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Gravity and the Flea 5.1 Gravity and the flea A gravitational approach to the measurement problem Loek van Rossem Bachelor Thesis in Mathematics and Physics & Astronomy Supervisor: Prof. dr. N.P. Landsman Radboud University Nijmegen The Netherlands August 20, 2018 Abstract In this text we study some possible solutions to the measurement problem in quantum mechanics. This problem is about an inherent contradiction between the Schr¨odinger equation and the Born postulate: the Schr¨odingerequation tells us how a system evolves, yet when it is applied to a measurement it does not reproduce the Born postulate. Alt- hough many solutions to the measurement problem have been proposed, to this day the problem remains unsolved. We first look at \the flea on Schr¨odinger'scat", which is based on the idea that small perturbations can have a large influence on macroscopic quantum mechanical systems. Therefore, the randomness in the Born postulate could be explained as a lack of knowledge about these perturbations. Some problems with this method are discussed, and an attempt will be made to address them. Then we will have a look at the Penrose approach, which tries to use gravity to solve the measurement problem. Inspired by the problems of these two solutions, we will try to combine them into a single hybrid solution, in an attempt to solve the issues. Page 2 Contents 1 Introduction 4 2 The measurement problem 5 2.1 A contradiction in quantum mechanics . 5 2.2 Interpreting measurements . 6 3 The flea on Schr¨odinger'scat 8 3.1 Instability of large quantum systems . 8 3.2 The measurement process . 10 3.3 Born probabilities . 12 3.4 Tunneling times . 13 3.5 Energy conservation . 14 3.6 Stability of outcomes . 15 3.7 Entanglement and locality . 16 4 The Penrose interpretation 18 4.1 Superpositions in space-time . 18 4.2 The Schr¨odinger-Newtonequation . 19 4.3 Collapse from the Schr¨odinger-Newtonequation . 21 5 Gravity and the flea 23 5.1 The measurement process 2.0 . 23 5.2 Applicability of the Schr¨odinger-Newtonequation . 25 5.3 Born probabilities . 27 5.4 Tunneling times . 31 5.5 Entanglement and locality . 32 6 Conclusion 33 References 33 Page 3 Chapter 1 Introduction The measurement problem is one of the oldest problems in quantum mechanics, and has been around since its discovery. Records of it go as far back as 1926, when Born questioned whether or not the outcomes of measurements in quantum mechanics are determined by hidden properties. The measurement problem is about a discrepancy between quantum mechanics and classical mechanics. Quantum mechanics predicts superpositions, even at the macroscopic scale, yet such things are completely absent in classical mechanics. The most famous formulation of the measurement problem is probably Schr¨odinger'scat, where quantum mechanics paradoxically predicts a cat to be in a superposition of being both alive and dead at the same time. Many solutions have been proposed over the years, such as the many-worlds interpretation, pilot wave theory, and Ghirardi-Rimini-Weber theory. Still, to this day, no completely satisfying solution has been found, despite the fundamental nature of the measurement problem. The aim of this text is to explore and discuss some of the solutions to the measurement problem. In chapter 2 we provide the mathematical details of the measurement problem and give a few examples of well known solutions. In chapter 3 we take a look at a more recent solution to the measurement problem: \the flea on Schr¨odinger'scat". We also discuss some of the issues it has and possible approaches to solving them. In chapter 4 we will look at a solution proposed by Penrose which claims that gravity is the crucial component causing wave function collapse. Finally, in chapter 5 we will propose a solution which makes use of ideas from both \the flea on Schr¨odinger'scat" and Penrose's solution. Page 4 Chapter 2 The measurement problem The measurement problem may be described as the failure of quantum theory to reproduce the macroscopic classical world.1 To put it briefly, quantum mechanics predicts macrosco- pic superpositions, yet none are ever observed. Despite the measurement problem being almost as old as quantum mechanics itself, no satisfying solution has been found. 