Merzbacher

Quantum Mechanics and the Copenhagen InterpretationQuantum Mechanics and the Copenhagen Interpretation

 

Eugen Merzbacher

University of North Carolina at Chapel Hill

Lecture presented at a CUNY symposium exploring scientific, historical and theatrical perspectives surrounding the events of Copenhagen,

a play by Michael Frayn

 

 

New York, 27 March, 2000

All his life, Niels Bohr struggled as he tried to express his thoughts and put them on paper, either in Danish or English or German. It did not help that he mumbled and often could not be heard, leave alone understood, by his listeners. Michael Frayn’s play “Copenhagen” admirably conveys the torment that Bohr and his friends put themselves through as they groped their way toward a formulation of quantum mechanics that eventually changed how we do, teach, and learn physics.

In the midst of this upheaval in physics it was crucial to realize that, radically new as the theory was, quantum mechanics does not totally supplant the classical concepts and goals of physics. In going from classical to quantum physics the trick has always been to know what to jettison and what to retain. When I teach quantum mechanics, I find this the greatest challenge of all. In some ways, quantum mechanics resembles the game of Jeopardy. We know the answers, but must learn to ask the right questions.

Bohr won the Nobel Prize in 1922, at age 37, “for his investigations of the structure of atoms and the radiations emanating from them”. In the following years, through the twenties and early thirties, he devoted most of his time and energy to a fuller understanding of the meaning of quantum mechanics. People began to speak of the “Copenhagen school”, the “Copenhagen interpretation” of quantum mechanics, the “Spirit of Copenhagen”, and more recently and sometimes unkindly of the “Copenhagen orthodoxy”.

Bohr set out to make sense out of a number of ideas and requirements that had come to be regarded as essential features of quantum mechanics:

1. The wave-particle duality (de Broglie)

2. The uncertainty or indeterminacy relation or principle (Heisenberg)

3. The statistical character of the predictions of the theory (Born and, ironically, Schrödinger)

4. The wholeness or indivisibility of quantum states (Bohr vs. Einstein)

The wave-particle duality is traditionally illustrated by the two-slit interference experiment. A stream of particles is directed at a screen with two slits. The particles are detected one by one far away from the screen with the two holes and in various locations, many of which they could not have reached if they followed classical orbits. This behavior is the hallmark of waves producing bright and dark interference fringes by superposition of the oscillations that spread out from the two slits. Yet, what is detected are individual single particles. These can be electrons, neutrons, whole atoms or molecules and even larger objects, but of course also photons, the massless particles of light (which Einstein postulated but Niels Bohr did not really fancy). In 1923 Louis de Broglie proposed that the wavelength that can be measured by examining the position of the interference fringes is linked to the velocity of the particles of mass m by a simple reciprocity law:

Image1

where Image2 is the speed of the particle, and h is Planck’s constant. The first equality applies to photons as well, since they have momentum, although no mass. Experiments immediately showed this to be correct.

At first there was the appearance of an internal contradiction here, but it was soon realized that the wave and particle aspects of matter and light are not incompatible. Instead they are inherent complementary features of one and the same thing. The reconciliation of the two seemingly contradictory properties ‚ waves and particles ‚ was achieved when it became clear that one must be extremely careful not to attribute long-familiar characteristics to the concepts “wave” and “particle”, just because they have those names. The waves are not as tangible as water or sound waves, and the particles are not tiny billiard balls. The recognition that the propositions of quantum mechanics are intrinsically statistical and probabilistic was the key to resolving the seeming paradox of the wave-particle duality. This was accomplished in 1926.

A perfect harmonic (sine) wave in space has a definite wavelength and thus represents a particle (photon, electron, neutron, atom, molecule, whatever) with a sharp and precise value of its momentum, or velocity. Since the amplitude of such a wave is constant and one maximum is indistinguishable from the next, the wave does not single out any particular location. If we want a wave form to localize a particle and peak at a certain position, we must superpose two or more perfect waves by adding them together. We gain spatial definition but lose sharp momentum and get a spread of velocities instead. This kind of loss in precision of one physical quantity at the expense of greater sharpness of the values of another quantity distinguishes quantum mechanics from classical physics, where it is assumed that one can know the values of all physical quantities simultaneously with infinite precision, at least in principle. Quantitatively, these ideas are expressed in the uncertainty (or indeterminacy, or Unbestimmtheits) relations, which Heisenberg derived in 1927 and which are frequently referred to in the play. The abandonment of the ideal of perfect precision in all knowledge of physical quantities inevitably constitutes a loss, but there is a compensating gain, since quantum mechanics presents us with a wealth of different states for the description of physical processes, far in excess of the toolbox of classical physics.

