Paper for Quantum Mechanics

1. Quantum Entanglement
Verschrankung
Jennifer Nalley
Quantum Mechanics
2008

2. Quantum Entanglement
Whatever happened to one particle would thus immediately affect the other particle, wherever in
the universe it may be. Einstein called this "Spooky action at a distance."
-Amir D. Aczel, Entanglement, The Greatest Mystery In Physics
Quantum Mechanics is not a topic that is easy to explain in words, or
even understand for that matter. One of the many famous quotations by the
great physicist Richard Feynman states,” Nobody knows how the quantum
world behaves the way it does, all we know is that it does behave the way it
does.” [1] So before delving too deeply into the subject of Quantum
Entanglement, a Quantum primer; (some information and history about
quantum mechanics itself) will be given. But first, in order to keep the subject
of Entanglement in mind, consider the infamous mystery concerning identical
twins:
For some time, there has been an interest in the possible telepathic
connection between identical twins. For instance: twins “knowing” when the
other is in danger and twins sharing physical pain. Tales of such ill understood
connections between twins have been studied by psychologists for years. In
the world of hard science, such scenarios have been looked upon with
skepticism, yet today, physicists deal with a phenomenon no less strange. It
seems to be the case that there is, “An instantaneous link between particles
that remains strong, secure, and undiluted no matter how far apart the
particles may be- even if they’re on opposite sides of the universe”. [2] As
darling puts it in the book Teleportation, “This quantum equivalent of
telepathy is demonstrated daily in laboratories around the world. It holds the
key to future high-speed computing and underpins the science of
teleportation*. Its name is entanglement [2]
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3. A Brief Description and History of Quantum Mechanics
Quantum mechanics is the physics that describes the behavior of the
very small. Although intriguing on its own, having some basic understanding
and history of quantum physics is a great asset for a good understanding our
topic, Quantum Entanglement .Quantum mechanics is unique in reality and
historically for a number of reasons. It is a very young concept, around a
century old, and continually baffles those who dare enter its world.
Additionally, it seems as though each time we become closer to understanding
quantum mechanics, there arises yet another set of mysteries. In contrast to
the well established, and conceptually comforting “Newtonian” physics, (those
laws that govern our macroscopic world), quantum mechanics is, to most
people, anything but intuitively sound. When the laws of quantum mechanics
proved undisputable, physicists were forced to let go of some previous beliefs,
and all that previously had known as truth needed to be reexamined. Many
physicists still refused to accept the implications of quantum physics, despite
the repeated evidence of its reality. What it was, and is still (for many), about
quantum mechanics that makes it hard to swallow, has to do with its apparent
incompatibility with the reality in which most of us live. Even Niels Bohr, one
of the primary pioneers of quantum mechanics admitted,” Anyone who is not
shocked by quantum theory has not understood it.” [1] Some of quantum
mechanics unique attributes will be made known as this story unfolds.
It’s a wave, it’s a particle, it’s well…both?
The door for quantum mechanics first opened when it became apparent
that there was not one, but two ways to explain the nature and behavior of
electromagnetic radiation, (later this proves to be true for all subatomic
particles). The wave nature of light had been shown with Young’s double–slit
experiment. The tell-tale interference patterns, characteristic of wave
behavior, created when light passed through the small slits verified this
aspect. [4].
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4. Around 1900, there was a conservative minded physicist named Max
Planck. His primary interest was thermodynamics, and he was dealing with
black-body radiation, and what was known at the time as the ultraviolet
catastrophe. [1, 5]
In short, Planck was looking to explain why, at high frequencies, black-body
radiation did not produce the “catastrophic” levels of high energies that
classical laws predicted. [1]
What Planck uncovered was that, “electric oscillations inside an atom
could only emit or absorb energy of a certain size.” [1] These “lumps” were
named quanta, and it so turned out that a quantum of energy is limited to be
“lumped” into parts which are in increments of , where  is the frequency
and  came to be known as Planck’s constant .The relationship E= was
established. In his the book In Search of Schrödinger’s Cat, Gribben notes
that,” The revolutionary aspect of Planck’s work in 1900 was that it showed a
limitation on classical physics.” [1](1) note
Enter Einstein. Albert Einstein saw the implications behind Planck’s
work, he simply applied the relationship E= to electromagnetic radiation,
(rather than small oscillations inside of an atom). He used this in his work on
the photoelectric effect, for which he would later receive the Nobel Prize. [1,
6]
The underlying importance of the Planck/ Einstein accounts above has to
do with its implications. From E= , it can be stated that light is not a
continuous wave, but instead it comes in definite packets, each that holds a
discreet energy, no more, no less. Today these packets are known as photons.
