Enough with the tease: The Schrödinger equation, as it is now universally known, is most often expressed mathematically as follows:−ħ2/2m ∂2Ψ/∂x2 + V(x, t) Ψ = i ħ ∂Ψ/∂t
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 ∂2Ψ/∂x2. 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* = Ψ2 (pronounced “sigh-squared”)—we obtain a new mathematical function, termed, imaginatively enough, Ψ2.26 Why would we want to do that? What physical interpretation should we give to the mathematical function Ψ2?
Schrödinger noticed that Ψ2 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 Ψ2 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 Ψ2 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. Ψ2 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 Ψ2 represented the “probability density” for the electron in the atom. That is, the function Ψ2 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.
Once I know how likely it is to find an electron throughout space, I can calculate its average distance from the proton in the nucleus, which we would call the size of the atom. The energy within the atom, its average momentum, and in fact anything I care to measure can be calculated using the Schrödinger equation. We know that the Schrödinger equation is a correct approach for determining the wave function Ψ for electrons in an atom, for it enables us to calculate properties of the atom that are in excellent agreement with experimental observations. This is the only true test of the theory and the only reason why we take seriously notions of matter-waves and wave functions.
Schrödinger first applied his equation to the simplest atom, hydrogen, with a nucleus of a single proton and only one orbiting electron. For the force acting on the electron he used the familiar law of electrostatic attraction, extensively confirmed as the correct description of the pull two opposite charges exert on each other. With no other assumptions or ad hoc guesses, his equation then yielded a series of possible solutions, that is, a set of different Ψ functions that corresponded to the electron having different probability densities. This is not unlike the set of different possible wavelengths that a plucked guitar string can assume. For each probability density there was a different average radius, and in turn a different energy. Obviously the solitary electron in the hydrogen atom could have only one energy value and only one average radius at any given time. What Schrödinger found was that there was a collection of possible energies that the electron could have, not unlike an arrangement of seats in a large lecture hall (shown schematically in Figure 15). There are some seats close to the front blackboard, some in the next row a little bit farther away, and so on all the way to seats so far from the front of the room that a student sitting here could easily sneak out of the class with very little effort.
An atom with only one electron is akin to a lecture hall with only one student attending. If there is a tilt to the hall, so that the fro
nt of the room is at a lower level than the rear (as in many audi-toriums), then the student would lower his or her energy by sitting in the front seat in the front row. This would be the lowest energy configuration for the student in the lecture hall, and similarly Schrödinger found that a single electron would have a probability density that corresponded to it having the lowest possible energy value. This configuration for an atom is termed the “ground state.” If the electron received additional energy, say, by absorbing light or through a collision with another atom, it could move from its seat closest to the front of the room to one farther away. With enough energy it could be promoted out of the auditorium entirely, and in this case it would be a free electron, leaving behind a positively charged ionized atom (which for the simple case of hydrogen would be just a single proton).
Figure 15: Cartoon sketch of the possible quantum states, represented as seats in a lecture hall that an electron can occupy, as determined by the Schrödinger equation for a single electron atom. In this analogy the front of the lecture hall, at the bottom of the figure, is where the positively charged nucleus resides. Upon absorbing or releasing energy, the electron can move from one row to another.
Schrödinger’s equation at once explained why atoms could absorb light only at specific wavelengths. There are only certain energies that correspond to valid solutions to the Schrödinger equation for an electron in an atom, pulled toward the positively charged nucleus by electrostatic attraction. Electrons ordinarily sit in the lowest energy level—the ground-state configuration. Upon absorbing energy from outside the atom, they can move from the row closest to the front of the hall to another row at a higher level, closer to the exit, provided the seats they move to are empty. Those who recall their high school chemistry may know that each “seat” can actually accommodate two electrons—this is discussed in some detail in Section 4. There is a precise energy difference between the row where the electron is sitting and the empty seat it will be promoted to. It must move from seat to seat and cannot reside between rows. Different atoms with different numbers of protons in their nucleus will have slightly different Ψ functions and different spacing between the rows of possible seats, similar to each string on a guitar having a different fundamental frequency as well as different overtones.
One intriguing consequence of the Schrödinger equation is that it explained that electrons in an atom could have only certain energy values, and that all other electronic energies were forbidden. Schrödinger’s solution has nothing to do with circular or elliptical orbits, but rather with the different possible probability densities the electron can have. Another way of stating this is that the probability of an electron having a “forbidden energy” is zero and thus will never be observed to occur.
Only if the energy supplied to the atom, for example, in the form of light, is exactly equal to this difference will the electron be able to move to this empty seat. Too much energy or too little would promote the electron between rows. Since the electron cannot be at these energy values (the probability of it occurring is zero), the electron cannot absorb the light of these energies. There will thus be a series of wavelengths of light that any given atom can absorb, or emit, when the electron moves from one row to another; all other light is ignored by the atom.
Now for the cool part. When Schrödinger used the electrostatic attraction between a proton and an electron for a simple hydrogen atom, he found a set of possible wave functions that corresponded to different probability densities. When he made sure that Ψ was “normalized”—which is a fancy way of saying that if you add up the probability density of the electron over all space, the total has to be 100 percent—then his equation yielded a set of possible energy levels for the electron that exactly corresponded to the energy transitions experimentally observed in hydrogen. No assumptions about orbits, no ignoring the fact that electromagnetism requires the electron to radiate energy in a circular trajectory. All you have to tell the Schrödinger equation is the nature of the interaction between the negatively charged electron and the positively charged nucleus in the atom, and it automatically tells you the possible line spectra of light for emission or absorption.
