FOUNDATIONS OF QUANTUM MECHANICS

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198 APPENDIX A. GLEASON’S THEOREM Iteration of (A. 52) yields, for n ∈ N + , nf (a) = f (na) for n a 1. (A. 53) Taking a = 1 n we see that ( 1 f = n) f (1) n and iterating again, we have or, indeed, ( m ) f = m n n = 1 , (A. 54) n for m, n ∈ N, m < n, (A. 55) f (a) = a ∀ a ∈ Q. (A. 56) From (2) we see that lim f (a) = sup f (a) = 0, (A. 57) a→0 a→0 and, using again (A. 52), lim f (a + b) = f (b) ∀ 0 b 1. (A. 58) a→0 Therefore, f is continuous, and f (a) = a ∀ a. □ (A. 59) A. 5 SUMMARY In this appendix we proved Gleason’s theorem for pure states, represented by extreme measures µ. In section A. 2 we proved that if Gleason’s theorem for pure states is true for any 3 - dimensional real Hilbert space, it is also true for any complex Hilbert space with dim H > 2. In A. 3. 1 we showed that µ is a nonincreasing function in θ, and in A. 3. 2 we proved that µ does not depend on ϕ. Finally, by proving the analytic lemma we showed that there can only be one form for the measure µ which satisfies these requirements for all P ∈ P (H) and that is the quantum mechanical one, i.e., in accordance with cos 2 θ.
WORKS CONSULTED Most subjects in these lecture notes are also found in Redhead (1987), Krips (1987), Hughes (1989), D’Espagnat (1989) and Bub (1997). Dickson (1998) is an accessible monograph. Jammer (1974) is a survey of the research in foundations of quantum mechanics in historical perspective from the beginnings of quantum mechanics until 1974. However, Jammer remains indispensable for every student seriously studying foundations of quantum mechanics. Bell (1987) contains his articles on quantum mechanics. Von Neumann’s Grundlagen (1932) is a masterpiece, which is still fully worth studying. Prugovečki (2006) is a modernized and more systematic version, but it evades subjects of interpretation and is mainly a mathematical reference book. Busch, Lahti and Mittelstaedt (1996) is a monograph on quantum mechanical measurement theory. Hooker (1975) is a collection of important articles of algebraic and logical signature. Wheeler and Zurek (1983) is an extensive collection of photocopies of important articles (EPR, Bohr, Bohm, Everett, etc.). Fine (1986) is the unequalled monograph on Einstein and quantum mechanics. Contributions to the research of foundations of quantum mechanics from Utrecht University are the work of Hilgevoord and Uffink and vice versa, of Dieks and Vermaas about the modal interpretation of quantum mechanics and Uffink’s thesis (1990) about uncertainty relations.
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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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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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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
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FOUNDATIONS OF QUANTUM MECHANICS

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