The Amazing Story of Quantum Mechanics (11 page)

Say there is a wave associated with us as we walk along the pedestrian pathway circling Lake Harriet in Minneapolis (and we do in fact have a “matter-wave” moving along with us, but of such a small wavelength that it is undetectable). Once we have returned to our starting point, our wave must join up smoothly and perfectly with the wave started when we left for our stroll. If the wave were at its highest point, a crest, when we began our walk, then as we walked around the lake, the wave would oscillate down to a valley, back up to a crest, and so on. Once we are back at our starting point, the wave must again be at a crest. This simple, commonsense aspect of waves (what would it look like if the cycle of the wave were such that our wave was at a valley when we returned to our starting point, where the original wave was a peak?) leads immediately to the consequence that only certain pathways—those that correspond to different wavelengths that start and end correctly—around the lake are possible.

Similarly, a wave associated with an electron in a closed orbit, returning to the same point after a full rotation, can assume only certain wavelengths. Once an orbit has been completed, the wave must be at the same point as when it left. This “single-valued” constraint, that at any point the wave can have only one amplitude (that is, it can’t be a peak and a valley at the same time), restricts the infinite range of possible wavelengths to a very small set of allowed orbits. Just as the guitar string has a lowest pitch—it is impossible to excite a wave on the clamped string lower than its fundamental oscillation—there is a lowest standing-wave electron orbit that can be constructed around the nucleus. Thus, electrons do not continuously lose energy and spiral into the nucleus with ever-decreasing radii, as there is no way for the apparently very real matter-wave to form standing waves with a lower wavelength than the lowest possible orbit. The de Broglie hypothesis of matter-waves saves the stability of atoms and also accounts in a natural way for the line spectra (as shown in Figure 12) of atoms.

Following a seminar presentation by Schrödinger of the matter-wave model of electrons in an atom, Pieter Debye, a senior physicist in the audience, pointed out that if electrons had waves, there should be a corresponding wave equation to describe them (just as Maxwell had found a wave equation for electric and magnetic fields that turned out to accurately describe the properties of electromagnetic waves, that is, light) and challenged Schrödinger to find it. Taking up this assignment, Schrödinger accomplished the deed in just six months. A consequence of Schrödinger’s work was the realization that the electrons could not really be considered as mini-planets in a nuclear solar system. The resulting equation, which now bears his name, would garner him a Nobel Prize, change the future, and lead to philosophical arguments over the nature of measurements of quantum systems that so disturbed Schrödinger that he would claim that he was sorry he brought the whole thing up.

The Schrödinger equation plays
the same role in atomic physics that Newton’s laws of motion play in the mechanics of everyday objects. Those with long memories may recall that back in the seventeenth century it was well-known that the application of external forces was required to change the motion of objects. What was lacking was a method by which one could calculate exactly
how
the motion of any given object would change as a consequence of the pushes and pulls of external forces. Newton found a remarkably elegant expression (the net force is equal to the mass multiplied by the acceleration, or F = m × a) that despite its surface simplicity could account for a wide range of complex motions.

As any student in an introductory physics class can tell you, it is not enough to identify the forces acting in a situation—one must be able to show how these forces will lead to a change in the object’s motion. Armed with Newton’s law, one can determine the trajectory of skiers and boaters, of runners and automobiles, of rockets fired at the moon, and of the moon itself (not to mention falling apples). We know that Newton’s laws of motion are correct, for comparisons of the theoretically calculated motion agree exactly with what is experimentally observed.

Similarly, in the quantum realm, it is not enough to say that there is a wave associated with the motion of all matter whose wavelength is inversely proportional to its momentum. One also needs a process—an equation—by which, if one knows the external forces acting on the object, the resulting behavior of its “matter-wave” can be determined.

Consider a simple hydrogen atom with one proton in its nucleus and one electron attracted to the proton electrostatically, the symbol that Dr. Manhattan etched into his forehead in the
Watchmen
comic and film. Given that we know the nature of the electrostatic attractive force between the negatively charged electron and the positively charged nucleus and that there is a wave associated with the motion of the electron in an atom, what we would like is a “matter-wave equation” that enables us to calculate the properties of the electron in the atom. These calculated properties, such as the average diameter of the atom, or the electron’s average momentum, could then be compared to experimental measurements, in order to test the correctness of the matter-wave approach. Any equation that does not yield testable results is useless from a physics point of view.

Schrödinger found just such an equation, though the procedure for calculating the wave function involves a formula
much
more complicated than Newton’s F = m × a. In fact, one often requires a computer in order to solve Schrödinger’s equation, except in a handful of simple cases. But the fact that the calculations can be difficult does not invalidate the Schrödinger approach. Applying Newton’s law of motion, F = m × a, to the motion of the more than trillion trillion air molecules in a room exceeds the calculating capability of the largest supercomputer. Even though we can’t do the sums, we know that in principle they could be done. The importance of Schrödinger’s equation is that the process by which one calculates an object’s wave function, given the relevant forces acting on it, is known. The math might be hard, but the path is clear.

We will not go into the process, involving mathematical trial and error, physical intuition, and a creative use of the principle of conservation of energy that enabled Schrödinger to develop his equation. It is, indeed, a fascinating story, filled with false starts, suspense, and plenty of sex!
²³
We are interested in how quantum mechanics brought about our futuristic lifestyle of light-emitting diodes, laptop computers, cell phones, and remote controls. Consequently, for our purposes, it is not important how the Schrödinger equation was developed, nor will we try to solve it, even for the simplest case of a single electron moving in a straight line in empty space (and it doesn’t get more plain vanilla than that). Schrödinger’s equation involves the rates of change of the wave function in both space and time; consequently it can’t be solved without calculus. In addition, it involves imaginary numbers (the term mathematicians use when referring to the square root of negative numbers), and thus calls upon considerable imagination to interpret. What we
will
do is discuss the significance of the equation and point out how various solutions lead, in some cases, to semiconductor transistors and, in other situations, to semiconductor lasers.

