FOUNDATIONS OF QUANTUM MECHANICS

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VIII. 4. 7 Modal interpretation . . . . . . . . . . . . 176 VIII. 4. 8 Decoherence . . . . . . . . . . . . . . 177 VIII. 5 Incompatible quantities . . . . . . . . . . . . 179 VIII. 6 Comments on the theory of measurement . . . . . . . . 181 A GLEASON’S THEOREM 183 A. 1 Introduction . . . . . . . . . . . . . . . 183 A. 2 Conversion to a 3 - dimensional real problem . . . . . . . 184 A. 2. 1 Step 1 . . . . . . . . . . . . . . . 185 A. 3 Formulation of the problem on the surface of a sphere . . . . . 186 A. 3. 1 Step 2 . . . . . . . . . . . . . . . 188 A. 3. 1. 1 Lemma 1 . . . . . . . . . . . . . 188 A. 3. 1. 2 Lemma 2 . . . . . . . . . . . . . 189 A. 3. 1. 3 Result of lemma 1 and 2 . . . . . . . . . . 191 A. 3. 2 Step 3 . . . . . . . . . . . . . . . 192 A. 4 An analytic lemma . . . . . . . . . . . . . 196 A. 4. 1 Step 4 . . . . . . . . . . . . . . . 196 A. 5 Summary . . . . . . . . . . . . . . . . 198 WORKS CONSULTED 199 BIBLIOGRAPHY 200
LIST <strong>OF</strong> FIGURES III. 1 A discontinuous measure for dim H = 2 . . . . . . . . . 48 III. 2 A rotated unit vector in the xz - plane . . . . . . . . . . 68 III. 3 Spin up for particle 1 along ⃗a, for particle 2 along ⃗ b . . . . . . 73 IV. 1 Heisenberg’s γ - microscope . . . . . . . . . . . . 79 IV. 2 The double slit interference experiment (Bohr 1949 ) . . . . . . 89 IV. 3 Contexts of measurement in which the interference of the particles is visible, and those in which the recoil of the screen is visible, exclude each other. (Bohr 1949 ) . . . . . . . . . . . . . . . . . . 90 IV. 4 Several perfect crystal neutron interferometers (Rauch and Werner 2000 ) . 93 IV. 5 The interference pattern in the neutron interferometer is acquired by measuring the intensity in the detectors at a variable optical path length difference. . 94 IV. 6 The probability distribution in position for a slit of width 2 a . . . . 101 IV. 7 The diffraction pattern for a small slit of width 2 a . . . . . . . 101 IV. 8 The probability distribution in position for a double slit, 2 a is the width of each slit and 2 A the distance between the slits . . . . . . . . . 104 IV. 9 The interference pattern for the double slit . . . . . . . . . 104 IV. 10 Moving screen . . . . . . . . . . . . . . . . 106 V. 1 A solution for dim H = 2 . . . . . . . . . . . . . 117 V. 2 a) Kochen - Specker diagram b) Conway - Kochen diagram . . . . 118 V. 3 M.C. Escher, Waterfall. Consider the 3 interpenetrating cubes on the top of the left pillar. Each cube has 4 lines from the mutual center to its vertices, 6 lines to the centers of its edges, and 3 lines to the centers of its faces. Three of the lines are shared by all three cubes, giving 3 · (4 + 6 + 3 ) − 6 = 33 lines. These are Peres’ vectors. (Text Meyer 2003 ) . . . . . . . . 119 V. 4 µ(P i ) = cos 2 θ . . . . . . . . . . . . . . . 120 VI. 1 VI. 2 The quantum potential for the two slit system as viewed from the screen, under assumption of a Gaussian distribution at the slits (Bohm 1989 ) . . . 131 A simulation of the double slit experiment in Bohmian mechanics. Each particle follows a certain path between the slits and the photographic plate. All particles coming from the upper slit arrive at the upper half of the photographic plate, likewise for the lower slit and lower half of the plate. The twists in the paths are caused by the quantum potential U. (Vigier et al. 1987 ) . . . 132 VII. 1 Thought experiment of Einstein, Podolsky and Rosen on the singlet . . . 140 VII. 2 A configuration in which the spin quantities violate the Bell inequality . . 142 VII. 3 The Bell inequality violated for every acute angle ϕ . . . . . . 143 VII. 4 the configuration giving the largest violation of the Bell inequality (all vectors in the same plane) . . . . . . . . . . . . . . . 143
Page 1 and 2: FOUNDATIONS OF QUANTUM MECHANICS JO
Page 3 and 4: CONTENTS I CONCEPTUAL PROBLEMS 7 I.