The two-slit interference experiment is the textbook illustration of the wave-particle duality. Together with the de Broglie reciprocal relation between velocity or momentum of a particle and the corresponding wavelength, it leads in two steps to the uncertainty relation. The distance  Image3 between the two slits, through which a single particle is sometimes said to be passing at once, gives us a measure of the uncertainty in position in the lateral direction. The deflection observed on the distant screen is a measure of the sideways momentum component,
Image4. The reciprocity of the two uncertainties is demonstrated by the superposition of waves emanating from two slits. As we bring the slits closer together, the interference pattern spreads out, and vice versa.

Several red herrings had to be disposed of in the effort to develop a unified framework to describe and predict the behavior of a vast range of physical systems. These extend from fundamental particles through nuclei, atoms and radiation to molecules and condensed matter, and into the mesoscopic domain. One of these red herrings was the heuristic notion that the theory should confine itself to dealing with observable quantities only. Since it led Heisenberg, with his incredible intuition, to the correct formulation of quantum mechanics, this mental crutch was of obvious value to the development of the theory, but it eventually became an impediment to a full understanding of quantum mechanics, especially by nonphysicists. (The history of the Aharonov-Bohm effect, around 1958-60, and even to this day, shows the danger of rigidly classifying physical quantities as observable and non-observable, and relegating the latter to second class status or worse.)

Another questionable notion is the claim that the human observer plays a more significant role in quantum physics than in classical physics, above and beyond the obvious fact that experimental tests must be prepared by humans in the laboratory, and that the entire scientific enterprise is an intellectual activity engaged in by conscious human beings. To this day, the role of the observer in quantum mechanics is often debated. Saying that “God does not play dice”, with the implied corollary that only humans do play dice, Einstein hoped that at a deeper level there would be a realistic non-statistical description of nature (with a capital “N”), with no reference to an observer. On the other hand, Bohr emphasized what he considered the indispensable importance of the observer and the measuring process in quantum phenomena. Today, we are able to interpret quantum mechanics less dogmatically than either Einstein or Bohr did. We know (or have become used to the idea) that wherever there are probabilities there are alternative outcomes of experiments and tests. Ultimately, these tests provide information to an observer, of course, but the information is of a statistical nature because Einstein’s “God” does indeed play dice without human intervention. Recent articles in Physics Today with titles such as Quantum theory without observers and Quantum theory needs no “interpretation” attest to the continuing interest in and concern about these issues.

While Heisenberg was developing his new mechanics, Schrödinger unintentionally (and Dirac intentionally) brought into being a theory that accounts for the probabilities in quantum physics. We call the object of this theory simply the state, because its knowledge provides us with all the information that quantum mechanics allows us to have about the system whose condition (or state) it describes. For reasons that need not concern us, we express the state mathematically as

Image5where the character inside the “ket” Image6 labels the particular state under consideration. This is an object to which the rules of vector algebra apply: Two states of a system can be added, and a state can be multiplied by a number. We will use the photon ‚ the particle of light ‚ to illustrate these concepts.

Since light can easily be linearly polarized and tested for its polarization (e.g., with polaroid filters), so can a photon. Photons that are polarized horizontally or vertically may be described by two basic states, Image7 . A general one-photon state is the superposition

Image8where a and b are (complex) numbers. Their squares, Image9 , are the probabilities that the superposition  Image10  represents the mutually exclusive outcomes: 100 percent, totally polarized photons in the H or V directions, respectively. For example, in the state

Image11our photon has 60% chance of being found polarized horizontally and 40% chance of being found polarized vertically. Casually, but misleadingly, one sometimes speaks of the photon occupying both Hand V states at once ‚ analogous to passing through two slits at once. The Copenhagen interpretation provided, and still provides, the appropriate language for these new concepts. Yet, language and words can be quite treacherous. As the play reminds us, Heisenberg insisted that it’s all in the mathematics. The availability of superpositions makes two-state quantum systems, or qubits, tempting tools for new methods of computation.

Our urge to see the world in terms of properties and attributes that particles “have” is strong. Even as we admit that quantum mechanics generally does not allow us to speak of a photon “having” a certain polarization, we cannot resist saying sometimes that the observation “puts” the system in a definite state of polarization and “collapses” the wave function or state. Purged of its extraneous baggage, the Copenhagen interpretation has moved our attention away from the particles that have certain properties to the properties themselves that are taken on by the particles. This may seem to be a trivial and awkward shift from a straightforward active language to a more convoluted passive mode of expression, but in quantum mechanics it makes all the difference, especially when the theory is extended to systems of several identical particles and their quantum fields. As a result, our entire concept of a physical system has had to undergo fundamental revision.

