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Philosophical Perspectives on the Exact Sciences and Their History” Master Projects (EPFL) “Philosophical perspectives on the exact sciences and their history” Some remarks on the measurement problem in quantum theory (Davide Romano) 1. An introduction to the quantum formalism In quantum mechanics, a generic physical state of a system is described by a unitary-norm vector � in Hilbert space ℋ. In the Dirac notation, this vector is called “ket” and it is represented by |� . ℋis a complex vector space with a norm given by � ∶= � � . The complex conjugated vector of |� is �| and it is called “bra”. Given the linearity of the Hilbert space, if two different physical states are possible for a quantum system, say |� and |� , then the linear superposition of these two states is another possible physical state for the system: |� =�!|� + �!|� Taken two generic ket, the inner product between them gives a complex number, which is a probability amplitude: � � = �!, with �! ∈ ℂ We can interpret the above expression as the probability amplitude of a physical transition from the state |� to the state |� Given a complete orthonormal basis in Hilbert space (we refer to the vectors of this basis as eigenvectors), we can represent any vector of that space as a generic linear superposition of eigenvectors. In QM, the most complete mathematical representation of the physical state of a system is given by its state-vector, which generally is a linear superposition of eigenstates. An eigenstate is an eigenvector of a hermitian operator (in the quantum formalism, each physical quantity or “observable” is described by a hermitian operator in Hilbert space): |� = ! �!|�! , where {�!} is a complete orthonormal basis of eigenvectors. 1 If we express a state-vector, i.e. |� , in the position basis representation (that is, we choose eigenstates of the position as eigenvector of the basis), we obtain the wave-function of the system: �! � = �! � ; wave function of the system � Mathematically, the wave function is a probability amplitude: its absolute square gives the density of probability about the position of a particle: � � ! ≔ �∗ � � � ; density of probability of the position of a particle. In order to have the probability to find the particle in a certain volume of space, we have to integer the density of probability in such a volume of space: !!!" ! ! � � �� ; probability to find the particle in the region �, � + �� 2. Time evolution and measurement of a system 2.1 Evolution of closed systems (linear evolution) The time evolution of a closed (or isolated) quantum system is given by the Schrödinger equation. The Schrödinger equation for a state-vector is: ! ! ! (1.a) �ℏ |� = + � |� ; � = + � is the Hamiltonian operator. !" !! !! The Schrödinger equation for a wavefunction is: ! !! ! (1.b) �ℏ � �, � = − + � � �, � !" !! !!! The Schrödinger equation is a linear differential equation: if we have two different solution �! and �!, then there is a linear superposition of them that is again a solution of the equation, � = � �! + � �!. This mathematical property is essential in order to hold in time the superposition principle of states that we have introduced before. The time evolution given by the Schrödinger equation is deterministic: given a state-vector (or a wavefunction) at a certain time and the initial conditions for 2 the system, we can predict the dynamical behavior of the quantum system for all times. So, where does the famous quantum probabilities come from? 2.2 The measure of a system (non linear evolution) When we perform a measure on a quantum system, or more generally when a quantum system interacts with a classical measurement apparatus (humans are only a subset of the set of all possible measurement apparata), the state-vector, which is described by a linear superposition of eigenstates, collapses in an istantaneous and random manner into one of its eigenstates (postulate of the collapse). The probability that the state-vector collapses into one definite eigenstate is given by the absolute square of the coefficient related to that eigenstate (Born rule): !"#$%&"!"'( ! ! (2) |� = ! �! |�! |�! ���ℎ � = �! ∶= �! � Now, we are ready to face the measurement problem! 3. The measurement problem Given a generic a quantum system |� = ! �! |�! , |� ∈ ℋ! and a classical apparatus |� ∈ ℋ!, we can describe the global system which is composed by the system and the apparatus through the state-vector |Ψ |Ψ = |� |� , |Ψ ∈ ℋ! ⊗ ℋ! The total state |Ψ is an entangled state between the system and the apparatus. At the initial time t=0, |Ψ is a factorized state, thus we can write |Ψ as a tensor product of two eigenstates which live in two different Hilbert spaces: 3 |Ψ 0 = |� 0 |� 0 = �! |�! 