Schrödinger’s equation destroyed the notion of a well-defined elliptical trajectory for the electron and replaced it with a smoothly varying probability of finding the electron at some point in space. In essence, physics had come full circle. At the start of the twentieth century, scientists thought that the atom consisted of a jelly of positive charges, in which electrons sat like marshmallows in a Jell-O dessert. In this model the atom emitted or absorbed light at very particular wavelengths as these wavelengths corresponded to the fundamental frequencies of oscillation of the trapped electrons. Rutherford demonstrated that the atom actually had a small positively charged nucleus, around which the electrons orbited. But this model could not explain what kept the electron from spiraling into the nucleus, nor did it account for the absorption line spectra. With Schrödinger the small positively charged nucleus remained, but the negatively charged electrons were like jelly smeared over the atom, in the form of a “probability density.” The source of the “uncertainty” in the electron’s location was accounted for by Werner Heisenberg, independently of Schrödinger, who was mixing business with pleasure in the Swiss Alps.
CHAPTER SEVEN
The Uncertainty Principle Made Easy
Just prior to the introduction of the Schrödinger equation, Werner Heisenberg was developing an alternative approach to atomic physics. The wavelike nature of matter is also at the heart of the famous Heisenberg uncertainty principle. Heisenberg took a completely different tack to the question of how de Broglie’s matter-waves could account for an atom’s optical absorption line spectra. As we’ve discussed, the conventional view that the electron was executing classical orbits around the nucleus could not be reconciled with the absence of light that should be continuously emitted by such an electron. Heisenberg’s struggle to envision what the electron was doing and where it was located within the atom finally convinced him to give up and not bother trying to figure out where the electron was. This turned out to be a winning strategy.
Physics is an experimental science, and the development of quantum mechanics was driven by a need to account for atomic measurements that were in conflict with what was expected from electromagnetic theory and thermodynamics. Heisenberg realized that any theory of the atom needed only to agree with measurements, rather than make predictions that could never be tested! Who cares where the electron “really was” inside the atom? Could you ever definitely measure its location to confirm or disprove this prediction? If not, then forget about it. What could be measured? For one, the wavelengths of the light emitted from an atom in a line spectrum. Well then, let’s construct a theory that describes the energy difference when an atom makes a transition from one state to another. Heisenberg’s model for the atom consisted of a large array of numbers that characterized the different states the electron could be in, and rules that governed when the electron could go from one state to another. When compared to the observed spectra of light absorbed or emitted from a hydrogen atom, Heisenberg’s approach agreed exactly with measurements.
In 1925 Heisenberg exiled himself to the German island of Helgoland in the southeast corner of the North Sea as he worked out the details of this approach. He had known for years that, in contrast to Schrödinger, he did his best work removed from any distractions. While the isolation suited his requirements for extended contemplation, his motivation for decamping to the island was a bit more mundane. Heisenberg suffered from severe allergies, and it was to escape the pollens of Göttingen that he traveled to the treeless Helgoland. The stark island’s only recreational option consisted of mountains, of which the hiking enthusiast Heisenberg availed himself as he struggled with his theory.
On this island Heisenberg constructed arrays of values for the different states in which electrons in an atom could be found, and the rules for how they would make the transition b
etween states. When he returned to Berlin and showed his preliminary efforts to Max Born, the older professor was initially confused. As he read and reread Heisenberg’s work, trying to understand these transition rules, he had a strong sense of familiarity. Eventually Born and Pascual Jordan, another physicist at Göttingen, recognized that Heisenberg had independently developed a mathematical notation known as a “matrix” to describe his theory—a notation that had been invented a hundred years earlier by mathematicians interested in solving series of equations that had many unknown variables. This branch of mathematics is called linear algebra, or matrix algebra, and Heisenberg had unknowingly reinvented a wheel that had been rounded years earlier.
(This is always happening, by the way. More often than not we find in physics that the necessary mathematics to solve a particularly challenging problem has already been developed, frequently no less than a century earlier. The mathematicians are just doing what mathematicians do and are not trying to anticipate or solve any physics problems. There are two notable exceptions: Newton had to invent calculus in order to test his predictions of celestial motion against observations, and modern string theorists are inventing the necessary mathematics simultaneously with the physics.)
Heisenberg’s approach using matrices is an alternative explanation for the behavior of electrons in atoms, for which he was awarded the Nobel Prize in Physics in 1932. Heisenberg’s theory was published in 1925. Less than a year later Schrödinger introduced his matter-wave equation to account for the interactions of electrons inside the atom. The two descriptions of the quantum world do not appear to have anything in common, aside from the fact that they both accurately predict the observed optical line spectra for atoms. In 1926 Schrödinger was able to mathematically translate one approach into the other, demonstrating that the two descriptions are in fact equivalent. Both theories rely on de Broglie’s matter-wave hypothesis, though neither takes the suggestion of elliptical electronic orbits literally. Both approaches make use of what we now would characterize as the electron’s wave function and have prescriptions for how one can mathematically calculate the average momentum, the average position, and other properties of an electron in an atom. For our purposes, we do not have to go too deeply into either theory, as our goal is to understand how the concepts of quantum mechanics underlie such wonders of the modern age as the laser and magnetic resonance imaging.
The Amazing Story of Quantum Mechanics Page 8