The Dreams That Stuff Is Made of: The Most Astounding Papers on Quantum Physics--and How They Shook the Scientific World, Stephen Hawking, 2011, 1071 The Dreams That Stuff Is Made of: The Most Astounding Papers on Quantum Physics--and How They Shook the Scientific World, Stephen Hawking, 2011, 1071 pages, Dewey 530.12 D81d, ISBN 9780762434343
Thirty-three foundational papers, lectures, and book excerpts on quantum mechanics, by Planck, Einstein, Rutherford, Bohr, Heisenberg, Schrödinger, Dirac, Pauli, Born, Bohm, Bell, Oppenheimer, Bethe, Feynman, Gamow and others, selected and introduced by Stephen Hawking.
The bulk of the book is graduate-level quantum mechanics. One exception, pp. 966-1001, Excerpts from Thirty Years that Shook Physics, by George Gamow, is a clear, detailed, not-too-technical explanation of what was discovered (1900-1929) and how.
Textbooks and professors teach the current understanding as received wisdom. The author of the original paper proposing a new idea does not teach it this way. He admits what is /not/ known at the time; where the current understanding fails to explain observed phenomena; points out the observations that suggest a new view; explores various possible ways of understanding the phenomenon; gives reasons for arriving at his conclusions. Everything written later will omit that indecision, and give only the conclusion as established fact. Original papers are irreplaceable sources of understanding what, and how, we know.
For most of the articles, the reader is presumed to understand calculus, differential equations, orthogonal functions, Fourier transforms, Green's functions, eigenvectors and eigenvalues, statistics, combinatorics, Hilbert spaces, thermodynamics, classical electrodynamics, optics, acoustics, special and general relativity, calculus of variations, Lagrangian and Hamiltonian formulations of classical mechanics, and chemistry. These eggheads were writing primarily to each other.
Many typographical errors.
CHAPTER 1. ENERGY IS QUANTIZED. 1901-1909. pp. 1-48.
1801. [Yes, 1801.] Thomas Young's two-slit experiment shows that light behaves as a wave, making an interference pattern. p. 1.
***** 1901. On the Law of Distribution of Energy in the Normal Spectrum, Max Planck (1858-1947 p. 976), p. 5
Planck explains blackbody radiation by supposing light energy to be quantized. pp. 1-15.
This is the paper where he derives the fact that the energy of a photon is Planck's constant times the frequency. p. 12. And computes the values of Planck's constant and the Boltzmann constant (to be 6.55*10^-27 erg sec and 1.346*10^-16 erg/deg, respectively; the current values are 6.626*10^-27 erg sec and 1.381*10^-16 erg/deg, respectively). pp. 15, 841-844, 980.
Errata: First line of p. 8: "energy E_N" should be "entropy S_N."
In the equation at the top of page 10, S^N should be S_N.
**** 1905. On a Heuristic Viewpoint Concerning the Production and Transformation of Light, Albert Einstein, p. 16
Einstein explains the photoelectric effect as a consequence of the quantization of light energy. pp. 3-4, 16-31.
Errata p. 22 title 4. "FROM" should be "FORM."
**** 1909. The Atomic Theory of Matter, Max Planck, p. 32
Planck points out that anything with entropy must be quantized: radiant energy included. p. 37.
Each individual atom interacts with its fellow in a /reversible/ process. Yet a gas comprising myriad atoms undergoes irreversible processes. pp. 38-39. The macro state is described only in mean values of the states of the myriad atoms. p. 40. Only the disordered states exist in nature. That gives us entropy. A single atom has no entropy. p. 41.
Two light rays never interfere, except when they originate in the same source of light. [See 1801, Thomas Young two-slit experiment.] p. 45.
Errata p. 47, equation in mid-page, the phi should be a subscript.
CHAPTER 2. ATOMIC STRUCTURE. 1909-1922. pp. 49-147
**** 1911. The Scattering of Alpha and Beta Particles by Matter and the Structure of the Atom, Ernest Rutherford, p. 52
This is the paper where Rutherford reports that atoms have nuclei.
Rutherford and Geiger shone a beam of alpha particles (which were not then known to be helium nuclei) on gold foil 400 nanometers thick. About 1 in 20,000 were deflected an average of 90 degrees. This could happen only if the gold atoms had their positive charges clumped tightly together. At the time, the atomic numbers of the elements were unknown.
An alpha particle deflected by an atomic nucleus moves in a hyperbola with the nucleus as an external focus.
Errata p. 56 drawing. SO should be longer than OA. S is the center of the nucleus, O the intersection of the asymptotes of the alpha particle's hyperbolic path, A the alpha particle's closest approach to the nucleus.
p. 56 equations. Every nu should be a v.
p. 56 first equation after "From conservation of energy," the minus sign should be a plus sign.
