Philosophers
Mortimer Adler Rogers Albritton Alexander of Aphrodisias Samuel Alexander William Alston Anaximander G.E.M.Anscombe Anselm Louise Antony Thomas Aquinas Aristotle David Armstrong Harald Atmanspacher Robert Audi Augustine J.L.Austin A.J.Ayer Alexander Bain Mark Balaguer Jeffrey Barrett William Barrett William Belsham Henri Bergson George Berkeley Isaiah Berlin Richard J. Bernstein Bernard Berofsky Robert Bishop Max Black Susanne Bobzien Emil du BoisReymond Hilary Bok Laurence BonJour George Boole Émile Boutroux F.H.Bradley C.D.Broad Michael Burke Lawrence Cahoone C.A.Campbell Joseph Keim Campbell Rudolf Carnap Carneades Ernst Cassirer David Chalmers Roderick Chisholm Chrysippus Cicero Randolph Clarke Samuel Clarke Anthony Collins Antonella Corradini Diodorus Cronus Jonathan Dancy Donald Davidson Mario De Caro Democritus Daniel Dennett Jacques Derrida René Descartes Richard Double Fred Dretske John Dupré John Earman Laura Waddell Ekstrom Epictetus Epicurus Herbert Feigl Arthur Fine John Martin Fischer Frederic Fitch Owen Flanagan Luciano Floridi Philippa Foot Alfred Fouilleé Harry Frankfurt Richard L. Franklin Michael Frede Gottlob Frege Peter Geach Edmund Gettier Carl Ginet Alvin Goldman Gorgias Nicholas St. John Green H.Paul Grice Ian Hacking Ishtiyaque Haji Stuart Hampshire W.F.R.Hardie Sam Harris William Hasker R.M.Hare Georg W.F. Hegel Martin Heidegger Heraclitus R.E.Hobart Thomas Hobbes David Hodgson Shadsworth Hodgson Baron d'Holbach Ted Honderich Pamela Huby David Hume Ferenc Huoranszki William James Lord Kames Robert Kane Immanuel Kant Tomis Kapitan Walter Kaufmann Jaegwon Kim William King Hilary Kornblith Christine Korsgaard Saul Kripke Thomas Kuhn Andrea Lavazza Christoph Lehner Keith Lehrer Gottfried Leibniz Jules Lequyer Leucippus Michael Levin George Henry Lewes C.I.Lewis David Lewis Peter Lipton C. Lloyd Morgan John Locke Michael Lockwood E. Jonathan Lowe John R. Lucas Lucretius Alasdair MacIntyre Ruth Barcan Marcus James Martineau Storrs McCall Hugh McCann Colin McGinn Michael McKenna Brian McLaughlin John McTaggart Paul E. Meehl Uwe Meixner Alfred Mele Trenton Merricks John Stuart Mill Dickinson Miller G.E.Moore Thomas Nagel Otto Neurath Friedrich Nietzsche John Norton P.H.NowellSmith Robert Nozick William of Ockham Timothy O'Connor Parmenides David F. Pears Charles Sanders Peirce Derk Pereboom Steven Pinker Plato Karl Popper Porphyry Huw Price H.A.Prichard Protagoras Hilary Putnam Willard van Orman Quine Frank Ramsey Ayn Rand Michael Rea Thomas Reid Charles Renouvier Nicholas Rescher C.W.Rietdijk Richard Rorty Josiah Royce Bertrand Russell Paul Russell Gilbert Ryle JeanPaul Sartre Kenneth Sayre T.M.Scanlon Moritz Schlick Arthur Schopenhauer John Searle Wilfrid Sellars Alan Sidelle Ted Sider Henry Sidgwick Walter SinnottArmstrong J.J.C.Smart Saul Smilansky Michael Smith Baruch Spinoza L. Susan Stebbing Isabelle Stengers George F. Stout Galen Strawson Peter Strawson Eleonore Stump Francisco Suárez Richard Taylor Kevin Timpe Mark Twain Peter Unger Peter van Inwagen Manuel Vargas John Venn Kadri Vihvelin Voltaire G.H. von Wright David Foster Wallace R. Jay Wallace W.G.Ward Ted Warfield Roy Weatherford C.F. von Weizsäcker William Whewell Alfred North Whitehead David Widerker David Wiggins Bernard Williams Timothy Williamson Ludwig Wittgenstein Susan Wolf Scientists Michael Arbib Walter Baade Bernard Baars Jeffrey Bada Leslie Ballentine Gregory Bateson John S. Bell Mara Beller Charles Bennett Ludwig von Bertalanffy Susan Blackmore Margaret Boden David Bohm Niels Bohr Ludwig Boltzmann Emile Borel Max Born Satyendra Nath Bose Walther Bothe Hans Briegel Leon Brillouin Stephen Brush Henry Thomas Buckle S. H. Burbury Melvin Calvin Donald Campbell Anthony Cashmore Eric Chaisson Gregory Chaitin JeanPierre Changeux Arthur Holly Compton John Conway Jerry Coyne John Cramer Francis Crick E. P. Culverwell Antonio Damasio Olivier Darrigol Charles Darwin Richard Dawkins Terrence Deacon Lüder Deecke Richard Dedekind Louis de Broglie Stanislas Dehaene Max Delbrück Abraham de Moivre Paul Dirac Hans Driesch John Eccles Arthur Stanley Eddington Gerald Edelman Paul Ehrenfest Manfred Eigen Albert Einstein Hugh Everett, III Franz Exner Richard Feynman R. A. Fisher David Foster Joseph Fourier Philipp Frank Steven Frautschi Edward Fredkin Lila Gatlin Michael Gazzaniga Nicholas GeorgescuRoegen GianCarlo Ghirardi J. Willard Gibbs Nicolas Gisin Paul Glimcher Thomas Gold A. O. Gomes Brian Goodwin Joshua Greene Dirk ter Haar Jacques Hadamard Mark Hadley Patrick Haggard J. B. S. Haldane Stuart Hameroff Augustin Hamon Sam Harris