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 Nancy Cartwright Gregg Caruso 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 Bas van Fraassen 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 Frank Jackson 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 Joseph Levine 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 David Albert 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 Jean Bricmont Hans Briegel Leon Brillouin Stephen Brush Henry Thomas Buckle S. H. Burbury Melvin Calvin Donald Campbell Sadi Carnot Anthony Cashmore Eric Chaisson Gregory Chaitin JeanPierre Changeux Rudolf Clausius 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 George F. R. Ellis 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 Basil Hiley Art Hobson Jesper Hoffmeyer Don Howard William Stanley Jevons Roman Jakobson E. T. Jaynes Pascual Jordan Ruth E. Kastner Stuart Kauffman Martin J. Klein William R. Klemm Christof Koch Simon Kochen Hans Kornhuber Stephen Kosslyn Daniel Koshland Ladislav Kovàč Leopold Kronecker Rolf Landauer Alfred Landé PierreSimon Laplace David Layzer Joseph LeDoux Gilbert Lewis Benjamin Libet David Lindley Seth Lloyd Hendrik Lorentz Josef Loschmidt Ernst Mach Donald MacKay Henry Margenau Owen Maroney Humberto Maturana James Clerk Maxwell Ernst Mayr John McCarthy Warren McCulloch N. David Mermin 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 Lord Rayleigh Jürgen Renn Juan Roederer Jerome Rothstein David Ruelle Tilman Sauer Jürgen Schmidhuber Erwin Schrödinger Aaron Schurger Sebastian Seung Thomas Sebeok Claude Shannon David Shiang Abner Shimony Herbert Simon Dean Keith Simonton B. F. Skinner Lee Smolin Ray Solomonoff Roger Sperry John Stachel Henry Stapp Tom Stonier Antoine Suarez Leo Szilard Max Tegmark Libb Thims William Thomson (Kelvin) Giulio Tononi Peter Tse Francisco Varela Vlatko Vedral Mikhail Volkenstein Heinz von Foerster Richard von Mises John von Neumann Jakob von Uexküll John B. Watson Daniel Wegner Steven Weinberg Paul A. Weiss Herman Weyl John Wheeler Wilhelm Wien Norbert Wiener Eugene Wigner E. O. Wilson Stephen Wolfram H. Dieter Zeh Ernst Zermelo Wojciech Zurek Konrad Zuse Fritz Zwicky Presentations Biosemiotics Free Will Mental Causation James Symposium 
The Schrödinger Equation
1. The Schrōdinger Equation. The fundamental equation of motion in quantum mechanics is Erwin Schrōdinger's famous wave equation that describes the evolution in time of his wave function ψ.
iℏ δψ / δt = H ψ (1)
Max Born interpreted the square of the absolute value of Schrōdinger's wave function ψ_{n} ^{2} (or < ψ_{n}  ψ_{n} > in Dirac notation) as providing the probability of finding a quantum system in a particular state n. As long as this absolute value (in Dirac braket notation) is finite,
< ψ_{n}  ψ_{n} > ≡ ∫ ψ* (q) ψ (q) dq < ∞, (2)
then ψ can be normalized, so that the probability of finding a particle somewhere < ψ  ψ > = 1, which is necessary for its interpretation as a probability. The normalized wave function can then be used to calculate "observables" like the energy, momentum, etc. For example, the probable or expectation value for the position r of the system, in configuration space q, is
< ψ  r  ψ > = ∫ ψ* (q) r ψ (q) dq. (3)
2. The Principle of Superposition. The Schrōdinger equation (1) is a linear equation. It has no quadratic or higher power terms, and this introduces a profound  and for many scientists and philosophers a disturbing  feature of quantum mechanics, one that is impossible in classical physics, namely the principle of superposition of quantum states. If ψ_{a} and ψ_{b} are both solutions of equation (1), then an arbitrary linear combination of these,
 ψ > = c_{a}  ψ_{a} > + c_{b}  ψ_{b} >, (4)
with complex coefficients c_{a} and c_{b}, is also a solution. Together with Born's probabilistic (statistical) interpretation of the wave function, the principle of superposition accounts for the major mysteries of quantum theory, some of which we hope to resolve, or at least reduce, with an objective (observerindependent) explanation of irreversible information creation during quantum processes. Observable information is critically necessary for measurements, though observers can come along anytime after the information comes into existence as a consequence of the interaction of a quantum system and a measuring apparatus. The quantum (discrete) nature of physical systems results from there generally being a large number of solutions ψ_{n} (called eigenfunctions) of equation (1) in its time independent form, with energy eigenvalues E_{n}.