**** 1913. On the Constitution of Atoms and Molecules, Niels Bohr, p. 75
Bohr points out that by standard electrodynamics, an orbiting electron should radiate its energy away--and that this doesn't happen. pp. 78-79, 840. However by supposing electron energy quantized in whole-number multiples of Planck's constant times frequency, we get stable atoms that absorb and emit light of characteristic frequencies corresponding to the differences between the allowed electron energies. This explains the hydrogen spectrum. pp. 82-103, 841.
Errata pp. 76, 94-95 random symbols are printed instead of accented letters.
p. 78 second paragraph "energy w" should be "energy W."
p. 79 third full paragraph, "bad" should be "had."
p. 80 there's no footnote corresponding to the asterisk.
pp. 80, 83-85, 88, 92, 97 all the A-hat characters should be deleted.
p. 80, first line after the central equations, -a should be a.
pp. 82-84, 86, 88, 90-91 "page 5" should be "page 80."
p. 87, central equation needs an end parenthesis after tau_1.
pp. 88, 90 "page 7" should be "pp. 82-83."
*** 1922. The Structure of the Atom, Niels Bohr, p. 104. This was Bohr's Nobel lecture.
Physical and chemical properties of atoms depend on the electrons; radioactivity depends on the nucleus. p. 105. "Isotopes" share chemical properties but differ in radioactivity. Rutherford has changed one element into another by breaking its nucleus by bombardment with alpha particles. p. 106.
"Atomic number" is an (electrically neutral) atom's number of electrons. pp. 108, 840.
Light comes in particles. How to reconcile that with its wave nature, such as interference patterns, is a mystery as of 1922. p. 112.
Studying the absorption and emission spectra of the elements led to insights into atomic structure: electrons are arranged in successive "shells" around the nucleus. How full of electrons the outer shell is, controls the element's physical and chemical properties, and gives rise to the periodicity of the periodic table. pp. 884-896.
Errata p. 111 first paragraph: "properties."
p. 114 (2), and again in bottom paragraph, v should be nu.
p. 115 first paragraph, delete hyphen.
p. 125, last paragraph, delete mid-sentence period.
p. 126 top, "WC" should be "we."
p. 126 penultimate paragraph, "we" should not be italicized.
p. 128 top, "d&rent" should be "different."
p. 132 last paragraph, delete hyphen.
CHAPTER 3. MATHEMATICAL FORMALISM (NON-RELATIVISTIC). 1926-1933. pp. 148-387.
The act of measurement alters the state of a system. p. 148. The more precisely we measure, the more we alter the state. p. 149.
**** 1929. Excerpts from The Physical Principles of the Quantum Theory, Werner Heisenberg, pp. 151-236.
"I have attempted to make the distinction between waves in spacetime and the Schrödinger waves in configuration space as clear as possible." p. 153.
Quantum theory doesn't divide the world into observer and observed, hence lacks clear causality. p. 156. To attribute an effect to a cause, we'd have to observe both without disturbing their interrelation. p. 203. It's not possible to distinguish an observed system from the observer's apparatus. They affect each other. p. 204.
Every experiment to determine some quantity renders the knowledge of others illusory. p. 156.
The most important concepts of atomic physics can be induced from five experiments:
(a) 1911. Wilson photographs of alpha and beta rays emitted by radioactive elements into supersaturated water vapor. The curvature of the tracks under electric and magnetic fields shows the mass and charge of the particles. pp. 157-158, 206-214.
(b) 1928. Diffraction of matter waves, Davisson and Germer. A beam of beta rays (electrons) passing through a crystal is diffracted as if it were a wave. The wavelength of a matter wave is Planck's constant divided by the momentum of the particle. pp. 158-159, 214-216.
(c) Diffraction of x-rays. The momentum of a photon is Planck's constant divided by its wavelength (Einstein 1905). Same as for matter waves. pp. 159-160, 214-216.
(d) 1925. Compton scattering. An x-ray photon, on collision with an electron, transfers kinetic energy and momentum to the electron, just as occurs in collisions between particles of matter. pp. 159-161, 226-228. https://en.m.wikipedia.org/wiki/Compt...
(e) 1913. Franck-Hertz experiment. An electron loses a quantized amount of energy in collision with an atom. The atom can absorb energy only in lumps of particular sizes. p. 161. https://en.m.wikipedia.org/wiki/Franc...
Light /sometimes/ behaves as a particle. Matter /sometimes/ behaves as a wave. Both are both things. p. 162.