Ralph Hartley Hyman Hartman JohnDylan Haynes Donald Hebb Martin Heisenberg Werner Heisenberg John Herschel Art Hobson Jesper Hoffmeyer E. T. Jaynes William Stanley Jevons Roman Jakobson Pascual Jordan Ruth E. Kastner Stuart Kauffman Martin J. Klein William R. Klemm Christof Koch Simon Kochen Hans Kornhuber Stephen Kosslyn Ladislav Kovàč Leopold Kronecker Rolf Landauer Alfred Landé PierreSimon Laplace David Layzer Joseph LeDoux Gilbert Lewis Benjamin Libet Seth Lloyd Hendrik Lorentz Josef Loschmidt Ernst Mach Donald MacKay Henry Margenau Humberto Maturana James Clerk Maxwell Ernst Mayr John McCarthy Warren McCulloch George Miller Stanley Miller Ulrich Mohrhoff Jacques Monod Emmy Noether Alexander Oparin Abraham Pais Howard Pattee Wolfgang Pauli Massimo Pauri Roger Penrose Steven Pinker Colin Pittendrigh Max Planck Susan Pockett Henri Poincaré Daniel Pollen Ilya Prigogine Hans Primas Henry Quastler Adolphe Quételet Jürgen Renn Juan Roederer Jerome Rothstein David Ruelle Tilman Sauer Jürgen Schmidhuber Erwin Schrödinger Aaron Schurger Thomas Sebeok Claude Shannon David Shiang Herbert Simon Dean Keith Simonton B. F. 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Dirac Three Polarizers Experiment
In his 1930 textbook The Principles of Quantum Mechanics, Paul Dirac introduced the uniquely quantum concepts of superposition and indeterminacy using polarized photons.
Dirac's examples suggest a very simple and inexpensive experiment to demonstrate the notions of quantum states, the projection or representation of a given state vector in another basis set of vectors, the preparation of quantum systems in states with known properties, and the measurement of various properties.
Measuring a system again after preparing it
Any measuring apparatus is also a state preparation system. We know that after a measurement of a photon which has shown it to be in a state of vertical polarization, for example, a second measurement with the same (vertical polarization detecting) capability will show the photon to be in the same state with probability unity. Quantum mechanics is not always uncertain. There is also no uncertainty if we measure the vertically polarized photon with a horizontal polarization detector. There is zero probability of finding the vertically polarized photon in a horizontally polarized state.
in a known state is a Pauli measurement of the first kind Since any measurement increases the amount of information, there must be a compensating increase in entropy absorbed by or radiated away from the measuring apparatus. This is the LudwigLandauer Principle. The natural basis set of vectors is usually one whose eigenvalues are the observables of our measurement system. In Dirac's bra and ket notation, the orthogonal basis vectors in our example are  v >, the photon in a vertically polarized state, and  h >, the photon in a horizontally polarized state. These two states are eigenstates of our measuring apparatus. The interesting case to consider is a third measuring apparatus that prepares a photon in a diagonally polarized state 45° between  v > and  h >. Dirac tells us this diagonally polarized photon can be represented as a superposition of vertical and horizontal states, with complex number coefficients that represent "probability amplitudes." Thus,
 d > = ( 1/√2)  v > + ( 1/√2)  h > (1)
Note that vector lengths are normalized to unity, and the sum of the squares of the probability amplitudes is also unity. This is the orthonormality condition needed to interpret the (squares of the) wave functions as probabilities, as first proposed by Max Born in 1927. When these complex number coefficients are squared (actually when they are multiplied by their complex conjugates to produce positive real numbers), the numbers represent the probabilities of finding the photon in one or the other state, should a measurement be made. Dirac's bra vector is the complex conjugate of the corresponding ket vector. It is the probability amplitudes that interfere in the twoslit experiment. To get the probabilities of finding a photon, we must square the probability amplitudes. Actually we must calculate the expectation value of some operator that represents an observable. The probability P of finding the photon in state ψ> at location (in configuration space) r is
P(r) = < ψ  r  ψ >
No single experiment can convey all the wonder and nonintuitive character of quantum mechanics. But we believe Dirac's simple examples of polarized photons can teach us a lot. He thought that his simple examples provided a good introduction to the subject and we agree.
