FOUNDATIONS OF QUANTUM MECHANICS

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118 CHAPTER V. HIDDEN VARIABLES Kochen and Specker (1967, p. 69) explicitly constructed a set of 117 spin quantities for which no consistent value assignment exists. This construction is depicted on the cover of Redhead (1987) and can be seen in figure V. 2 a. It shows that every value assignment in accordance with function rule (V. 21) leads to contradictions. Kochen and Conway only needed 31 quantities in the so - called Peres cube of 33 points (Peres 1993). This construction is depicted in figure V. 2 b. □ Figure V. 2: a) Kochen - Specker diagram b) Conway - Kochen diagram (Redhead 1987 ) (Tkadlec 2000 )
V. 3. KOCHEN AND SPECKER’S THEOREM 119 Figure 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 ) It is interesting to see what the measure (V. 29), according to Von Neumann the probability measure of quantum mechanics, looks like in this case. For a pure state W = |ψ⟩ ⟨ψ|, with P i = |χ⟩ ⟨χ| the measure (V. 29) is µ(P i ) = Tr P i W = ⟨ψ | P i | ψ⟩ = |⟨χ | ψ⟩| 2 (V. 32) so that in a real space we have µ(P i ) = |⟨χ | ψ⟩| 2 = cos 2 θ, (V. 33)
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FOUNDATIONS OF QUANTUM MECHANICS JO
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CONTENTS I CONCEPTUAL PROBLEMS 7 I.
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VI BOHMIAN MECHANICS 127 VI. 1 Intr
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LIST OF FIGURES III. 1 A discontinu
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I CONCEPTUAL PROBLEMS Anyone who is
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I. 1. INTRODUCTION 9 of affairs. [.
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I. 2. INCOMPLETENESS AND LOCALITY 1
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II THE FORMALISM As far as the laws
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II. 1. FINITE - DIMENSIONAL HILBERT
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II. 2. OPERATORS 21 representation
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II. 2. OPERATORS 23 An example of a
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II. 3. EIGENVALUE PROBLEM AND SPECT
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II. 4. FUNCTIONS OF NORMAL OPERATOR
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II. 4. FUNCTIONS OF NORMAL OPERATOR
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II. 5. DIRECT SUM AND DIRECT PRODUC
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II. 5. DIRECT SUM AND DIRECT PRODUC
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II. 6. ADDENDUM: INFINITE - DIMENSI
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II. 6. ADDENDUM: INFINITE - DIMENSI
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The angular momentum operator II. 6
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III THE POSTULATES The sciences do
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III. 1. VON NEUMANN’S POSTULATES
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III. 2. PURE AND MIXED STATES 45 Ad
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III. 2. PURE AND MIXED STATES 47 Ea
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III. 2. PURE AND MIXED STATES 49 Th
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III. 3. THE INTERPRETATION OF MIXED
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ut the probability to find the syst
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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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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
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Page 79 and 80: IV THE COPENHAGEN INTERPRETATION It
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Page 95 and 96: IV. 4. NEUTRON INTERFEROMETRY 93 If
Page 97 and 98: IV. 4. NEUTRON INTERFEROMETRY 95 al
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Page 111 and 112: V HIDDEN VARIABLES While we have th
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Page 141 and 142: VII BELL’S INEQUALITIES There is
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Page 149 and 150: dµ A(⃗a, λ, µ) dν B( ⃗ b,
Page 151 and 152: Likewise we calculate the following
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Page 163 and 164: VII. 7. MISCELLANEA 161 Therefore,
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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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