Heisenberg's Uncertainty Principle: the product of the uncertainties of a particle's position and momentum must be greater than or equal to Planck's constant. Derived by considering the particle to be a wave packet. p. 165. Another derivation, pp. 166-168, yields the product of the uncertainties greater than or equal to Planck's constant divided by 2 pi. Page 246 gives the product of the two probable errors as greater than or equal to Planck's constant divided by 4 pi.
**** 1933. The Development of Quantum Mechanics, Werner Heisenberg, pp. 237-250.
*** 1928. Quantisation as an Eigenvalue Problem, Parts I-IV, Erwin Schrödinger, pp. 251-387
Errata: p. 252 line 7. "is" should be "it."
p. 252 line 8. "junction" should be "function."
p. 252, second paragraph, penultimate sentence, the clause beginning "such that" needs a verb.
p. 252, 4th paragraph, line 2 should begin, "continuous."
CHAPTER 4. RELATIVISTIC QUANTUM MECHANICS. 1928-1946. pp. 388-444
*** 1928. The Quantum Theory of the Electron, Paul A.M. Dirac, p. 391
Relativistic quantum mechanics: Dirac equation replaces Schrödinger equation. Predicts that antimatter must exist. p. 394. Relativistic effects are the source of "spin." pp. 388, 391-408.
*** 1940. On the Connection between Spin and Statistics, Wolfgang Pauli, pp. 409-423.
Particles with half-integer spin, "fermions," obey Fermi-Dirac statistics: Pauli exclusion principle applies: only one particle of a given set of quantum numbers can be in one place at one time. This is why white dwarf stars don't collapse: their electrons can't get too near one another. pp. 388-389. Integer-spin particles, "bosons," obey Bose-Einstein statistics: any number of particles can be in the same quantum state. pp. 388-390.
**** 1946. Exclusion Principle and Quantum Mechanics, Wolfgang Pauli, p. 424-444
This was Pauli's Nobel lecture.
The periodic table has groups of 2, 8, 18, 32, …, 2*n^2 elements, n integer. p. 424.
Electrons, protons and neutrons all have spin 1/2. p. 434.
Matter/antimatter particle pairs can be generated and annihilated. pp. 439, 441.
Errata p. 439 Delta x > c/v should be Delta x > c/nu.
CHAPTER 5. NONDETERMINISM. PROBABILITIES ONLY. 1921-1954. pp. 445-567.
Errata p. 445 line 5 "is remain" should be "remains."
**** 1954. The Statistical Interpretation of Quantum Mechanics, Max Born, p. 448
This was Born's Nobel lecture.
Planck's constant is the quantum of action (= energy*time = momentum*distance). p. 448.
Position-coordinate q and its corresponding momentum p are not scalars but noncommuting matrices such that p*q - q*p = h/2*pi*i (where i^2 = -1 and h is Planck's constant).
Concepts corresponding to no conceivable observation should be eliminated from physics. --Einstein, Heisenberg. p. 459.
The determinism of classical physics is an illusion. pp. 458-459.
It is necessary to redefine what is meant by "a particle." p. 461.
Errata p. 449-450 v should be nu in the formulas for energy h*nu, frequency nu, and momentum h*nu/c.
p. 454 Missing superscript in the text corresponding to the footnoted Schrödinger paper.
p. 455 "anew's" should be "anew."
p. 458 each nu should be v, referring to speed.
* The Present Situation in Quantum Mechanics, Erwin Schrödinger, p. 462
Schrödinger's cat: Put a cat in a box with a vial of poison that equally probably will or won't be broken, based on whether a radioactive decay does or doesn't occur. Schrödinger says that quantum mechanics says the cat is simultaneously alive and dead, because the state of the cat hasn't been "measured." No, a tree falling in the forest makes a sound whether anyone hears it or not. The issue is whether radiation has or has not interacted with matter--not whether someone is aware of it. A particle or wave goes through both slits of the two-slit experiment--and makes a black dot in a particular spot on the glass plate, having hit a particular molecule of silver nitrate--whether anybody sees it or not.
* 1935. Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? Albert Einstein, Boris Podolsky, and Nathan Rosen, pp. 463-470.
Refuted by Bohr 1935. pp. 471-483.
***** 1935. Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? Niels Bohr, pp. 471-483.
Quantum-mechanical uncertainty is not merely our ignorance of the values of certain quantities, but the impossibility of unambiguously defining these quantities. p. 477.
1952. A Suggested Interpretation of the Quantum Theory in Terms of Hidden Variables, David Bohm, p. 484
Errata p. 484 No accent on "leads."
p. 487. Superscript 1 should be superscript 10 for the footnote.