Using superpositions of the basis states Image7 , we can introduce two different new basis states, such as

Image12

or, conversely

Image13

Here, D and D’ stand for the two 45° or “diagonal” directions. Substituting these expressions into the original state, we get

Image14which means that our state is 99% Image15  and 1% Image16 , so its polarization is very close to the diagonal (or 45°) direction.

Next, we consider a two-photon state, labeling the two distinguishable photons 1 and 2 (or left and right, or red and blue). Evidently, there are four basic states:

Image17

The first of these states signifies a physical situation in which both photons are polarized along the horizontal, etc. Quantum mechanics allows, and indeed demands, that we should be able to construct from these basis states more general two-photon states by superposition, just as we did for one photon:

Image19

where again the (absolute value) squares of the numbers a, b, c, d, are the probabilities of detecting the particular two-photon basis state, when a measurement is made. For certain values of the coefficients the two-photon state is factorable:

Image20

If this factorization is possible, we can regard the two photons as carrying their physical information independently, without influencing one another. Generally, however, a two-particle state cannot be factored into two one-particle states. The state is then said to be entangled (verschränkt in Schrödinger’s German). Entangled states are frequently referred to as weird and involving spooky actions-at-a-distance.

Schrödinger’s allegorical cat. whom we also encounter in the play, is the notorious caricature of an entangled state. A radioactive nucleus, which is initially undecayed but finally decayed, takes the place of photon 1, and the cat, which is either alive or dead, symbolizes photon 2. The entire state is a superposition, not just of the live and dead states of the cat, but of two distinguishable “two-particle” states which correlate the cat states with the initial and final states of a radioactive nucleus. No such superposition has ever been observed, because of extremely rapid decoherence processes induced by even the gentlest of interactions with the environment.

Schrödinger’s radioactive nucleus coupled to the cat illustrates one of the simplest examples of an entangled two-photon polarization state:

Image21

with fifty-fifty probability of finding the photons both with horizontal or vertical polarization, but never with one of them polarized along the horizontal and the other one along the vertical. It is important to remark that the same two-photon state can also be written in terms of the diagonal basis as:

Image22

Again the entanglement is apparent.

At this point, the counterfactual EPR (Einstein-Podolsky-Rosen) argument (in the version first presented by Bohm) kicks in: If, in our entangled two-photon state, photon 1 is found to have horizontal (vertical) polarization, then photon 2 is certain to be polarized the same way, horizontally (vertically). This is so even if the photons are very far apart and incapable of interacting with each other at the time of these polarization measurements, as was the case in a recent experiment, conducted in the suburbs of Geneva with a relativistic wrinkle. The doctrine of local realism, advocated by Einstein, demands that, whether it is subjected to measurement or not, photon 2 must, in an anthropomorphic way of speaking, “know” that its polarization is horizontal (vertical), and this information must be encoded in it (e.g., as the value of a hidden variable). Since we could equally well have elected to test photon 1 for polarization in the diagonal directions Image23, bisecting the horizontal and vertical directions, local realism requires that photon 2 carries the information of its potential polarization direction unambiguously with it, although the two photons may be separated by several kilometers. In the given entangled state, Einstein’s local realism then implies that photon 2 has both 100% sharp polarization in the horizontal (vertical) direction and also along one of the diagonal directions. But in quantum theory, no such one-photon state can possibly exist! Einstein, who thought that the predictions of quantum mechanics have only statistical validity for ensembles of particles, argued that quantum mechanics is (correct but) incomplete and must allow for a more detailed deterministic and realistic description of the states of single particles (photons). Bell’s inequality showed this hypothesis to be incompatible with some (although not all) of the predictions of quantum mechanics. The entangled two-photon state is, as Bohr might say, an indivisible whole, and cannot be thought of as a composite of two distinct one-photon states, which would retain their individuality and which could each carry their own sharply defined polarizations. The latter view is not consistent with the formalism of quantum mechanics.

While it is certainly important and interesting to know that quantum mechanics and local realism are incompatible, it is even more important to know what actual experiments tell us. Can entangled states (of two or more particles) be made in the laboratory, and do they in tests obey the predictions of quantum mechanics? In the past twenty years, it has finally become possible to address these questions experimentally. Several years ago, at the fiftieth anniversary celebration of the American Institute of Physics, Edward Purcell said that he was glad to be alive when Alain Aspect and his group showed that an entangled two-photon state behaves as quantum mechanics predicts (violating Bell’s inequality). In recent times, the predictions of quantum mechanics, analyzed in terms of the Copenhagen interpretation, have been confirmed experimentally for ever more entangled states. This is the topic of Anton Zeilinger’s subsequent talk in this symposium. The play Copenhagen has provided us once more with a memorable opportunity for examining these fundamental issues.