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 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 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 Donald Campbell Anthony Cashmore Eric Chaisson Gregory Chaitin JeanPierre Changeux Arthur Holly Compton John Conway John Cramer E. P. Culverwell Olivier Darrigol Charles Darwin Richard Dawkins Terrence Deacon Lüder Deecke Richard Dedekind Louis de Broglie Max Delbrück Abraham de Moivre Paul Dirac Hans Driesch John Eccles Arthur Stanley Eddington Gerald Edelman Paul Ehrenfest Albert Einstein Hugh Everett, III Franz Exner Richard Feynman R. A. Fisher Joseph Fourier Philipp Frank Steven Frautschi Edward Fredkin Lila Gatlin Michael Gazzaniga GianCarlo Ghirardi J. Willard Gibbs Nicolas Gisin Paul Glimcher Thomas Gold A. O. Gomes Brian Goodwin Joshua Greene Jacques Hadamard Mark Hadley Patrick Haggard Stuart Hameroff Augustin Hamon Sam Harris 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 Simon Kochen Hans Kornhuber Stephen Kosslyn Ladislav Kovàč Leopold Kronecker Rolf Landauer Alfred Landé PierreSimon Laplace David Layzer Benjamin Libet Seth Lloyd Hendrik Lorentz Josef Loschmidt Ernst Mach Donald MacKay Henry Margenau James Clerk Maxwell Ernst Mayr John McCarthy Warren McCulloch Ulrich Mohrhoff Jacques Monod Emmy Noether 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 Adolphe Quételet Jürgen Renn/a> Juan Roederer Jerome Rothstein David Ruelle Tilman Sauer Jürgen Schmidhuber Erwin Schrödinger Aaron Schurger Claude Shannon David Shiang 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 William Thomson (Kelvin) Giulio Tononi Peter Tse Vlatko Vedral Heinz von Foerster John von Neumann John B. Watson Daniel Wegner Steven Weinberg Paul A. Weiss 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 
Formulations of Quantum Mechanics
We must try to distinguish a "formulation" of quantum mechanics from an "interpretation" of quantum mechanics, although it is difficult sometimes.
For example, David Bohm's 1952 PilotWave theory provided "hidden variables" in the form of a "quantum potential" that changes instantaneously (infinitely faster than light speed) throughout all space, in order to restore a deterministic view of quantum mechanics, which Bohm thought that Einstein wanted. Einstein was appalled. Many writers describe this as the "pilotwave interpretation" and it is both a formulation and an interpretation. Different formulations in both classical and quantum mechanics provide exactly the same predictions and experimental results. But the mathematics of one formulation may yield solutions of a specific physical situation much more quickly and easily than others. Some are more easily generalized to relativistic mechanics. Some help us to "visualize" what is going on  the socalled "elements of reality"  in particular problems. Some are better at eigenvalues than transition probabilities, some better for bound/periodic systems than for "free" particles. Some find "constants of the motion" better. Some are better for quantum field theory than for quantum electrodynamics and so on. Just as there are many formulations of classical mechanics, there are many formulations (some analogous) of quantum mechanics.
Formulations
Old Quantum Theory (BohrSommerfeld, 1913)
Bohr's original work is no longer an active formulation of quantum mechanics, because it explains relatively little, but that little was enough to find the various "quantum numbers" (principal, angular, magnetic, and spin) behind atomic structure and atomic spectral lines. With the help of Sommerfeld (angular momentum) and later Pauli (spin), the old quantum theory discovered the "allowed stationary states" and the "selection rules" for allowed transitions between those states in various atoms and later molecules. It could not calculate "transition probabilities" needed to explain the intensities of the spectral lines.
Bohr quantized the electron orbits (visualized as planetary electrons traveling around a Rutherford nucleus), treating them as periodic bound systems. He did not quantize the "free" radiation emitted or absorbed when electrons "jump" from one orbit to another. Bohr thought it was classical electromagnetic radiation until at least 1925, ignoring Einstein's hypothesis of light quanta (1905), their connection to waves (1909), and their emission and absorption, along with his discovery of "stimulated" emission (1916). Bohr's "correspondence principle" allowed him to match up the transitions for states with large quantum numbers to classical behavior at large distances from the nucleus, which helped him to determine some physical constants (e.g., Rydberg). In the mid1920's, Bohr's assistant Hendrik Kramers analyzed the allowed orbits into their Fourier components and developed a matrix of transitions between states. He showed that transition probabilites were proportional to the squares of the Fourier component waves. Werner Heisenberg helped Kramers with the calculations in a joint paper, and then returned to Göttingen where he extended the matrix idea to reformulate old quantum theory as "matrix mechanics." With the help of Max Born and Pascual Jordan, he could correctly calculate transition probabilities, explaining the strengths of spectral lines. Wolfgang Pauli used matrix mechanics to calculate the structure of the hydrogen atom, reproducing Bohr's results.
