FOUNDATIONS OF QUANTUM MECHANICS

More documents

Recommendations

Info

VII. 5 Unit spheres for a n , b n and a n b n . In the shaded areas of the larger sphere a n b n is positive, in the unshaded areas a n b n is negative. . . . . . . . 144 VII. 6 Comparison of the quantum mechanical expectation values and those for the local deterministic HVT . . . . . . . . . . . . . . 145 VII. 7 Violation of the Bell inequality again . . . . . . . . . . 149 VII. 8 The Mermin pentagon . . . . . . . . . . . . . . 159 VII. 9 Minkowski diagram of the EPRB experiment, where λ is in the past light cones of both A and B . . . . . . . . . . . . . . . 160 VIII. 1 Schrödinger’s cat paradox (DeWitt 1970 ) . . . . . . . . . 170 A. 1 Construction of a 3 - dimensional subspace E . . . . . . . . 185 A. 2 Rotation of s to s 0 and t to t ′ along a great circle around axis r . . . 188 A. 3 Projection of points on a great circle onto a plane P through the north pole 189 A. 4 Projection of meridians, circles with constant latitude, and a great circle . 190 A. 5 Spiral representing a projected path from s to t along subsequent great circles, each time starting at their most northern point . . . . . . . . 190 A. 6 Path from t to v, having the same longitude . . . . . . . . 191 A. 7 A strictly in - or decreasing curve C . . . . . . . . . . 192 A. 8 Great circle C, coordinate system (p, q, t), and rotating pair (s, s ⊥ ) . . 193 A. 9 Great circle C and tilted great circles C ′ and C ′′ . . . . . . . 194 A. 10 Two continuous curves on S 2 , intersecting in q . . . . . . . 195
I CONCEPTUAL PROBLEMS Anyone who is not shocked by quantum theory has not understood it. — Niels Bohr I think it is safe to say that no one understands quantum mechanics. — Richard Feynman I. 1 INTRODUCTION Quantum mechanics emerged at the beginning of the 20 th century from an attempt to understand the interaction between atoms and radiation. The presence of discrete lines in the emission and absorption spectra of chemical elements indicates that this interaction takes the form of discrete quanta. When, in the years 1925 and 1926, a coherent theory was developed by the unified efforts of Werner Heisenberg, Paul Dirac, Max Born, Pascual Jordan, Wolfgang Pauli and Erwin Schrödinger, and this theory was axiomatized seven years later by John von Neumann, the question about the physical interpretation of the mathematical symbols of the theory arose. The central mathematical concept in quantum mechanics is ψ, in the form of a wave function ψ(q) in Schrödinger’s wave mechanics, or of a vector |ψ⟩ in Hilbert space, à la Von Neumann. According to Born, its physical meaning is that ψ determines probabilities for results of measurements, and a key question is then how such probabilities must be interpreted. By means of four examples we will give an idea of the conceptual problems raised by quantum mechanics. (i) Consider as a first example the decay of radioactive nuclei of a certain kind, as discussed by Einstein (P.A. Schilpp (1949, p.667, ff). We see the unstable nuclei