Quantum Mechanics

Heroic & Dark Fantasy and Science Fiction Character created by Kevin L. O'Brien

The Lady of the Lake, © by Dorian Cleavenger

Ashliegh, © by Dorian Cleavenger

Quantum Mechanicsodern cosmology is so inextricably linked to quantum mechanics that a full understanding of the former is not possible without a proper understanding of the latter. This essay explores the history and theory of quantum mechanics, but this discussion will involve some of the most bizarre concepts and theories in modern science. Not even theoretical physicists understand them completely. Yet it is well worth the effort to try, so if anyone is adventuresome enough to join us in our sojourn into Wonderland, we would welcome the company.

Blackbody radiation spectraQuantum mechanics actually began in 1900, when a series of papers by Max Planck explained blackbody radiation; this is the spectrum of light emitted by objects that have been heated until they glow. Before this, calculations based on classical mechanics and electrodynamics had predicted the "ultraviolet catastrophe"; that is, the spectrum of light would include such intense ultraviolet radiation that the glowing object would blind anyone looking directly at it. The fact that it never happened in real life was a cause for major concern among scientists, because it suggested they did not have a complete picture of how the world operated.

Planck was able to derive the correct spectrum in his papers, which showed that the wavelengths of the spectrum and the strength of its emission were determined by temperature, and that long before most objects could become hot enough to emit intense ultraviolet light they would melt. However, in order to achieve this success he had to assume that energy was emitted in discrete chunks that he called quanta (singular: quantum). This idea was so bizarre that even he wasn't sure it as right, and most scientists at the time doubted it.

A photon of lightOne who did not was Albert Einstein, and he used the idea to explain the photoelectric effect. This is the phenomenon where light shining on a metal plate could cause an electrical current to flow; the more intense the light, the stronger the current. It is the theoretical basis for modern photovoltaic cells. Einstein claimed that light was actually made up of quanta that he called photons, and that it was the action of the photons hitting the atoms on the metal plate that caused the current to flow. With this theory he pretty much proved to everyone that quanta were real. (Incidentally, the Nobel Prize he won was for explaining the photoelectric effect, not for relativity.)

Another problem perplexing scientists at that time was the stability of atoms. Ernest Rutherford had revolutionized atomic theory by proposing that atoms were mini-solar systems, with electrons orbiting a central nucleus. The problem with this otherwise excellent model was that, according to prevailing electromagnetic theory, an orbiting electron would continuously radiate away its energy until it fell into the nucleus. Calculations showed that this would happen in only a trillionth of a second, yet the hydrogen atom was known to be very stable. This problem in fact constituted the worst failure in physics up to that time.

The Rutherford model of the atomNiels Bohr solved the problem by invoking quanta. He proposed that the angular momentum of an orbiting electron also came in discrete units, which would confine it to a specific set of orbits. It could only radiate energy if it jumped to an orbit of lower energy, during which it would then give off a photon. However, once it reached the orbit with the lowest energy, the ground state orbital, it could go no further, since there would be no further lower-energy orbits to jump to. It would then become trapped, unable to fall into the nucleus, and the result would be a stable atom.

Using this theory, Bohr was able to explain another mystery of physics, the pattern of spectral lines that all elements gave off. This phenomenon is what happens when you heat up, say, hydrogen or helium gas and then use a prism to split the light of the resulting glow into its constituent colors. Instead of getting a continuous spectrum like you do with sunlight, what you get are a finite set of discrete colored lines, each with its own wavelength. Bohr was able to show that each line corresponded to a specific orbit, and the colored light that produced that line came from electrons jumping from the ground state orbital to a higher orbital, then falling back and radiating away the energy as a photon of a specific wavelength. Unfortunately, he couldn't explain the spectrum of any other element, and he almost didn't publish his findings until Rutherford convinced him that hydrogen and helium would be enough to convince his colleagues he was right.