H ψ_{n} = E_{n} ψ_{n}, (5)
The discrete spectrum energy eigenvalues E_{n} limit interactions (for example, with photons) to specific energy differences E_{n}  E_{m}. In the old quantum theory, Bohr postulated that electrons in atoms would be in "stationary states" of energy E_{n}, and that energy differences would be of the form E_{n}  E_{m} = hν, where ν is the frequency of the observed spectral line. Einstein, in 1916, derived these two Bohr postulates from basic physical principles in his paper on the emission and absorption processes of atoms. What for Bohr were assumptions, Einstein grounded in quantum physics, though virtually no one appreciated his foundational work at the time, and few appreciate it today, his work eclipsed by the Copenhagen physicists. The eigenfunctions ψ_{n} are orthogonal to each other
< ψ_{n}  ψ_{m} > = δ_{nm} (6)
where the "delta function"
δ_{nm} = 1, if n = m, and = 0, if n ≠ m. (7)  φ > = ∑ _{n = 0} ^{n = ∞} c_{n}  ψ_{n} >. (8) The expansion coefficients are c_{n} = < ψ_{n}  φ >. (9) In the abstract Hilbert space, < ψ_{n}  φ > is the "projection" of the vector φ onto the orthogonal axes ψ_{n} of the ψ_{n} "basis" vector set. 2.1 An example of superposition. Dirac tells us that a diagonally polarized photon can be represented as a superposition of vertical and horizontal states, with complex number coefficients that represent "probability amplitudes." Horizontal and vertical polarization eigenstates are the only "possibilities," if the measurement apparatus is designed to measure for horizontal or vertical polarization. Thus,
 d > = ( 1/√2)  v > + ( 1/√2)  h > (10)
The vectors (wave functions) v and h are the appropriate choice of basis vectors, the 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. When these (in general complex) number coefficients (1/√2) are squared (actually when they are multiplied by their complex conjugates to produce positive real numbers), the numbers (1/2) represent the probabilities of finding the photon in one or the other state, should a measurement be made on an initial state that is diagonally polarized. Note that if the initial state of the photon had been vertical, its projection along the vertical basis vector would be unity, its projection along the horizontal vector would be zero. Our probability predictions then would be  vertical = 1 (certainty), and horizontal = 0 (also certainty). Quantum physics is not always uncertain, despite its reputation. The axiom of measurement depends on the idea of "observables," physical quantities that can be measured in experiments. A physical observable is represented as an operator A that is "Hermitean" (one that is "selfadjoint"  equal to its complex conjugate, A* = A). The diagonal n, n elements of the operator's matrix, < ψ_{n}  A  ψ_{n} > = ∫ ∫ ψ* (q) A (q) ψ (q) dq, (11)
The molecule suffers a recoil in the amount of hν/c during this elementary process of emission of radiation; the direction of the recoil is, at the present state of theory, determined by "chance"...
The offdiagonal n, m
elements describe the uniquely quantum
property of interference between
wave functions and provide a measure of
the probabilities for transitions between
states n and m.
The weakness of the theory is, on the one hand, that it does not bring us closer to a linkup with the wave theory; on the other hand, it also leaves time of occurrence and direction of the elementary processes a matter of "chance."
It speaks in favor of the theory that the statistical law assumed for [spontaneous] emission is nothing but the Rutherford law of radioactive decay.
It is the intrinsic quantum probabilities that provide the ultimate source of indeterminism, and consequently of irreducible irreversibility, as we shall see. Transitions between states are irreducibly random, like the decay of a radioactive nucleus (discovered by Rutherford in 1901) or the emission of a photon by an electron transitioning to a lower energy level in an atom (explained by Einstein in 1916). The axiom of measurement is the formalization of Bohr's 1913 postulate that atomic electrons will be found in stationary states with energies E_{n}. In 1913, Bohr visualized them as orbiting the nucleus. Later, he said they could not be visualized, but chemists routinely visualize them as clouds of probability amplitude with easily calculated shapes that correctly predict chemical bonding. The offdiagonal transition probabilities are the formalism of Bohr's "quantum jumps" between his stationary states, emitting or absorbing energy hν = E_{n}  E_{m}. Einstein explained clearly in 1916 that the jumps are accompanied by his discrete light quanta (photons), but Bohr continued to insist that the radiation was classical for another ten years, deliberately ignoring Einstein's foundational efforts in what Bohr might have felt was his area of expertise (quantum mechanics). The axiom of measurement asserts that a large number of measurements of the observable A, known to have eigenvalues A_{n}, will result in the number of measurements with value A_{n} that is proportional to the probability of finding the system in eigenstate ψ_{n}. Quantum mechanics is a probabilistic and statistical theory. The probabilities are theories about what experiments will show. Experiments provide the statistics (the frequency of outcomes) that confirm the predictions of quantum theory  with the highest accuracy of any theory ever discovered! The third novel idea of quantum theory is often considered the most radical. It has certainly produced some of the most radical ideas ever to appear in physics, in attempts by various "interpretations" to deny it. The projection postulate is actually very simple, and arguably intuitive as well. It says that when a measurement is made, the system of interest will be found in (will instantly "collapse" into) one of the possible eigenstates of the measured observable. We have several possibilities for eigenvalues. We can calculate the probabilities for each eigenvalue. Measurement simply makes one of these actual, and it does so, said Max Born, in proportion to the absolute square of the probability amplitude wave function ψ_{n}.