The Three Polarizers
We use three squares of polarizing sheet material to illustrate Dirac's explanation of quantum superposition of states and the collapse of a mixture of states to a pure state upon measurement or state preparation.
Here are the three polarizing sheets. They are a neutral gray color because they lose half of the light coming though them. The lost light is absorbed by the polarizer, converted to heat, and this accounts for the (Boltzmann) entropy gain required by our new information (Shannon entropy) about the exact polarization state of the transmitted photons.
In figure 2, polarizers A and B are superimposed to show that the same amount of light comes through two polarizers, as long as the polarizing direction is the same. The first polarizer prepares the photon in a given state of polarization. The second is then certain to find it in the same state. Let's say the direction of light polarization is vertical when the letters are upright.
If one polarizer, say B, is turned 90°, its polarization direction will be horizontal and if it is on top of vertical polarizer A, no light will pass through it, as we see in figure 3.
The Wonder and Mystery of the Oblique Polarizer
As you would expect, any quantum mechanics experiment must contain an element of “Wow, that’s impossible!” or we are not getting to the nonintuitive and unique difference between quantum mechanics and the everyday classical mechanics. So let’s look at the amazing aspect of what Dirac is getting to, and then we will see how quantum mechanics explains it. We turn the third polarizer C so its polarization is along the diagonal. Dirac tells us that the wave function of light passing through this polarizer can be regarded as in a mixed state, a superposition of vertical and horizontal states. As Einstein agreed, the information as to the exact state in which the photon will be found following a measurement does not exist. We can make a measurement that detects vertically polarized photons by holding up the vertical polarizer A in front of the oblique polarizer C. Either a photon comes through A or it does not. Similarly, we can hold up the horizontal polarizer B in front of C. If we see a photon, it is horizontally polarized. From equation (1) we see that the probability of detecting a photon diagonally polarized by C, if our measuring apparatus (polarizer B) is measuring for horizontally polarized photons, is 1/2. Similarly, if we were to measure for vertically polarized photons, we have the same 50% chance of detecting a photon. Going back to polarizers A and B crossed at a 90° angle, we know that no light comes through when we cross the polarizers. If we hold up polarizer C along the 45 degree diagonal and place it in front of (or behind) the cross polarizers, nothing changes. No light is getting through. But here is the amazing, impossible part. If you insert polarizer C between A and B, some light now gets through. Note that C is slipped between A (in the rear) and B (in front).
Let's start by removing the A polarizer from the back. This is the polarizer that prepared the state of the photons in vertical polarization state  v >. The light now comes through polarizer C first, which prepares it in a state of diagonal polarization  d >. We recall from equation (1) that  d > is a superposition of the basis vectors  v > and  h >.
Only some of the diagonally polarized light from C gets through the horizontal polarizer B.
If A crossed with B blocks all light, how can adding another polarization filter add light? It is much less light than through C alone. We shall see why.
The Quantum Physics Explanation
Let’s start with the A polarizer in the back. It prepares the the photons in the vertical polarization state  v >. If we now had just polarizer B, it would measure for horizontal photons. None are coming through A, so no photons get through B. When we interpose C at the oblique angle, it measures for diagonal photons. The vertically polarized photons coming through A can be considered in a superposition of states at a 45 degree angle and a 45 degree angle. Photons at 45 degrees are absorbed by C. Those at +45 degrees pass through C. C makes a measurement of 45 degree photons. It can also be viewed as a preparation of 45 degree photons. Only half the photons come through polarizer C, but they have been prepared in a state of diagonal polarization  d >. The original vertical photons coming through A had no chance of getting through B, but the diagonal photons passing through C (half the original photons) can now be regarded as in a linear superposition of vertical and horizontal photons, and the horizontal photons can now pass through B. Those vertically polarized will get absorbed by B, as usual. Recall from equation (1) that  d > is a superposition of the basis vectors  v > and  h >, with coefficients 1/√2, which when squared give us probabilities 1/2. Fifty percent of these photons emerging from C will pass though B. One quarter or 25% of the original A photons make it through. This happens if we send just one photon through at a time, just as with the twoslit experimant. Just as we can not say that the photon passes through slit A or B (only probabilities are moving in von Neumann’s process 2), we cannot say that our photons are in one state or another. They are in the mysterious linear combination that can collapse instantaneously into one state when a measurement is made.
Dirac's Description
In chapter 1 of his book The Principles of Quantum Mechanics, Paul Dirac describes our experiment. (Complete text of Chapter 1) from section 2, The Polarization of photons, pp.57 Dirac describes the superposition of states with further comments on indeterminacy.