Now we are left with two opposing truths. The Young interference
experiment defines light as a wave. E= shows light as a particle. Which is
the real truth? Well, the answer is both. In fact it is true for all quantum
particles, (not light as we know it) alone. There is a wave-particle duality in
the quantum world. This not only forced physicists and mankind alike to tweak
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5. the way they viewed the world in which they lived, but it gave physicists a
whole new set of problems to be solved and questions to be answered, from
scratch. [B] see appendix
And Quantum Mechanics is Born
According to Robert Laughlin,” Quantum mechanics is the deterministic
law of motion of very small things- atoms, molecules, and the subatomic
particles of which they are made. It was discovered in the 1920’s by
physicists trying to reconcile numerous strange and highly embarrassing
experimental facts that seemed incompatible with Newton’s clockwork.” [7]
This definition may seem a bit demeaning to quantum mechanics, but let us
look at what was indeed going on at the time.
Going back to the Young slit experiments; it did not take long for it to
become apparent that something very strange was going on when “particles”
were sent through the slit interface one at a time. Over time, one would
expect individual particles to add up as particles do, but instead, they
continued to produce a wave interference pattern. It was as though the
particles had a means of communication or as though the wave-particle
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6. traveled simultaneously through both slits. [5] Next comes something more
unusual. Given a set-up like the above, and while attempting to make a
measurement of a particle, (i.e. position, momentum, any “observable”),
something odd happens. It seems as though the act of measuring one
observable makes it impossible to measure any other observable. As Darling
states it,” The act of measurement somehow prompts nature to make a
choice, that is a choice made randomly with the probabilities (i.e. finding a
particle somewhere) determined by the wave function (see Schrödinger’s
wave equation below). [2] Another testimony of quantum mechanic’s
counterintuitive nature can be found on the first page of the Quantum
Mechanics volume of the Richard Feynman’s Lectures. It is devoted to the
double slit experiment. When he was asked why this was, Feynman replied,”
Why? Because this is a phenomenon absolutely impossible to explain in any
classical way, and which has in it the heart of quantum mechanics.” [1].
• see appendix a] for more details on wave and particle addition
The quantum pioneers of the time were able to successfully come up with a
very good mathematical description for quantum mechanics, methods and
notation which is still used in quantum mechanics textbooks. There is the
Schrödinger wave equation, where the Greek letter psi  represents the wave
function. Max Born showed the benefit of thinking in terms of a probability
density adding clarity to the mathematics. There is the famous Heisenberg
Uncertainty principle, which sets limits on the certainty of measuring multiple
observables. Paul Dirac, a mathematical champion created a shorthand
notation for quantum mathematics. It is considered to be the standard form of
expression in quantum mathematics. Despite that there exists such
mathematical methods for quantum mechanics, it must be realized that these
methods, as sophisticated as they are, yield results such as expectation
values, probability densities, possible values and uncertainties. Still noting is
absolute. Max Born himself said,” In between our attempts to pin it (about an
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7. electron) down, it has no existence or substance, outside the mathematics of
the wave function.” [2]
Approaching the Entangled “Is the Moon there if Nobody Looks?
If you are asking yourself what any of this has to do with quantum
entanglement, we must go back to the beginning. The man, whose early
works set the stage for quantum mechanics, came to be one of the theories
biggest critics. As many good physics stories do, this one comes back to
Albert Einstein. It should be made known that contrary to popular belief;
Einstein did not hate, or disbelieve in quantum mechanics. He did however
believe that the theory was incomplete, and that there were “hidden
variables” yet to be accounted for. [*]
Moreover, Einstein was uncomfortable not particularly with the
indeterminism that quantum theory indicated, but the theory’s lack of
conditions of separability and locality for composite systems. [10]. Debates
about the topic, which started at a conference in Geneva between Einstein
and Bohr, are a great historical documentation of the time. The famous line,”
God does not play dice”, hallmarks Einstein’s discomfort with the new theory.
Bohr’s response was, “Einstein, stop telling God what to do.” [*]
Einstein, along with two colleagues, Podolsky and Rosen, addressed their
concern openly in the form of a paper which was printed in the 1932 Physical
Review Journal. The content of this article came to be known as the EPR
paradox. [8, 10] It addressed the concerns about quantum mechanics by
describing a hypothetical system, a “thought experiment”, which intended to
make apparent a lack of logical reasoning in quantum theory. [10] In
particular, these illogical implications that Einstein and his colleagues thought
of, would in a sense eventually become what is now known as entanglement.
In his book, “The Fabric of the Cosmos”, Brian Greene elaborates on what the
1932 work set out to accomplish, “They wanted to show that every particle
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8. does possess a definite position and velocity at any given instance in time,
and thus they wanted to conclude that the uncertainty principle reveals a
fundamental limitation on the quantum mechanical approach.” Greene later
states that the EPR papers intended to demonstrate that quantum theory,” …
was therefore an incomplete theory of physical reality and, perhaps, merely a
stepping-stone toward a deeper framework waiting to be discovered.” [6]
Later it will be seen just how Einstein and his colleagues would eventually be
proven wrong. Or at least unsuccessful in debunking that which they intended.