Enough with the tease: The Schrödinger equation, as it is now universally known, is most often expressed mathematically as follows:
where
ħ
represents our old friend, Planck’s constant (the angled bar through the vertical line in the letter
h
is a mathematical shorthand that indicates that the value of Planck’s constant here should be divided by 2π, that is,
ħ
=
h
/2π); m is the mass of the object (typically this would be the electron’s mass) whose wave function Ψ (pronounced “sigh”) we are interested in determining; V is a mathematical expression that reflects the external forces acting on the object; and
i
is the square root of -1.
²⁴
The rate of change of Ψ in time is represented by ∂Ψ/∂t, while the rate of change of the rate of change of Ψ in space is represented by ∂
²
Ψ/∂x
²
. As messy and complicated as this equation may seem, I have taken it easy and written this formula in its simpler, one-dimensional form, where the electron can move only along a straight line. The version of this equation capable of describing excursions through three dimensions has a few extra terms, but we won’t be solving that equation either.

The Schrödinger equation requires us to know the forces that act on the atomic electrons in order to figure out where the electrons are likely to be and what their energies are. The term labeled V in the Schrödinger equation represents the work done on the electron by external forces, which can change the energy of the atomic electron. For reasons that are not very important right now, V is referred to as the “potential” acting on the electron.
²⁵

In the most general case, these forces can change with distance and time. The fact that these forces, and hence the potential V, can be time t- and space x-dependent is reflected in the notation V(x,t) in this equation, which implies that V can take on different values at different points in space x and at different times t. Sometimes the forces do not change with time, as in the electrical attraction between the negatively charged electron and the positively charged protons in the atomic nucleus. In this case we need only to know how the electrical force varies with the separation between the two charges, to determine the potential V at all points in space.

I’ve belabored the fact that V can take on different values depending on which point in space x we are, and at what time t we consider, because what is true of V is also true of the wave function Ψ. Mathematically, an expression that does not have one single value, but can take on many possible values depending on where you measure it or when you look, is called a “function.” Anyone who has read a topographic map is familiar with the notion of functions, where different regions of the map, sometimes denoted by different colors, represent different heights above sea level. This is why Ψ is called a “wave
function.
” It is the mathematical expression that tells me the value of the matter-wave depending on
where
the electron is (its location in space x) and
when
I measure it (at a time t). Though as we’ll see in the next chapter with Heisenberg, where and when get a little fuzzy with quantum objects.

For an electron near a proton, such as a hydrogen atom, the only force on the electron is the electrostatic attraction. Since the nature of the electrical attraction does not change with time, the potential V will depend only on how far apart in space the two charges are from each other. One can then solve the Schrödinger equation to see what wave function Ψ. will be consistent with this particular V. The wave function Ψ will also be a mathematical function that will take on different values depending on the point in space. Now, here’s the weird thing (among a large list of “weird things” in quantum physics). When Schrödinger first solved this equation for the hydrogen atom, he incorrectly interpreted what Ψ represented—in his own equation!

Schrödinger knew that Ψ itself could not be any physical quantity related to the electron inside the atom. This was because the mathematical function he obtained for Ψ involved the imaginary number
i.
Any measurement of a real, physical quantity must involve real numbers, and not the square root of -1. But there are mathematical procedures that enable one to get rid of the imaginary numbers in a mathematical function. Once we know the wave function Ψ, then if we square it—that is, multiply it by itself soΨ×Ψ* = Ψ
²
(pronounced “sigh-squared”)—we obtain a new mathematical function, termed, imaginatively enough, Ψ
²
.
²⁶
Why would we want to do that? What physical interpretation should we give to the mathematical function Ψ
²
?

Schrödinger noticed that Ψ
²
for the full three-dimensional version of the matter-wave equation had the physical units of a number divided by a volume. The wave function Ψ itself has units of 1/(square root (volume)). This is what motivated the consideration of Ψ
²
rather than Ψ—for while there are physically meaningful quantities that have the units of 1/volume, there is nothing that can be measured that has units of 1/(square root (volume)). He argued that if Ψ
²
were multiplied by the charge of the electron, then the result would indicate the charge per volume, also known as the charge density of the electron. Reasonable—but wrong. Ψ
²
does indeed have the form of a number density, but Schrödinger himself incorrectly identified the physical interpretation of solutions to his own equation.

Within the year of Schrödinger publishing his development of a matter-wave equation, Max Born argued that in fact Ψ
²
represented the “probability density” for the electron in the atom. That is, the function Ψ
²
tells us the probability per volume of finding the electron at any given point within the atom. Schrödinger thought this was nuts, but in fact Born’s interpretation is accepted by all physicists as being correct.

What Schrödinger discovered was the quantum analog of Newton’s force = (mass) × (acceleration). Newton said, in essence, you tell me the net forces acting on an object, and I can tell you where it will be (how the motion will change) at some other point in space and at some later time. Schrödinger’s equation asserts that if you know the potential V (which can be found by identifying the forces acting on the electron), then I can tell you the probability per volume of finding the electron at some point in space and time, now and in the past and future.