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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 Belsham
Henri Bergson
George Berkeley
Isaiah Berlin
Richard J. Bernstein
Bernard Berofsky
Robert Bishop
Max Black
Susanne Bobzien
Emil du Bois-Reymond
Hilary Bok
Laurence BonJour
George Boole
Émile Boutroux
F.H.Bradley
C.D.Broad
Michael Burke
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
John Martin Fischer
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
Jaegwon Kim
William King
Hilary Kornblith
Christine Korsgaard
Saul Kripke
Andrea Lavazza
Keith Lehrer
Gottfried Leibniz
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
Friedrich Nietzsche
John Norton
P.H.Nowell-Smith
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
Jean-Paul Sartre
Kenneth Sayre
T.M.Scanlon
Moritz Schlick
Arthur Schopenhauer
John Searle
Wilfrid Sellars
Alan Sidelle
Ted Sider
Henry Sidgwick
Walter Sinnott-Armstrong
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
William Whewell
Alfred North Whitehead
David Widerker
David Wiggins
Bernard Williams
Timothy Williamson
Ludwig Wittgenstein
Susan Wolf

Scientists

Michael Arbib
Bernard Baars
Gregory Bateson
John S. Bell
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
Jean-Pierre Changeux
Arthur Holly Compton
John Conway
John Cramer
E. P. Culverwell
Charles Darwin
Terrence Deacon
Louis de Broglie
Max Delbrück
Abraham de Moivre
Paul Dirac
Hans Driesch
John Eccles
Arthur Stanley Eddington
Paul Ehrenfest
Albert Einstein
Hugh Everett, III
Franz Exner
Richard Feynman
R. A. Fisher
Joseph Fourier
Lila Gatlin
Michael Gazzaniga
GianCarlo Ghirardi
J. Willard Gibbs
Nicolas Gisin
Paul Glimcher
Thomas Gold
A.O.Gomes
Brian Goodwin
Joshua Greene
Jacques Hadamard
Patrick Haggard
Stuart Hameroff
Augustin Hamon
Sam Harris
Hyman Hartman
John-Dylan Haynes
Martin Heisenberg
Werner Heisenberg
John Herschel
Jesper Hoffmeyer
E. T. Jaynes
William Stanley Jevons
Roman Jakobson
Pascual Jordan
Ruth E. Kastner
Stuart Kauffman
Martin J. Klein
Simon Kochen
Stephen Kosslyn
Ladislav Kovàč
Rolf Landauer
Alfred Landé
Pierre-Simon Laplace
David Layzer
Benjamin Libet
Seth Lloyd
Hendrik Lorentz
Josef Loschmidt
Ernst Mach
Donald MacKay
Henry Margenau
James Clerk Maxwell
Ernst Mayr
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
Juan Roederer
Jerome Rothstein
David Ruelle
Erwin Schrödinger
Aaron Schurger
Claude Shannon
David Shiang
Herbert Simon
Dean Keith Simonton
B. F. Skinner
Roger Sperry
John Stachel
Henry Stapp
Tom Stonier
Antoine Suarez
Leo Szilard
William Thomson (Kelvin)
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
H. Dieter Zeh
Ernst Zermelo
Wojciech Zurek

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 Pilot-Wave 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 "pilot-wave 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 so-called "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.

Classical Mechanics Quantum Mechanics
Newtonian
Lagrangian
Hamiltonian
Hamilton's Principle
Hamilton-Jacobi
Maupertuis' Principle of Least Action
Poisson Brackets
Louville Equation
Old Quantum Theory (Bohr-Sommerfeld, 1913)
Matrix Mechanics (Heisenberg-Born-Jordan, 1925)
Wave Mechanics (Schrödinger, 1926)
Poisson Bra-kets, Transformation Theory (Dirac, 1927)
Creation-Destruction Operators (Dirac-Jordan-Klein, 1927)
Density Matrix (Von Neumann, 1927)
Variational-Hamilton's Principle (Jordan-Klein, 1927)
Phase Space Distribution (Wigner, 1932)
Path Integral-Sum over Histories (Feynman, 1948)
Pilot-Wave (De Broglie-Bohm, 1952)
Hamilton-Jacobi/Action-Angle (Leacock-Padgett, 1983)

Formulations
Old Quantum Theory (Bohr-Sommerfeld, 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 mid-1920'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 (Heisenberg-Born-Jordan, 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 non-commuting operators (for which pq ≠ qp). Max Born knew that matrices have this non-commuting 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 (off-diagonal elements are all zero).

Identifying the energy levels En 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 single-row 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 time-independent, and the vectors, called wave functions, evolve in time. The time-dependent 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 = [ - ℏ22 / 2 m + V (r, t) ] ψ (r, t)         (3)

He could even write a two-particle wave function (important for the "entangled" particles in the EPR experiments), now in a six-dimensional space, δψ (r1, r2, t).

But Schrödinger's easily "visualizable" wave functions were not the vectors of n-dimensional "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 Copenhagen-Göttingen 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 "wave-particle 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 "ghost-field" 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 "ghost-field") and light particles (by then being called photons).

Poisson Bra-kets, Transformation Theory (Dirac, 1927)

Creation-Destruction Operators (Dirac-Jordan-Klein, 1927)

Density Matrix (Von Neumann, 1927)

Variational-Hamilton's Principle (Jordan-Klein, 1927)

Phase Space Distribution (Wigner, 1932)

Path Integral-Sum over Histories (Feynman, 1948)

Pilot-Wave (De Broglie-Bohm, 1952)

Hamilton-Jacobi/Action-Angle (Leacock-Padgett, 1983)

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