Page 5: VI BOHMIAN MECHANICS 127 VI. 1 Intr
Page 9 and 10: I CONCEPTUAL PROBLEMS Anyone who is
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
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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
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III. 4. COMPOSITE SYSTEMS 55 Proof
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III. 4. COMPOSITE SYSTEMS 57 With (
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III. 4. COMPOSITE SYSTEMS 59 ◃ Re
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Now consider an operator W of the f
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III. 5. PROPER AND IMPROPER MIXTURE
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III. 6. SPIN 1/2 PARTICLES 65 In th
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III. 6. SPIN 1/2 PARTICLES 67 we ha
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III. 6. SPIN 1/2 PARTICLES 69 and w
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III. 6. SPIN 1/2 PARTICLES 71 EXERC
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III. 6. SPIN 1/2 PARTICLES 73 The t
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III. 6. SPIN 1/2 PARTICLES 75 We ar
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IV THE COPENHAGEN INTERPRETATION It
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IV. 1. HEISENBERG AND THE UNCERTAIN
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IV. 1. HEISENBERG AND THE UNCERTAIN
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IV. 2. BOHR AND COMPLEMENTARITY 83
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IV. 3. DEBATE BETWEEN EINSTEIN EN B
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IV. 3. DEBATE BETWEEN EINSTEIN EN B
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IV. 4. NEUTRON INTERFEROMETRY 93 If
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IV. 4. NEUTRON INTERFEROMETRY 95 al
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IV. 5. THE UNCERTAINTY RELATIONS 97
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V HIDDEN VARIABLES While we have th
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V. 2. NON - CONTEXTUAL HIDDEN VARIA
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V. 2. NON - CONTEXTUAL HIDDEN VARIA
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V. 3 KOCHEN AND SPECKER’S THEOREM
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V. 3. KOCHEN AND SPECKER’S THEORE
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V. 3. KOCHEN AND SPECKER’S THEORE
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V. 4. CONTEXTUAL HIDDEN VARIABLES 1
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128 CHAPTER VI. BOHMIAN MECHANICS s
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130 CHAPTER VI. BOHMIAN MECHANICS E
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132 CHAPTER VI. BOHMIAN MECHANICS W
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134 CHAPTER VI. BOHMIAN MECHANICS
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136 CHAPTER VI. BOHMIAN MECHANICS B
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VII BELL’S INEQUALITIES There is
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VII. 1. LOCAL DETERMINISTIC HIDDEN
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VII. 1. LOCAL DETERMINISTIC HIDDEN
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VII. 2. LOCAL DETERMINISTIC CONTEXT
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dµ A(⃗a, λ, µ) dν B( ⃗ b,
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Likewise we calculate the following
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VII. 4. THE DERIVATION OF EBERHARD
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VII. 5. STOCHASTIC HIDDEN VARIABLES
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VII. 6. AN ALGEBRAIC PROOF WITHOUT
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VII. 7. MISCELLANEA 161 Therefore,
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VIII THE MEASUREMENT PROBLEM [. . .
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VIII. 2. MEASUREMENT ACCORDING TO C
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VIII. 3. MEASUREMENT ACCORDING TO Q
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VIII. 3. MEASUREMENT ACCORDING TO Q
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VIII. 4. THE MEASUREMENT PROBLEM IN
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VIII. 5. INCOMPATIBLE QUANTITIES 17
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VIII. 6. COMMENTS ON THE THEORY OF
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A GLEASON’S THEOREM Proofs really
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A. 2. CONVERSION TO A 3 - DIMENSION
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A. 3. FORMULATION OF THE PROBLEM ON
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A. 4. AN ANALYTIC LEMMA 197 To prov
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WORKS CONSULTED Most subjects in th
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202 BIBLIOGRAPHY Bohm, D.J., Aharon
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204 BIBLIOGRAPHY Daneri, A., Loinge
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206 BIBLIOGRAPHY Frank, P.G. (1949)
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208 BIBLIOGRAPHY Isham, C.J. (1995)
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210 BIBLIOGRAPHY Pauli, W.E. (1933)
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212 BIBLIOGRAPHY Suppes, P., Zanott
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FOUNDATIONS OF QUANTUM MECHANICS

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