Additionally, in a sense, the EPR paradox worked to defeat its own purpose.
The EPR brought up new things to think about, like entanglement, and what
its existence entailed. The paper brought attention to many things that would
eventually manifest new, real quantum oddities.
Erwin Schrödinger himself had much to say about this peculiar quantum
behavior which Einstein did not like(and labeled “Spooky action at a Distance”
[3]. In fact it was Schrödinger who coined the term ‘entanglement’ to describe
this peculiar connection between quantum systems. He explained, “When two
systems, of which we know the states by their respective representatives,
enter into temporary physical interaction due to known forces between them,
and when after a time of mutual influence the systems separate again, then
they can no longer be described in the same way as before, viz. by endowing
each of them with a representative of its own. I would not call that one but
rather the characteristic trait of quantum mechanics, the one that enforces its
entire departure from classical lines of thought. By the interaction the two
representatives [the quantum states] have become entangled”.
*A detailed look at the possible and realized implications will be discussed in detail.
Thirty years later, an Irish physicist names John Bell enters the picture.
Bell set out to once and for all prove The EPR Paradox correct in the form of
Bell’s theorem. The Stanford Encyclopedia of Philosophy states, “In John S.
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9. Bell's pioneering paper of 1964 the realism consisted in postulating in addition
to the quantum state a ‘ “complete state” ’, which determines the results of
measurements on the system, either by assigning a value to the measured
quantity that is revealed by the measurement regardless of the details of the
measurement procedure, or by enabling the system to elicit a definite
response whenever it is measured, but a response which may depend on the
macroscopic features of the experimental arrangement or even on the
complete state of the system together with that arrangement.” [10] To sum it
up, Bell theoretically showed that local realism leads to a requirement for
certain types of phenomena that are not present in quantum mechanics. This
requirement is called Bell's inequality. [5-12]
It was a brilliant step of logic that if tested, would prove the EPR. Bell’s
theorem was presented as a logical statistical analysis of light particles going
through polarized lenses. In 1964, when the Bell theorem emerged,
technology was not ready to actually test it. The actual experiment was
eventually carried out by first by John F Clauser. [11] To make a long story
short, Bell’s Theorem, designed and expected to finally prove Einstein and his
colleagues correct, failed. In fact, the repeated experimental results indicated
that quantum mechanics carried with it many of those unbelievable
implications that Einstein mentioned as evidence to debunk the theory. In
general, these quantum implications fall under the notion of quantum
entanglement.
With some history established, as well as a general feel to what quantum
entanglement is, here is a summary. Quantum Entanglement, (as a concept),
can roughly be explained and summarized in a paragraph. Schrödinger
discovered it through his newly formed quantum theory, when he examined
the mathematical descriptions of two quantum particles that bump into one
other. After the interaction, it is impossible to tease apart the two particles'
characteristics. Once they are entangled, it makes no sense to talk about the
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10. properties of just one of them. All the information about the particles, such as
their momentum and spin, lies only in their joint properties, (recall the twins).
This also means that if something affects the quantum state of one particle, it
will inevitably affect the quantum state of the other, no matter how far apart
they are. It is this that gives entanglement the "spooky" character that
Einstein found so distasteful. [1, 10]
After thirty years, technology has finally reached a point that has enabled
scientists to actualize, that which until now, had been the theoretical
possibilities of entanglement. Much of this is going on right now, and the
progress is quite amazing. The progress in quantum entanglement research,
as far as applications are concerned, is rapidly making way. Current physics
publications are filled with entanglement articles. Some of entanglement
applications are not only extremely useful, but conceptually astounding and
exciting [*]. Currently, the hot topics related to Quantum Entanglement seem
to be: Quantum Cryptography, Quantum Computing, and Teleportation. [*]
Although much has already been done, if scientists were to fully harness the
power of quantum entanglement, it would be an enormous technological leap.
Imagine cryptography systems that are crack proof. Quantum computing has
already established the name q-bit (quantum bit) for its memory. Such
computers would render the microchip obsolete.
Surprisingly, the application of quantum entanglement that has made the
most progress is teleportation. Discover magazine featured an article on the
matter, “Yet it is entanglement that opens the door to teleportation.” In fact,
using the EPR experiment, scientists at IBM found that it was possible to
teleport atoms. “Two particles with the same information are identical, so
teleporting the information is essentially the same as teleporting the particle
itself” [13]. Since the, scientist have actually successfully teleported photons
and entire cesium atoms. [13] It is estimated that within a decade, it will be
possible to do the same with DNA and viruses.
