks of thinking and skiing, he succeeded. Schrödinger's equation opened the door to the quantum world, not only answering the many paradoxes that had arisen, but also providing a method for calculating the structure of atoms, molecules, and solids, and for understanding the structure of all matter. Schrödinger's creation, called wave mechanics, precipitated a genuine revolution in science. Almost simultaneously, a totally different formulation of quantum theory was created by Werner Heisenberg: matrix mechanics. The two theories looked different but turned out to be fundamentally equivalent. Often, they are simply referred to as "quantum mechanics." Schrödinger and Heisenberg were awarded the Nobel Prize in 1932 for their theories.

In wave mechanics, our knowledge about a system is embodied in its wavefunction. A wavefunction is the solution to Schrödinger's equation that fits the particular circumstances. For instance, one can speak of the wavefunction for a particle moving freely in space, or an electron bound to a proton in a hydrogen atom, or a mass moving under the spring force of a harmonic oscillator.

To get some insight into the quantum description of nature, let's consider a mass M, moving in one dimension, bouncing back and forth between two rigid walls separated by distance L. We will refer to this idealized one-dimensional system as a particle in a box. The wavefunction must vanish outside the box because the particle can never be found there. Physical waves cannot jump abruptly, so the wavefunction must smoothly approach zero at either end of the box. Consequently, the box must contain an integral number of half-wavelengths of the particle's de Broglie wave. Thus, the de Broglie wavelength must obey , where L is the length of the box and n = 1, 2, 3... . The integer n is called the quantum number of the state. Once we know the de Broglie wavelength, we also know the particle's momentum and energy. math icon See the math

The first three allowed de Broglie wave modes for a particle in a box.

Figure 17: The first three allowed de Broglie wave modes for a particle in a box.

Source: © Daniel Kleppner. More info

The mere existence of matter waves suggests that in any confined system, the energy can have only certain discrete values, that is the energy is quantized. The minimum energy is called the ground state energy. For the particle in the box, the ground state energy is (hL)2/8M. The energy of the higher-lying states increases as n2. For a harmonic oscillator, it turns out that the energy levels are equally spaced, and the allowed energies increase linearly with n. For a hydrogen atom, the energy levels are found to get closer and closer as n increases, varying as 1/n2.

If this is your first encounter with quantum phenomena, you may be confused as to what the wavefunction means and what connection it could have with the behavior of a particle. Before discussing the interpretation, it will be helpful to look at the wavefunction for a system slightly more interesting than a particle in a box.

The harmonic oscillator

A simple harmonic oscillator (bottom) and its energy diagram (top).

Figure 18: A simple harmonic oscillator (bottom) and its energy diagram (top).

Source: © Daniel Kleppner. More info

In free space, where there are no forces, the momentum and kinetic energy of a particle are constant. In most physically interesting situations, however, a particle experiences a force. A harmonic oscillator is a particle moving under the influence of a spring force as shown in Figure 18. The spring force is proportional to how far the spring is stretched or compressed away from its equilibrium position, and the particle's potential energy is proportional to that distance squared. Because energy is conserved, the total energy, E = K + V, is constant. These relations are shown in the energy diagram in Figure 18.

The energy diagram in Figure 18 is helpful in understanding both classical and quantum behavior. Classically, the particle moves between the two extremes (-a, a) shown in the drawing. The extremes are called "turning points" because the direction of motion changes there. The particle comes to momentary rest at a turning point, the kinetic energy vanishes, and the potential energy is equal to the total energy. When the particle passes the origin, the potential energy vanishes, and the kinetic energy is equal to the total energy. Consequently, as the particle moves back and forth, its momentum oscillates between zero and its maximum value.

Low-lying energy levels of a harmonic oscillator.

Figure 19: Low-lying energy levels of a harmonic oscillator.

Source: © Daniel Kleppner. More info

Solutions to Schrödinger's equation for the harmonic oscillator show that the energy is quantized, as we expect for a confined system, and that the allowed states are given by where is the frequency of the oscillator and n = 0, 1, 2... . The energy levels are separated by , as Planck had conjectured, but the system has a ground state energy 1/2 , which Planck could not have known about. The harmonic oscillator energy levels are evenly spaced, as shown in Figure 19.

What does the wavefunction mean?

If we measure the position of the mass, for instance by taking a flash photograph of the oscillator with a meter stick in the background, we do not always get the same result. Even under ideal conditions, which includes eliminating thermal fluctuations by working at zero temperature, the mass would still jitter due to its zero point energy. However, if we plot the results of successive measurements, we find that they start to look reasonably orderly. In particular, the fraction of the measurements for which the mass is in some interval, s, is proportional to the area of the strip of width s lying under the curve in Figure 20, shown in blue. This curve is called a probability distribution curve. Since the probability of finding the mass somewhere is unity, the height of the curve must be chosen so that the area under the curve is 1. With this convention, the probability of finding the mass in the interval s is equal to the area of the shaded strip. It turns out that the probability distribution is simply the wavefunction squared.

The ground state wavefunction of a harmonic oscillator (left) and the corresponding probability distribution (right).

Figure 20: The ground state wavefunction of a harmonic oscillator (left) and the corresponding probability distribution (right).

Source: © Daniel Kleppner. More info

Here, we have a curious state of affairs. In classical physics, if one knows the state of a system, for instance the position and speed of a marble at rest, one can predict the result of future measurements as precisely as one wishes. In quantum mechanics, however, the harmonic oscillator cannot be truly at rest: The closest it can come is the ground state energy 1/2 . Furthermore, we cannot predict the precise result of measurements, only the probability that a measurement will give a result in a given range. Such a probabilistic theory was not easy to accept at first. In fact, Einstein never accepted it.

Aside from its probabilistic interpretation, Figure 20 portrays a situation that could hardly be less like what we expect from classical physics. A classical harmonic oscillator moves fastest near the origin and spends most of its time as it slows down near the turning points. Figure 20 suggests the contrary: The most likely place to find the mass is at the origin where it is moving fastest. However, there is an even more bizarre aspect to the quantum solution: The wavefunction extends beyond the turning points. This means that in a certain fraction of measurements, the mass will be found in a place where it could never go if it obeyed the classical laws. The penetration of the wavefunction into the classically forbidden region gives rise to a purely quantum phenomenon called tunneling. If the energy barrier is not too high, for instance if the energy barrier is a thin layer of insulator in a semiconductor device, then a particle can pass from one classically allowed region to another, tunneling through a region that is classically forbidden.

The quantum description of a harmonic oscillator starts to look a little more reasonable for higher-lying states. For instance, the wavefunction and probability distribution for the state n = 10 are shown in Figure 21.

The wavefunction (left) and probability distribution (right) of a harmonic oscillator in the state n = 10.

Figure 21: The wavefunction (left) and probability distribution (right) of a harmonic oscillator in the state n = 10.

Source: © Daniel Kleppner. More info

Although the n = 10 state shown in Figure 21 may look weird, it shows some similarities to classical behavior. The mass is most likely to be observed near a turning point and least likely to be seen near the origin, as we expect. Furthermore, the fraction of time it spends outside of the turning points is much less than in the ground state. Aside from these clues, however, the quantum description appears to have no connection to the classical description of a mass oscillating in a real harmonic oscillator. We turn next to showing that such a connection actually exists.


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