decay at various times, one almost immediately, another only after a long time; the α - particles are radiated in ever different directions. Quantum mechanics describes these nuclei by a non-stationary wave function, and using this function one can calculate the expected lifetime of the nuclei. A natural reaction is to assume that the nuclei differ from each other, and that this difference is the cause of the mutually different individual life spans and the different directions the α - particles are radiated in. In this view, the quantum mechanical expectation value would be comparable to the average life span in a population. However, this does not fit in a natural way in the quantum mechanical description. Quantum mechanics describes all nuclei by the same wave function. If this description is complete, the fact that quantum mechanics gives only expected life spans is not due to a lack of knowledge. Rather, there simply is nothing more to know concerning the nuclei than their wave function and the probabilities that follow from it. On the other hand, we see before our eyes that the nuclei do not behave the same way, they decay at different times and send the α - particles in ever different directions. This suggests that more can
Page 1 and 2: FOUNDATIONS OF QUANTUM MECHANICS JO
Page 3 and 4: CONTENTS I CONCEPTUAL PROBLEMS 7 I.
Page 5 and 6: VI BOHMIAN MECHANICS 127 VI. 1 Intr
Page 7: LIST OF FIGURES III. 1 A discontinu
Page 11 and 12: I. 1. INTRODUCTION 9 of affairs. [.
Page 13 and 14: I. 2. INCOMPLETENESS AND LOCALITY 1
Page 19 and 20: II THE FORMALISM As far as the laws
Page 21 and 22: II. 1. FINITE - DIMENSIONAL HILBERT
Page 23 and 24: II. 2. OPERATORS 21 representation
Page 25 and 26: II. 2. OPERATORS 23 An example of a
Page 27 and 28: II. 3. EIGENVALUE PROBLEM AND SPECT
Page 29 and 30: II. 4. FUNCTIONS OF NORMAL OPERATOR
Page 31 and 32: II. 4. FUNCTIONS OF NORMAL OPERATOR
Page 33 and 34: II. 5. DIRECT SUM AND DIRECT PRODUC
Page 35 and 36: II. 5. DIRECT SUM AND DIRECT PRODUC
Page 37 and 38: II. 6. ADDENDUM: INFINITE - DIMENSI
Page 39 and 40: II. 6. ADDENDUM: INFINITE - DIMENSI
Page 41 and 42: The angular momentum operator II. 6
Page 43 and 44: III THE POSTULATES The sciences do
Page 45 and 46: III. 1. VON NEUMANN’S POSTULATES
Page 47 and 48: III. 2. PURE AND MIXED STATES 45 Ad
Page 49 and 50: III. 2. PURE AND MIXED STATES 47 Ea
Page 51 and 52: III. 2. PURE AND MIXED STATES 49 Th
Page 53 and 54: III. 3. THE INTERPRETATION OF MIXED
Page 55 and 56: ut the probability to find the syst
Page 57 and 58: III. 4. COMPOSITE SYSTEMS 55 Proof
Page 59 and 60:
III. 4. COMPOSITE SYSTEMS 57 With (
Page 61 and 62:
III. 4. COMPOSITE SYSTEMS 59 ◃ Re
Page 63 and 64:
Now consider an operator W of the f
Page 65 and 66:
III. 5. PROPER AND IMPROPER MIXTURE
Page 67 and 68:
III. 6. SPIN 1/2 PARTICLES 65 In th
Page 69 and 70:
III. 6. SPIN 1/2 PARTICLES 67 we ha
Page 71 and 72:
III. 6. SPIN 1/2 PARTICLES 69 and w
Page 73 and 74:
III. 6. SPIN 1/2 PARTICLES 71 EXERC
Page 75 and 76:
III. 6. SPIN 1/2 PARTICLES 73 The t