Atomic orbitalsDespite these early successes, though, scientists really didn't know what to make of the quantum idea. It seemed very bizarre and counterintuitive, yet it also worked so well. One major stumbling block was that there was no guiding principle to allow them to make sense of it. Then Louis de Broglie proposed that all particles — from electrons to cannon balls, including photons — act like standing waves. A standing wave is one that doesn't move, it just sits in one place and vibrates. He explained that such waves can only vibrate at certain specific and discrete (i.e., quantized) frequencies. It did the trick, but eventually people began asking, if a particle is a wave, what would the wave equation be? Erwin Schrödinger accepted the challenge and derived a generalized equation that could be applied to any particle, which in turn became the master key for explaining much of modern physics. (Max Born, Pascual Jordan, and Werner Heisenberg independently derived equivalent equations around the same time, which confirmed Schrödinger's equation.) With this, quantum theory took off like gangbusters: within just a few years — this was in the late twenties and early thirties mind you! — scientists were able to explain a whole host of mysteries, including the spectra of heavier elements and the properties of chemical reactions. It continues to bear ripe theoretical fruit even to this day.

However, Schrödinger's equation did open one can of worms that would turn out to be a deep mystery, even today. The answer it gives, when it is used to calculate the properties of a particle, is called a wave function. The mystery was, and largely still is, what the heck does this wave function mean? What is it really telling us, besides certain obvious physical properties about the particle? Max Born proposed the beginnings of an answer when he argued that the wave function should be seen in terms of probabilities. For example, say a scientist wanted to find the location of an electron. By this time, Werner Heisenberg had already described his uncertainty principle, which simply stated that you can never know both the velocity and the position of an electron simultaneously (for the simple reason that the act of trying to determine either one changes both). As such, he would have to use the Schrödinger equation to calculate a wave function for that electron for each region it might be in at the time of the measurement. The probability that the electron actually was in a specific region during the time of the measurement would be determined by the strength of the wave function for each region; the stronger the wave function, the higher the probability. Hence, the scientist can never really know exactly where the electron was at the time of the measurement, all he can calculate is the probabilities of where it might have been. And there is no guarantee that it was actually located in the region with the highest probability. The same principle applies to any application of the Schrödinger equation: the resulting wave function describes a series of probabilities for various possible outcomes, but never a definitive answer.

Atomic spectral linesThe problem with this, of course, is that it suggests there is a fundamental randomness built into the laws of nature. Einstein never liked this idea, and according to a popular, but true, story he complained that he couldn't believe that God plays dice with the universe. (According to another story, possibly apocryphal, when he had said it once too often, an exasperated Bohr finally told him to stop telling God what to do.)

Of greater importance for this essay, however, is that Born's interpretation of the Schrödinger equation created another more serious problem. Schrödinger realized that the wave function not only described probabilities of possible outcomes, it could also describe a state in which all possible outcomes existed simultaneously. Going back to our electron-location example above, the wave function isn't just telling us in which region the electron might be, it is also telling us that the electron is in all these regions simultaneously, and its actual position was not pinned down until our hypothetical scientist performed his measurement. More bizarre still, Schrödinger claimed that if this was true of electrons and atoms, it was also true of macroscopic objects, since they are all made up of atoms.

A Schrödinger wave function solutionSchrödinger called this phenomenon superposition, and he demonstrated it with his now famous thought experiment. Seal a cat inside a box without windows, along with a vial of nerve gas set to release its poison when a single atom of radium decays. Since radioactive decay is a completely random process, we can never know when the atom of radium decays, and so we can never know from moment to moment whether the cat is dead or alive. According to Schrödinger, the atom enters a state of superposition in which it is both decayed and undecayed, and the cat itself enters a state of superposition in which it is both dead and alive. And this isn't just an abstract illusion. If everything about quantum mechanics up to this point is correct, the cat really is both dead and alive, and it won't be one or the other until we actually open the box and look in (wearing gas masks, of course!).

A simpler, and infinitely more humane, thought experiment is to imagine we have an ideal playing card that responds only to quantum stimuli. According to classical physics, if you can balance it on one edge on top of a table, it will stay balanced forever. According to quantum physics, however, it will fall within a few seconds, no matter how well balanced, but it will fall in both directions at once. The result would be a superposition with two cards lying on the table, one face up and the other face down, in perfect consistency with the prediction of its wave function.