Note that Einstein saw the chance in quantum theory at least ten years before Born
In this way, ontological chance enters physics, and it is partly this fact of quantum randomness that bothered Einstein ("God does not play dice") and Schrōdinger (whose equation of motion for the probabilityamplitude wave function is deterministic). The projection postulate, or collapse of the wave function, is the element of quantum mechanics most often denied by various "interpretations." The sudden discrete and discontinuous "quantum jumps" are considered so nonintuitive that interpreters have replaced them with the most outlandish (literally) alternatives. The famous "manyworlds interpretation" substitutes a "splitting" of the entire universe into two equally large universes, massively violating the most fundamental conservation principles of physics, rather than allow a diagonal photon arriving at a polarizer to suddenly "collapse" into a horizontal or vertical state. 4.1 An example of projection. Given a quantum system in an initial state  φ >, we can expand it in a linear combination of the eigenstates of our measurement apparatus, the  ψ_{n} >.
 φ > = ∑ _{n = 0} ^{n = ∞} c_{n}  ψ_{n} >. (8)
In the case of Dirac's polarized photons, the diagonal state  d > is a linear combination of the horizontal and vertical states of the measurement apparatus,  v > and  h >. When we square the (1/√2) coefficients, we see there is a 50% chance of measuring the photon as either horizontal or vertically polarized.
 d > = ( 1/√2)  v > + ( 1/√2)  h > (10)
4.2 Visualizing projection. When a photon is prepared in a vertically polarized state  v >, its interaction with a vertical polarizer is easy to visualize. We can picture the state vector of the whole photon simply passing through the polarizer unchanged. The same is true of a photon prepared in a horizontally polarized state  h > going through a horizontal polarizer. And the interaction of a horizontal photon with a vertical polarizer is easy to understand. The vertical polarizer will absorb the horizontal photon completely. The diagonally polarized photon  d >, however, fully reveals the nonintuitive nature of quantum physics. We can visualize quantum indeterminacy, its statistical nature, and we can dramatically visualize the process of collapse, as a state vector aligned in one direction must rotate instantaneously into another vector direction.
When we have only one photon at a time, we never get onehalf of a photon coming through the polarizer. Critics of standard quantum theory sometimes say that it tells us nothing about individual particles, only ensembles of identical experiments. There is truth in this, but nothing stops us from imagining the strange process of a single diagonally polarized photon interacting with the vertical polarizer. There are two possibilities. We either get a whole photon coming through (which means that it "collapsed" or the diagonal vector was "reduced to" a vertical vector) or we get no photon at all. This is the entire meaning of "collapse." It is the same as an atom "jumping" discontinuously and suddenly from one energy level to another. It is the same as the photon in a twoslit experiment suddenly appearing at one spot on the photographic plate, where an instant earlier it might have appeared anywhere. We can even visualize what happens when no photon appears. We can imagine that the diagonal photon was reduced to a horizontally polarized photon and was completely absorbed. Why can we see the statistical nature and the indeterminacy? First, statistically, in the case of many identical photons, we can say that half will pass through and half will be absorbed. The indeterminacy is that in the case of one photon, we have no ability to know which it will be. This is just as we cannot predict the time when a radioactive nucleus will decay, or the time and direction of an atom emitting a photon. This indeterminacy is a consequence of our diagonal photon state vector being "represented" (transformed) into a linear superposition of vertical and horizontal photon state vectors. Thus the principle of superposition together with the projection postulate provides us with indeterminacy, statistics, and a way to "visualize" the collapse of a superposition of quantum states into one of the basis states. For Teachers
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