Page 77 and 78:
III. 6. SPIN 1/2 PARTICLES 75 We ar
Page 79 and 80:
IV THE COPENHAGEN INTERPRETATION It
Page 81 and 82:
IV. 1. HEISENBERG AND THE UNCERTAIN
Page 83 and 84:
IV. 1. HEISENBERG AND THE UNCERTAIN
Page 85 and 86:
IV. 2. BOHR AND COMPLEMENTARITY 83
Page 87 and 88:
Page 89 and 90:
Page 91 and 92:
IV. 3. DEBATE BETWEEN EINSTEIN EN B
Page 93 and 94:
IV. 3. DEBATE BETWEEN EINSTEIN EN B
Page 95 and 96:
IV. 4. NEUTRON INTERFEROMETRY 93 If
Page 97 and 98:
IV. 4. NEUTRON INTERFEROMETRY 95 al
Page 99 and 100:
IV. 5. THE UNCERTAINTY RELATIONS 97
Page 101 and 102:
Page 103 and 104:
Page 105 and 106:
Page 107 and 108:
Page 109 and 110:
Page 111 and 112:
V HIDDEN VARIABLES While we have th
Page 113 and 114:
V. 2. NON - CONTEXTUAL HIDDEN VARIA
Page 115 and 116:
V. 2. NON - CONTEXTUAL HIDDEN VARIA
Page 117 and 118:
V. 3 KOCHEN AND SPECKER’S THEOREM
Page 119 and 120:
V. 3. KOCHEN AND SPECKER’S THEORE
Page 121 and 122:
V. 3. KOCHEN AND SPECKER’S THEORE
Page 123 and 124:
V. 4. CONTEXTUAL HIDDEN VARIABLES 1
Page 125 and 126:
Page 127:
Page 130 and 131:
128 CHAPTER VI. BOHMIAN MECHANICS s
Page 132 and 133:
130 CHAPTER VI. BOHMIAN MECHANICS E
Page 134 and 135:
132 CHAPTER VI. BOHMIAN MECHANICS W
Page 136 and 137:
134 CHAPTER VI. BOHMIAN MECHANICS
Page 138 and 139:
136 CHAPTER VI. BOHMIAN MECHANICS B
Page 141 and 142:
VII BELL’S INEQUALITIES There is
Page 143 and 144:
VII. 1. LOCAL DETERMINISTIC HIDDEN
Page 145 and 146:
VII. 1. LOCAL DETERMINISTIC HIDDEN
Page 147 and 148:
VII. 2. LOCAL DETERMINISTIC CONTEXT
Page 149 and 150:
dµ A(⃗a, λ, µ) dν B( ⃗ b,
Page 151 and 152:
Likewise we calculate the following
Page 153 and 154:
VII. 4. THE DERIVATION OF EBERHARD
Page 155 and 156:
VII. 5. STOCHASTIC HIDDEN VARIABLES
Page 157 and 158:
Page 159 and 160:
Page 161 and 162:
VII. 6. AN ALGEBRAIC PROOF WITHOUT
Page 163 and 164:
VII. 7. MISCELLANEA 161 Therefore,
Page 165 and 166:
VIII THE MEASUREMENT PROBLEM [. . .
Page 167 and 168:
VIII. 2. MEASUREMENT ACCORDING TO C
Page 169 and 170:
VIII. 3. MEASUREMENT ACCORDING TO Q
Page 171 and 172:
VIII. 3. MEASUREMENT ACCORDING TO Q
Page 173 and 174:
VIII. 4. THE MEASUREMENT PROBLEM IN
Page 175 and 176:
Page 177 and 178:
Page 179 and 180:
Page 181 and 182:
VIII. 5. INCOMPATIBLE QUANTITIES 17
Page 183 and 184:
VIII. 6. COMMENTS ON THE THEORY OF
Page 185 and 186:
A GLEASON’S THEOREM Proofs really
Page 187 and 188:
A. 2. CONVERSION TO A 3 - DIMENSION
Page 189 and 190:
A. 3. FORMULATION OF THE PROBLEM ON
Page 191 and 192:
Page 193 and 194:
Page 195 and 196:
Page 197 and 198:
Page 199 and 200:
A. 4. AN ANALYTIC LEMMA 197 To prov
Page 201:
WORKS CONSULTED Most subjects in th
Page 204 and 205:
202 BIBLIOGRAPHY Bohm, D.J., Aharon
Page 206 and 207:
204 BIBLIOGRAPHY Daneri, A., Loinge
Page 208 and 209:
206 BIBLIOGRAPHY Frank, P.G. (1949)
Page 210 and 211:
208 BIBLIOGRAPHY Isham, C.J. (1995)
Page 212 and 213:
210 BIBLIOGRAPHY Pauli, W.E. (1933)
Page 214 and 215:
212 BIBLIOGRAPHY Suppes, P., Zanott
show all

FOUNDATIONS OF QUANTUM MECHANICS

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?