The Schrödinger's cat thought experimentOf course, if we perform this experiment in real life, we don't see this superposition; we only see one card either face up or face down, we do not see two with one face up and one face down lying beside each other. The reason why is still the most profound mystery in quantum mechanics. A possible explanation, however, grew out of the Schrödinger cat experiment during a series of conversations between Bohr and Heisenberg. It is called the Copenhagen interpretation, because they both were living in Copenhagen at that time. Remember that according to the thought experiment, the cat was both living and dead as long as the box remained sealed. Only when it was opened did the cat become either alive or dead. This led them to suggest that as long as an event remains unobserved, its possible outcome remains in superposition and its wave function evolves in a continuous and smooth manner consistent with the Schrödinger equation. In mathematics, this kind of evolution is described as being unitary, and Bohr and Heisenberg explained the predicted superposition as being a result of the unitary evolution of the wave function.

The act of observation, however, disrupts this unitary evolution and causes the wave function to collapse to one specific outcome, usually but not always the most probable at the time of the observation. So going back to our balancing card experiment, if we use a robot to balance the card on a table in a sealed room, the card will fall as predicted by its wave function and end up in a state of superposition with two cards lying beside each other, one face up and the other face down, by unitary evolution. However, as soon as we enter the room to look at the cards, we disrupt the superposition, the wave function collapses, and one card disappears at random, according to the probabilities dictated by the wave function at that moment, leaving the other card in place.

Wave function collapseOne thing that needs to be kept in mind about the unitary evolution of a wave function and its collapse during observation is that the former is completely deterministic. There are absolutely no random elements operating during unitary evolution, the entire process is guided by physical law and the Schrödinger equation. The only time random processes become involved is during collapse, and even then all they do is select which of the possible outcomes established by the wave function appears, based on the probabilities; they have no influence on which outcomes are available or how they developed. This may seem surprising, considering that the modern popular view of quantum mechanics is that of a random process that is largely unpredictable, but in fact the randomness largely comes in with the Born and Copenhagen interpretations of the Schrödinger equation. Quantum mechanics is actually statistical, and while that does involve probabilities, one can still make accurate predictions of most quantum processes, thanks in large part to the Schrödinger equation itself.

The Copenhagen interpretation proved to be remarkably successful at accurately calculating the observed outcomes of experiments and real world events. Using it, quantum mechanics went on to achieve such successes as predicting antimatter, understanding radioactivity (which led to the development of nuclear power), explaining how semiconductors and superconductors work, and describing the interaction between light and matter (which led to the development of lasers) and radio waves and atomic nuclei (which led to the development of magnetic resonance imaging). Meanwhile, an extension of quantum mechanics called quantum field theory forms the basis for elementary particle physics, which is the foundation of such contemporary studies as neutrino oscillation, supersymmetry (a form of string theory), and the search for the Higgs boson (the theoretical mediator of mass energy).

Quantum superpositionYet it also has one major problem: no one has yet derived a formula for calculating when and how a wave function collapses. Until this can be done, the problem posed by superposition is not really solved; it just gets pushed back a step or two, especially with regard to real-world events. That is, if a tree falls in a forest and no one observes it, has it truly fallen, or does it really exist in a state of fallen and not fallen? And if the latter, just what does that mean, exactly? We still do not know. This problem has even been used by some ambitious theists as the ultimate proof of God's existence. Being omnipresent, He can observe everything and thus force all wave functions to collapse, thereby avoiding quantum paradoxes. And this is not just some insignificant abstract speculation to keep eggheads entertained at cocktail parties; the resolution of this mystery will have a profound impact on the very nature of reality.

Of course, one could suppose that it is the concept of superposition that is wrong, but in fact there are two arguments against this. The first is theoretical: if superposition is fantasy, then there is something seriously wrong with our understanding of quantum mechanics, and that seems unlikely considering its successes. The second is experimental: superposition has been demonstrated with photons and subatomic particles like electrons. It has not been directly observed of course, but experiments have been done whose results are consistent with superposition being real. In other words, if superposition wasn't real, the experiments would have given different results.

Unitary evolution of EarthSo it would seem that superposition is here to stay, but then how do we get out of our paradox? By jumping out of the frying pan and into the fire. Hugh Everett III asked the question, what if the universe is evolving in a unitary fashion? After all, if quantum mechanics describes how the universe works, then the universe itself should be describable by a wave function, albeit an extraordinarily complex one. Everett proposed that in fact that wave function did evolve in a smooth, continuous, deterministic fashion that left no room for random nonunitary collapse. Instead, he assumed that microscopic superpositions would rapidly build up into incredibly complex macroscopic superpositions. This means that the quantum card would really be in two positions (face up and face down) at once; it also means that any observer would enter a state of mental superposition, where he sees one or the other outcome, simultaneously.