2.1 A contradiction in quantum mechanics In quantum mechanics, states evolve according to the Schr¨odingerequation: @ i¯h j (t)i = H j (t)i : (2.1) @t On the other hand, measurements occur according to the Born postulate, which says that the possible measurement outcomes for measuring an observable A are its eigenvectors, and the probability pφ of obtaining such an eigenvector jφi as an outcome is 2 pφ = j hφj i j ; (2.2) where j i is the state before the measurement. After the measurement, the state of the system has been reduced to jφi, and we say the wave function has collapsed. Using these postulates, most of modern day physics can be derived, yet they also lead to a contradiction. After all, measurement apparatuses are physical objects, and so they too will evolve according to the Schr¨odingerequation. Thus we can also use the Schr¨odingerequation to see what the outcome of a measurement is. In general, this does not lead to the same anwser as the Born postulate. For instance, consider the spin of an electron in the following superposition: 1 p (j"i + j#i): (2.3) 2 Here j"i represents the state where the spin is in the positive z direction (i.e. σz j"i = j"i), and is j#i the state where the spin is in the negative z direction (σz j#i = − j#i). If we measure the spin in the z direction, then according to the Born postulate there is a 50% chance of getting the outcome spin up and a 50% chance of getting spin down. Now we investigate what should happen according to the Schr¨odingerequation. Let j0iA be the state initially describing the measurement apparatus. Then our total initial state is: 1 j (0)i = p (j"i + j#i) ⊗ j0i : (2.4) 2 A 1Many different formulations of the measurement problem are possible [5]. Page 5 CHAPTER 2. THE MEASUREMENT PROBLEM 2.2 The Schr¨odingerequation says that time evolution is a unitary operator U(t), so 1 j (t)i = U(t) p (j"i + j#i) ⊗ j0i 2 A 1 1 = p U(t)(j"i ⊗ j0i ) + p U(t)(j#i ⊗ j0i ) : (2.5) 2 A 2 A Since the measurement apparatus measures the spin, we have U(T )(j"i ⊗ j0iA) = j"i ⊗ j"iA ; U(T )(j#i ⊗ j0iA) = j#i ⊗ j#iA ; (2.6) where j"iA is the state of the measurement apparatus indicating spin up, j#iA is the state of the measurement apparatus indicating spin down, and T is a point in time large enough such that the measurement can be considered complete. Combining (2.5) and (2.6), we obtain 1 1 j (T )i = p j"i ⊗ j"i + p j#i ⊗ j#i : (2.7) 2 A 2 A So instead of collapsing the superposition of the electron, the measurement apparatus joins the electron in the superposition. This is in contradiction with the Born postulate; we now have a 100% chance of getting the state p1 j"i ⊗ j"i + p1 j#i ⊗ j#i instead of a 2 A 2 A 50% chance of getting j"i ⊗ j"iA and a 50% chance of getting j#i ⊗ j#iA. This paradox is probably best known in the form of Schr¨odinger'scat. There, a scientist puts a cat in a box, along with a radioactive atom. The box is set up so that it will release a deadly poison if the atom decays. Since the atom is a quantum system, it will enter a superposition of being decayed and not decayed. This will result in the cat being in a superposition of dead and alive, which is not the expected macroscopic behavior. Here, the atom plays the role of the electron and the cat plays the role of the measurement apparatus. 2.2 Interpreting measurements This paradox is known as the measurement problem. The different ways of resolving it correspond to the interpretations of quantum mechanics. We will now give a few examples of these interpretations. One of the better known interpretations is the many-worlds interpretation, also known as the Everett interpretation. According to this, the Schr¨odingerequation always gives the correct outcome. Thus, when a measurement takes place, a macroscopic superposition like (2.7) really does occur physically. The interpretation says that the two parts of the wave function each represent their own branch of the universe, one in which the outcome was spin up and the device registered spin up, and one in which the outcome was spin down and the device registered spin down. An observer would read off spin up or spin down, so it would appear to him that the wave function collapsed. Another interpretation is the pilot wave theory, or de Broglie{Bohm theory. In pilot wave theory the particle and its wave function are separate entities. The wave function guides the behavior of the particle according to the so called guiding equation. This equation is constructed in such a way as to recover standard quantum mechanics. The particle always has a well-defined position (as opposed to a superposition), so no collapse happens during a measurement. Finally, there is Ghirardi{Rimini{Weber theory. Ghirardi{Rimini{Weber theory is an example of an objective collapse theory, which means that wave function collapse is a Page 6 CHAPTER 2.
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