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 Bois-Reymond Hilary Bok Laurence BonJour George Boole Émile Boutroux Daniel Boyd 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 Tom Clark 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 Austin Farrer 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 Arthur O. Lovejoy E. Jonathan Lowe John R. Lucas Lucretius Alasdair MacIntyre Ruth Barcan Marcus Tim Maudlin James Martineau Nicholas Maxwell 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.Nowell-Smith Robert Nozick William of Ockham Timothy O'Connor Parmenides David F. Pears Charles Sanders Peirce Derk Pereboom Steven Pinker U.T.Place 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 John Duns Scotus Arthur Schopenhauer John Searle Wilfrid Sellars David Shiang Alan Sidelle Ted Sider Henry Sidgwick Walter Sinnott-Armstrong Peter Slezak 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 Marcello Barbieri Gregory Bateson Horace Barlow 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 Jean-Pierre 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 Bernard d'Espagnat 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 Benjamin Gal-Or Howard Gardner Lila Gatlin Michael Gazzaniga Nicholas Georgescu-Roegen GianCarlo Ghirardi J. Willard Gibbs James J. Gibson 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 Jeff Hawkins John-Dylan Haynes Donald Hebb Martin Heisenberg Werner Heisenberg John Herschel Basil Hiley Art Hobson Jesper Hoffmeyer Don Howard John H. Jackson William Stanley Jevons Roman Jakobson E. T. Jaynes Pascual Jordan Eric Kandel 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 Pierre-Simon Laplace Karl Lashley David Layzer Joseph LeDoux Gerald Lettvin Gilbert Lewis Benjamin Libet David Lindley Seth Lloyd Werner Loewenstein Hendrik Lorentz Josef Loschmidt Alfred Lotka Ernst Mach Donald MacKay Henry Margenau Owen Maroney David Marr Humberto Maturana James Clerk Maxwell Ernst Mayr John McCarthy Warren McCulloch N. 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Dirac's Principles of Quantum Mechanics
In his great textbook The Principles of Quantum Mechanics, P. A. M. (Paul) Dirac gave us his three basic concepts from which all of non-relativistic quantum mechanics follows.
He actually gave us just one principle, one axiom, and one postulate.
1. 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
ψa and ψb are both solutions of equation
(1), then an arbitrary linear combination of
these,
| ψ > = ca | ψa > + cb | ψb >, (4)
with complex coefficients ca and cb, 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
(observer-independent) 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 ψn
(called eigenfunctions) of equation (1) in
its time independent form, with energy eigenvalues
En.
H ψn = En ψn, (5)
The discrete spectrum energy eigenvalues En
limit interactions (for example, with
photons) to specific energy
differences En - Em.
In the old quantum theory, Bohr postulated that electrons in atoms would be in "stationary states" of energy En, and that energy differences would be of the form En - Em = 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
ψn are orthogonal to each other
< ψn | ψm > = δnm (6)
where the "delta function"
δnm = 1, if n = m, and = 0, if n ≠ m. (7)
Once they are normalized, the ψn form an orthonormal set of
functions (or vectors) which can serve as a basis for
the expansion of an arbitrary wave function φ
| φ > = ∑ n = 0 n = ∞ cn | ψn >. (8)
The expansion coefficients are
cn = < ψn | φ >. (9)
In the abstract Hilbert space, < ψn | φ > is the "projection" of the vector φ onto the orthogonal axes ψn of the ψn "basis" vector set.
1.1 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.
2. The Axiom of Measurement.
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 "self-adjoint" - equal to its complex conjugate, A* = A).
The diagonal n, n elements of the operator's
matrix,
< ψn | A | ψn > = ∫ ∫ ψ* (q) A (q) ψ (q) dq, (11)
are interpreted as giving the expectation
value for An (when we make a
measurement).
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 weakness of the theory is, on the one hand, that it does not bring us closer to a link-up 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.
The off-diagonal 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.
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 En. 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 off-diagonal transition probabilities are the formalism of Bohr's "quantum jumps" between his stationary states, emitting or absorbing energy hν = En - Em. 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
An, will result in the number of measurements
with value An that is proportional
to the probability of finding the system
in eigenstate ψn.
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!
3. The Projection Postulate.
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 ψn.
Albert Einstein, 1916
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 probability-amplitude 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 non-intuitive that interpreters have replaced them with the most outlandish (literally) alternatives. The famous "many-worlds 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 | ψn >.
| φ > = ∑ n = 0 n = ∞ cn | ψn >. (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 non-intuitive 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.
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