This might be a bit hard to get your brain around, so let's try to explain it this way. Just as Schrödinger predicted, when the quantum card falls, it lands both face up and face down at the same time, producing two cards, each of which shows one or the other outcome (that is, one card is face up and one card is face down). At the exact same moment, our minds also undergo a superposition, where two personalities appear. Otherwise identical, one personality sees the face up card only, while the other sees the face down card only. As such, each personality of the superposition sees a result that is consistent with the classical view of reality, and more importantly they both perceive an apparent randomness consistent with the classical rules of probability. Again, however, this is no illusion: the two copies of the observer really exist, just as the two cards really exist, simultaneously.

The Many-Worlds interpretationThough formally known as the relative-state formulation, this conceptualization is known popularly as the many-worlds interpretation, because each component of a specific superposition perceives its own world. It solves the problem of having to determine when and how a wave function collapses, but it creates a whole new problem, namely that all these parallel perceptions of reality are equally real. This forms the basis for a Level 3 multiverse (ML3), which simply assumes that these parallel realities in a superposition are not just conceptually real but are in fact physically real. So far this interpretation has survived a number of tests and challenges, so it does seem to be true, but there are two important questions that need to be answered before it can become the accepted view of reality.

The first is, why can we not perceive macroscopic superpositions, especially those that involve our own mind? The answer, first proposed by H. Dieter Zeh and further developed by him and others, is that a process similar to wave function collapse occurs, except that it involves no observers and is controlled by the Schrödinger equation itself, rather than some as yet unidentified external influence. According to this idea, an ideal pristine superposition is defined as being coherent. This can be illustrated using a mathematical picture called a density matrix. A density matrix for a quantum card superposition would look like a square plane with four peaks of equal height, one at each corner. Two of the peaks, in opposite corners, represent the two possible classical outcomes of the card falling, landing face up or face down. The other two peaks represent the fact that these two outcomes can interfere with each other, preventing a classical one-card result and creating the quantum two-card superposition.

A coherent density matrixIn contrast, a typical example of classical randomness, a coin toss, would produce a density matrix with just two peaks, in opposing corners, representing the two outcomes, heads or tails. The interference peaks do not exist, so there is nothing to prevent one specific outcome and thus create a superposition instead. In fact, this density matrix actually tells us that, when the coin is flipped and caught but not yet revealed, it is not in a state of quantum superposition at all, in which two coins, one heads, one tails, actually exist simultaneously, but it really exists as one coin, either heads or tails (we simply do not know which yet). Such a situation is defined as being decoherent.

The bridge between a coherent quantum superposition and a decoherent classical outcome is a process called decoherence. Decoherence theory proposes that the interference peaks seen in a density matrix are actually eliminated by tiny interactions with the environment destroying the superposition. Even a single photon or gas molecule bouncing off the fallen card(s) can destroy the superposition and create a classical outcome of only one card. And even if we could repeat the experiment with the card completely isolated from the environment, our own neurons would enter a superposition state, then decohere within 10^-20 seconds, so we would still see just one card, as predicted by classical physics.

The second important question is, why does decoherence only produce results consistent with real-world probabilities? In other words, why only a face-up or face-down result; why nothing truly Hilbert spacebizarre, like a single card with half of it face up and half of it face down? The answer seems to be that the classical outcomes are the most robust and can survive the decoherence process, whereas the non-classical outcomes are destroyed by it.

Decoherence theory does not completely eliminate the need for an interpretation of the wave function, such as the Copenhagen or many-worlds interpretations, but it does tend to greatly reduce their problems. Decoherence can act as the mechanism for collapse in the former, by making the entire environment itself the observer, whereas it explains why the superpositions of the latter cannot be perceived. It also allows us to present an analogous explanation for the nature of Hilbert space, the domain where the parallel universes of an ML3 reside.

Sources / Further Reading

"100 Years of Quantum Mysteries" by Max Tegmark and John Archibald Wheeler, Scientific American, Vol. 284, No. 2, February 2001, pp. 68-75

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