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11. Quantum cryptography deals heavily with the polarization of light. For
example, a light beam is a composed of a stream of photons. The direction of
light's electric field is its direction of polarization. [Text and lab] The
polarization direction of a photon can be at any particular angle, for example
"vertical" or "horizontal". It is possible to generate a pair of entangled photons
if, for example, a laser is shone at a crystal. In that case a single photon can
split to become two photons. Each photon produced in this way will always
have a polarization orthogonal to the other photon. In the case that someone
wants to send a “secret message” entangled photons could theoretically be
used. This has to do with the initial polarized orientation, and the fact that any
intruder trying to eavesdrop with a polarizer, will just mess up the message
Once again, there is an apparent connection between the particles, no matter
how far apart they are taken. [**see reference]
At this moment, technologies which are based on quantum entanglement
are making more than progress. Because of this phenomenon, some
astounding, “Sci-Fi” ideas are becoming realities. To backtrack, quantum
entanglement can be traced back to the EPR paper/ paradox in 1935. An
attempt by Einstein and his two colleagues to show at the time, that quantum
theory was incomplete, (Even when unintentional or accidental, it seems everything is ultimately
traced back to Einstein). Considering that it has only been in the last decade or so,
that these theories have been testable, the present is an exciting time for
quantum entanglement.
It is now and tomorrow that it will unfold, just what all does entanglement
hold?
(yes you are supposed to think that last part is lame)
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12. References- (in order of appearance)
[1] J. Gibbon, In Search of Schrodinger’s Cat: Quantum Physics and Reality (Bantam Books, New
York, NY, 1984).
[2] D. Darling, Teleportation (John Wiley & Sons, Inc., Hoboken, NJ, 2005).
[3] Schrödinger, E. (1935) "Discussion of Probability Relations Between Separated Systems,"
Proceedings of the Cambridge Philosophical Society 31 (1935): 555-563; 32 (1936): 446-451
[4] R.L. Liboff, Introductory Quantum Mechanics (Addison Wesley, San Francisco, CA, 2003).
[5] R. Penrose, The Road to Reality (Alfred A. Knopf, New York, 2005).
[6] B. Greene, The Fabric of the Cosmos (Vintage Books, New York, NY, 2004).
[7] R.B. Laughlin, A Different Universe (Basic Books, New York, NY, 2005)
[8] A.Einstein, B. Podolsky, N. Rosen, Phys. Rev. 68, 777 (1932).
[9] Copenhagen Interpretation of Quantum Mechanics, Stanford Encyclopedia of Philosophy,
http://plato.stanford.edu/entries/qm-copenhagen #3, accessed 12 Feb 2008.
[10] Quantum Entanglement and Information, Stanford Encyclopedia of Philosophy,
http://plato.stanford.edu/entries/qt-entangle, accessed 12 Feb 2008.
[11] E.H. Walker, The Physics of Consciousness (Basic Books, New York, NY, 2000).
[12] D. Lindley, Where Does the Weirdness Go? Basic Books, New York, NY, 1996)
[13] M. Kaku, Through the Wormhole, Discover, Special Ed. 38, (2008)
[*] no one source, gathered from combination of many books previously read
[**] In Technical writing class , I focused on quantum cryptography the entire time, so where I
learned every detail is not easy to track.
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13. The QUANTUM Cast
Year Researcher Quantum Mechanics Concept
1901 Planck Blackbody radiation
1905 Einstein Photoelectric effect
1913 Bohr Quantum theory of spectra
1922 Compton Scattering of photons off electrons
1924 Pauli Exclusion principle
1925 De Broglie Matter waves
1926 Schroedinger Wave equation
1927 Heisenberg Uncertainty principle
1927 Davison /Germer Wave properties of electrons
1927 Born Interpretation of the wave function
Appendix and additional notes Jennifer Nalley
Entanglement
[A] A note about water waves and particle behavior to clarify the Young
interference experiments, and the duel behavior of electromagnetic radiation
(wave/ particle):
If a demonstration were to be set up to show how typical particles and typical
waves differ in their behavior, it would be easy to visualize the following
known phenomenon; When two non interacting beams of particles combine in
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14. the same region of space, their intensities add. Imagine a target at a gun
range that has been shot multiple times.
When waves interact, their amplitudes, (wave height) add. In the wave case,
imagine, or go observe ocean waves. When two ocean waves are in sync, a
larger wave is created. They add up. Conversely, when out of sync, water
waves cancel out one another.
[B] It should be noted here that Louis De Broglie made considerable
contributions to the duality theory, and should not be overlooked.
(1) *=6.6261X10 34−
Js
* later  (which is  /2) would be a common occurrence in quantum mechanical mathematics
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