**Quantum 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: where is s
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 [Image] 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, [Image].
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

[Image] where the character inside the "ket" [Image]
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, [Image]. A general one-photon state is the superposition

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

[Image]

our 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 [Image],
we can introduce two different new basis states, such as [Image] or,
conversely [Image]

Here, D and D’ stand for the two 45° or "diagonal" directions.
Substituting these expressions into the original state, we get [Image]
which means that our state is 99% [Image] and 1% [Image], 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: [Image] 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:

[Image][Image] 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:

[Image]

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:

[Image][Image]

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:

[Image]

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 informationmust 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
[Image], 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.