FOUNDATIONS OF QUANTUM MECHANICS

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90 CHAPTER IV. THE COPENHAGEN INTERPRETATION and therefore we have to know the momentum of the screen with an uncertainty δp d . (IV. 7) λ l Gaining such an exactitude is, however, only possible if the screen is movable. But in that case it is no longer possible to fulfil its function as a screen which determines an exact position for the slit. It is therefore no longer part of the original measuring context, as can be seen in figure IV. 3. Figure 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 ) Actually, because now we will perform a measurement on the screen, the screen itself has to be considered an object. This means that quantum mechanics applies to it, and the screen is, therefore, also subject to an uncertainty relation δq λ l . (IV. 8) d But this is an indefiniteness of the same order of magnitude as the distance between the interference bands. Bohr concludes that under these circumstances interference can no longer be seen. With this reasoning he was able to transform Einstein’s objection to an affirmation of his idea of complementarity; as soon as we try to carry out a closer analysis of the phenomenon, we have to modify the experimental setup in such a way that the phenomenon changes unrecognizably. Nowadays an alternative of this thought experiment can actually be carried out in a laboratory, as we will discuss in section IV. 4. IV. 3. 2 THE PHOTON BOX At the 6 th Solvay Conference in 1930 in Brussels, Einstein gave another example, which is known under the name ‘the photon box’. It concerns an isolated box filled with radiation and equipped with a clock mechanism which opens a shutter during a very short interval. It is assumed that in advance the box is weighed meticulously.
IV. 3. DEBATE BETWEEN EINSTEIN EN BOHR 91 Upon closure of the shutter we have, according to Einstein, a choice: either we weigh the box again and determine how much mass has vanished so that we can, using the relation E = m c 2 , retrieve the energy of the escaped photon, or we open the box and read off the clock mechanism to determine when the shutter has been opened, which enables us to predict the time of exit of the photon and therefore its time of arrival at a remote detector. We can choose between both options long after the photon has left. Bohr’s answer is not entirely clear. It may be assumed that he did not understand Einstein’s intentions correctly. 1 He explains Einstein’s objection as an attempt to refute the uncertainty relation between energy and time; he shows that both determinations cannot possibly be made at the same time. Bohr reasons as follows. Assume that the box hangs in equilibrium from a spring in a gravitational field. When in a time interval T a mass δm escapes, it receives an upward impulse F ∆t of magnitude g δm T. (IV. 9) We can keep T finite by, at some moment, hanging a small weight to the box to compensate for the loss of mass. Suppose we want to determine the mass of the photon by measuring this momentum transfer then, again, the momentum of the box at the start of the experiment must be exactly known, δp g δm T. (IV. 10) But now the same argument applies as used in the double slit experiment. This precise determination of momentum is only possible if the fixation of the position of the box is given up. The box itself must be considered a quantum mechanical object, and therefore the uncertainty relation δ pδ q h applies to it. The position of the box is unknown with an uncertainty of magnitude δq g δm T (IV. 11) from which it follows that the gravitational potential ϕ g to which the clock is exposed is also uncertain, δϕ g ≃ g δq . (IV. 12) δm T But according to the red shift formula from the general theory of relativity (!) the pace of a clock is influenced by the gravitational potential, ∆T T = δϕ g , (IV. 13) c2 therefore, the pace of the clock is also uncertain, and consequently the time of opening of the clock is unknown. Under the circumstances in which we can determine the energy of the photon, we cannot retrieve its exit time exactly. Although Bohr seems to rebuke Einstein with his own theory, Bohr’s answer evokes, among other things, the question whether it is appropriate that the correctness of quantum mechanics relies on the correctness of the general theory of relativity, which is a classical theory, and is, strictly spoken, contradictory to quantum mechanics. 1 That Einstein indeed had the intention to point out the freedom of choice is apparent in a letter to Bohr from Paul Ehrenfest, who heard the argument from Einstein earlier.
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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
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
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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 85 and 86: IV. 2. BOHR AND COMPLEMENTARITY 83
Page 91: IV. 3. DEBATE BETWEEN EINSTEIN EN B
Page 95 and 96: IV. 4. NEUTRON INTERFEROMETRY 93 If
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Page 99 and 100: IV. 5. THE UNCERTAINTY RELATIONS 97
Page 111 and 112: V HIDDEN VARIABLES While we have th
Page 113 and 114: V. 2. NON - CONTEXTUAL HIDDEN VARIA
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Page 117 and 118: V. 3 KOCHEN AND SPECKER’S THEOREM
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Page 123 and 124: V. 4. CONTEXTUAL HIDDEN VARIABLES 1
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Page 141 and 142: 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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