Matrix Mechanics (HeisenbergBornJordan, 1925)
While alone on an island recovering from an allergy attack, Werner Heisenberg wrote a paper about a new method to calculate transition probabilities and predict intensities for spectral lines. When Max Born looked at the paper, he recognized Heisenberg's work as matrix multiplications. Born and Pascual Jordan developed the mathematics for the first consistent formulation of quantum mechanics that could explain (and calculate transition probabilities for) Bohr's "quantum jumps."
Heisenberg looked for quantitities that he called "observables," as opposed to Bohr's visualization of circular and elliptical orbits for the electrons. One of these is the system's energy, for example, which he could calculate without reference to an orbit. Heisenberg could also calculate position and momentum "observables," but these involved noncommuting operators (for which pq ≠ qp). Max Born knew that matrices have this noncommuting property. Heisenberg showed that in general, the "quantum conditions" are that pq  qp = ih, which later was written as a minimal condition known as the "uncertainty principle," pq  qp ≥ ih. Heisenberg's matrices are Hermitian, so its eigenvalues are real, and Heisenberg identified those eigenvalues with the possible values of an observable. The matrix elements of the Hamiltonian are diagonal (offdiagonal elements are all zero). Identifying the energy levels E_{n} in an atom as observables which provide no knowledge of the internal dynamics (e.g., electron position, momentum, periodicity, etc.), Heisenberg declared the "underlying reality" as unknowable in principle. In any case, matrix mechanics gives us no "visualization" of what is going on "really." The matrix elements of the polarization are periodic functions of the time that provide the frequencies and intensities of the spectral lines. Heisenberg's square (n x n) matrices are operators that operate on an n x 1 singlerow or column vector. In the Heisenberg picture these "state vectors" (  ψ > in Dirac notation) are constants, and the operators A evolve in time.
iℏ dA / dt = [ A (t), H] + δA / δt (1)
The operator H is the system Hamiltonian, the total (kinetic plus potential) energy.
Wave Mechanics (Schrödinger, 1926)
In the Schrödinger picture, the operators are timeindependent, and the vectors, called wave functions, evolve in time. The timedependent Schrödinger equation is a linear partial differential equation very similar to Heisenberg's equation (1)
iℏ δψ / δt = H ψ (2)
For a single particle of mass m moving in an electric (but not magnetic field), Schrödinger could write his equation in ordinary physical space as
iℏ δψ (r, t) / δt = [  ℏ^{2} ∇^{2} / 2 m + V (r, t) ] ψ (r, t) (3)
He could even write a twoparticle wave function (important for the "entangled" particles in the EPR experiments), now in a sixdimensional space, δψ (r_{1}, r_{2}, t). But Schrödinger's easily "visualizable" wave functions were not the vectors of ndimensional "configuration space" of Heisenberg's matrix mechanics. Von Neumann called it a Hilbert space, in which n can be infinite and range over both discrete and continuous eigenvalues. The CopenhagenGöttingen school was interested in energy levels and "quantum jumps." Heisenberg had successfully explained the relative intensities of spectral lines in terms of transition probabilities. Pauli used matrix mechanics to derive the structure of the hydrogen atom. Schrödinger, on the other hand, built his "wave mechanics" with an emphasis on the "waveparticle duality" that Einstein had been advocating for twenty years. For Einstein. "reality" consisted of individual light quanta emitted and absorbed by the atoms. When there are large numbers of such quanta, the wavelike properties show up. Einstein imagined the waves to be a "ghostfield" that guided the light quanta to exhibit classical interference phenomena. Crests in the waves would have more quanta than the nodes. Louis de Broglie accepted Einstein's view that light waves consist of particles. De Broglie hypothesized that material particles might have wave characteristics that guide the particles. He proposed "pilot waves" with a wavelength related to the moomentum p of the particle.
λ = h / p (4)
Schrödinger's breakthrough was finding the wave equation to describe de Broglie matter waves. Schrödinger and de Broglie argued that the electronic structure of atoms could be visualized as standing waves that fit an integer number of de Broglie wavelengths around each orbit. Schrödinger claimed he had found a natural explanation for integer quantum numbers that had merely been postulated by Bohr. Einstein hoped the wave theory might restore a continuous field explanation. Schrödinger interpreted an electron wave ψ as the actual electric charge density spread out in space. Einstein had strong reasons for objecting to this view as early as 1905 for light quanta, and Schrödinger gave up that interpretation. A few weeks after Schrödinger's final paper, Max Born offered his statistical interpretation, in which ψ is a probability amplitude (generally a complex number, which supports interference with itself), whose absolute square ψ*ψ is the probability of finding the electron somewhere. (cf. Kramers' transition amplitude, which was squared to provide the transition probability. ) Born acknowledged Einstein's similar view for the relation between light waves (the "ghostfield") and light particles (by then being called photons).
Poisson Brakets, Transformation Theory (Dirac, 1927)
CreationDestruction Operators (DiracJordanKlein, 1927)
Density Matrix (Von Neumann, 1927)
VariationalHamilton's Principle (JordanKlein, 1927)
Phase Space Distribution (Wigner, 1932)
Path IntegralSum over Histories (Feynman, 1948)
PilotWave (De BroglieBohm, 1952)
HamiltonJacobi/ActionAngle (LeacockPadgett, 1983)
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