1964. On the Einstein-Podolsky-Rosen Paradox, John Bell. pp. 559-567.
CHAPTER 6. QUANTUM ELECTRODYNAMICS. 1927-1947. pp. 568-666.
*** 1927. The Quantum Theory of the Emission and Absorption of Radiation, Paul A.M. Dirac, p. 570
The light quantum (photon) has the peculiarity that it apparently ceases to exist when it is in one of its stationary states. p. 591.
Errata pp. 571-572, 575, 593-596. h should be hbar, referring to h/(2*pi).
*** 1933. The Lagrangian Method in Quantum Mechanics, Paul A.M. Dirac, pp. 598-606.
The Lagrangian method uses coordinates and velocities; the Hamiltonian uses coordinates and momenta. pp. 598, 848.
*** 1927 (dated 1927 but cites papers through 1932). On Quantum Electrodynamics, Paul A.M. Dirac, V.A. Flock, and Boris Podolsky, p. 607-619.
Errata p. 607 delete period after "and."
**** 1934. Foundations of the New Field Theory, Max Born and Leopold Infeld, p. 620-651.
Errata p. 624. first equation. first x^2 should be x^1.
*** 1950. Electron Theory, J. Robert Oppenheimer, p. 652-666.
CHAPTER 7. QUANTUM ELECTRODYNAMICS "RENORMALIZED" TO REMOVE INFINITE ENERGY. 1947. pp. 667-679.
**** 1947. Fine Structure of the Hydrogen Atom by a Microwave Method, Willis Lamb and Robert Rutherford, p. 669-674.
**** The Electromagnetic Shift of Energy Levels, Hans Bethe, p. 675-679.
Errata p. 675. Retherford should be Rutherford.
CHAPTER 8. RELATIVISTIC QUANTUM ELECTRODYNAMICS. 1946-1949. pp. 680-831.
A positron moving forward in time is an electron moving backward in time. p. 681.
*** 1946. On a Relativistically Invariant Formulation of the Quantum Theory of Wave Fields, Sin-Itiro Tomonaga, p. 682-698.
*** 1949. Space-Time Approach to Quantum Electrodynamics, Richard Feynman, p. 699-761.
Errata p. 699 last paragraph "Co" should be "to."
p. 701 first paragraph, delete "7."
*** 1949. The Theory of Positrons, Richard Feynman, p. 762-794.
Errata p. 762. No accent on "transition."
*** 1949. The Radiation Theories of Tomonaga, Schwinger and Feynman, Freeman Dyson, pp. 795-831.
Errata p. 796. "simplexity" should be "simplicity."
CHAPTER 9. HISTORY. pp. 832-
General-relativistic quantum theory or quantum gravity is still an unsolved problem, as of 2011.
**** 1926. Problems of Atomic Dynamics, Max Born, p. 834-965.
A system of electric charges cannot be in stable equilibrium. p. 840.
Errata p. 838 bottom, pointe should be points.
p. 841 m*nu^2 /2 should be m*v^2/2.
p. 931 top line should begin, "systems of"
***** Excerpts from Thirty Years That Shook Physics (Chapters I and IV), George Gamow, p. 966
Max Planck and light quanta (photons: blackbody radiation), 1900. pp. 967-980.
Einstein explains the photoelectric effect, 1905. pp. 980-985.
Compton Effect: photons have momentum and kinetic energy, 1923. pp. 985-990.
DeBroglie waves: electrons behave as waves, both in atomic orbits and in moving through space. 1925. Davisson and Germer verified it. Stern showed that atoms also diffract as waves. pp. 987-993.
Schrödinger's wave equation, 1926. Explains Bohr's quantum orbits. pp. 993-1001.
Some scattering of light by light must be expected because of virtual electron-pair formation. p. 974.
De Broglie could speak excellent English, but at home in Paris would speak only French to a visitor. p. 992.
Errata p. 979. "hv" should be "h nu."
pp. 988-989. Every nu should be a v. These are all speeds, not frequencies.
**** Excerpts from Lectures on Quantum Mechanics, Paul A.M. Dirac, p. 1002
This, I think, will be terrific for a middle-school kid. Benjamin expresses enthusiasm for, and enjoyment of, math. Conversational tone, a little humor.
Trigonometry, Pythagorean triples, trig functions, finding heights of trees and mountains, law of cosines, law of sines, Hero's formula, trigonometric identities, radians, graphs of trig functions
Imaginary number i, Euler's number e, e^(i theta) notation, polyhedrons, complex arithmetic, compound interest, exponential functions, logarithms
Calculus: optimization, differentiation, derivatives of products of functions, quotients of functions, polynomials, exponentials, trig functions; chain rule; Taylor series
The sum of the cubes of the first n integers is the square of the sum of the first n integers: Sum[i^3, {i, 1, n}] = (Sum[i, {i, 1, n}])^2 = (1/4)(n^2)(n + 1)^2 https://www.wolframalpha.com/input?i=... p. 12
How to find the day of the week for any date. pp. 65-70.
Sierpinski triangle: a fractal pattern of the odd (black) and even (white) numbers in Pascal's triangle: https://www.wolframalpha.com/input?i=... pp. 92-94. The larger the triangle, the more nearly white it is.
Wilson's theorem: n is a prime number if and only if (n - 1)! + 1 is a multiple of n. p. 144.
If p is an odd prime number, then 2^( p - 1) - 1 is a multiple of p . (Fermat. p. 145) A number that has this property, but is not prime, is called a pseudoprime.
Connecting the midpoints of any quadrilateral always produces a parallelogram. p. 150.
There are over 300 proofs of the Pythagorean Theorem. p. 174. Benjamin gives us five of them.
Three points on a circle, two of them forming the diameter, are a right triangle. Proof p. 185.
Central angle theorem: For any two points X and Y on a circle centered at O, the angle XPY at /any/ point P on the major (larger) arc of the circle, from X around to Y, will be half of the angle XOY. The angle XQY at any point Q on the minor arc of the circle, from X to Y, will be 180 degrees minus the angle XPY. p. 186.
Area of circle of radius r = pi r^2. Proofs pp. 187-188.
Ellipse, semimajor and semiminor axes a and b: (x/a)^2 + (y/b)^2 = 1 p. 189.
Drawing an ellipse: p. 190.
Approximate formula for the circumference of an ellipse, semimajor and semiminor axes a and b: pi( 3a + 3b - sqrt( (3a + b)(3b + a) ) ) Notice that if a = b = r, it's a circle, circumference 2 pi r. Srinivasa Ramanujan (1887-1920). p. 191.
Volume of sphere = (4/3) pi r^3
Area of sphere = 4 pi r^2
Cone of height h, on a circle of radius r, slant height s (s^2 = r^2 + h^2):
Volume of cone = pi r^2 h/3
Area of cone = pi r s p. 192.
Volume of a pizza, radius z, thickness a, V = pi z z a p. 193.
Pythagorean triples: For a right triangle, short sides a and b, hypotenuse c: Where a, b, c are whole numbers, they're called a "Pythagorean triple."
For any two positive integers (m, n), m > n, the numbers a = m^2 - n^2 b = 2mn c = m^2 + n^2 are a Pythagorean triple. Notice that a^2 + b^2 = c^2. p. 205.
Every Pythagorean triple can be created by some choice of (m, n). p. 205.
Law of cosines: for any triangle, sides a, b, c, angle C opposite side c: c^2 = a^2 + b^2 - 2 a b cos C proof p. 216.
Area of any triangle, sides of lengths a, b form angle C: area = (1/2) a b sin C proof p. 217.
Law of sines: for any triangle, sides a, b, c, opposite angles A, B, C: (sin A)/a = (sin B)/b = (sin C)/c proof pp. 217-218.
Hero's formula: for any triangle, sides a, b, c, semiperimeter s = (a + b + c)/2, area of triangle = Sqrt[ s(s - a)(s - b)(s - c) ] p. 219.
Trigonometric identities sin^2 x + cos^2 x = 1 many others p. 225.
For any polyhedron with a number F of flat faces, a number E of straight-line edges, and a number V of vertices, F + V = 2 + E p. 232
Multiplying complex numbers: The magnitude (length) of the product is the product of the magnitudes. The argument (angle) of the product is the sum of the arguments. Dividing complex numbers, divide the lengths and subtract the angles. p. 240.
e^(i theta) = cos theta + i sin theta e^x = 1 + x + x^2/2! + x^3/3! + x^4/4! + … cos x = 1 - x^2/2! + x^4/4! - x^6/6! + … sin x = x - x^3/3! + x^5/5! - x^7/7! + … p. 251
A Crash Course in Mathematica, Stephan Kaufman, 1999, 200 pages, for Mathematica version 3, ISBN 3764361271, Library-of-Congress QA 76.95 K3913 1999 SA Crash Course in Mathematica, Stephan Kaufman, 1999, 200 pages, for Mathematica version 3, ISBN 3764361271, Library-of-Congress QA 76.95 K3913 1999 Steenbock Library. Translated from German by Katrin Gygax. Originally /Mathematica--kurz und bündig/, 1998.
This is a good primer. The basics this book covers are still current.
Thomas Ray
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