0 |� 0 ! During the time interval in which the experiment is performed, the state-vector of the global system evolves according to the Schrödinger equation. Because of the linearity of the equation, in the final state each eingenstate of the system is entangled with a different macroscopic physical state of the apparatus: !"!!ö!"#$%& !"#$%&'#( |Ψ 0 = ! �! |�! 0 |� 0 |Ψ � = ! �!|�! |�! �! represents different macroscopic physical states of the classical apparatus: for example, a pointer which points to the left or to the right, a display which indicates a number, … These are also called pointer states and they are the analogous for the apparatus of the eigenstates for the system: in other words, they are all the possible physical states in which the apparatus can be after the measure. Do you see the problem? The final state of the global system is not a definite physical state, but it is a linear superposition of different macroscopic states. This is not in agreement with our experience: when we perform a measure on a quantum system, we always end up with a unique definite macroscopic result. This is the core of the measurement problem! 3.1 A pragmatic solution for standard quantum mechanics Solution in Standard QM: when a measure on a quantum system is performed by a classical apparatus, the state-vector of the quantum system collapses in one of its eigenstates (postulate of the collapse) with probability given by the square norm of the coefficient (Born’s rule). If we introduce in the quantum formalism the postulate of the collapse and the Born rule, then the theoretical predictions of the measurement outcomes performed on a quantum system are in perfect agreement with all the experimental results. So, what’s the problem? 4 3.2 Beyond the standard QM: quantum theory without observer For All Practical Purposes, there are no problems at all (and, in fact, for the vast majority of the physicists the measurement problem is not a real problem!) Nevertheless, if we are interested in the conceptual foundations of QM, we have to deal with the following issues: 1) With the introduction of the collapse postulate, QM is divided into two different regimes, which are governed by two different laws: when a quantum system is isolated, it evolves according to the Schrödinger equation, whereas when a system is “measured” or “observed”, then it collapses randomly in one of its eigenstates with probability given by the Born rule. But there is nothinG in the quantum formalism itself that defines what counts as an observer in the theory! If we don’t know what exactly counts as an observer, we don’t know precisely when the postulate of the collapse applies on the quantum systems, and the quantum formalism remains vague and ambiguous. 2) It seems quite strange that, before the observer enters into the picture, the global state does not have a definite result! This could easily lead to some paradox, like SchrödinGer’s cat: before we open the box, the cat is in a macroscopic superposition to be dead and alive at the same time (this paradox strongly lies on the validity of the eigenvalue-eigenstate link. For a critical discussion of such a postulate, see Wallace (2012)) . But this is absurd: a fundamental physical theory should give a precise and definite account of the physical state of the system even before the observer acts upon the system. Can really the cat be in a macroscopic superposition before we open the box? The reality which is described by a fundamental physical theory should be observer-independent. Note: Einstein’s main critique to the Copenhagen interpretation of QM was about the problem of the realism of that theory, not about the probabilistic nature of it (as it commonplace to think). 3) The collapse postulate introduces a distinction between a macroscopic world (measuring devices, observers, …) where the rules of classical mechanics hold (so, no superposition of states) and a microscopic world, which is described by the rules of quantum mechanics (so that superposition of states is permitted…). See, for example, the solution proposed by Griffiths’ textbook on QM for the Schrödinger’s cat paradox: “the essence of a measure is that a 5 macroscopic system has been modified. We have a meaure when a microscopic system (which is described by the laws of quantum mechanics) interacts with a macroscopic system (which is described by the rules of classical mechanics), leaving on the latter a permanent record. The macroscopic system itself is not authorized to be in a linear superposition of distinct states” (Griffiths, Introduction to Quantum Mechanics, section 12.4: Schrödinger’s cat). The distinction between microscopic and macroscopic systems seems to be artificial (ad hoc) and arbitrary (the boundary line between these two levels is not specifyed by the theory itself). Indeed, it is not easy to define precisely when a system, foe example, becomes macroscopic: how many particles we need in order to have a macroscopic system?. Moreover, the fact that a system is macroscopic is not always a guarantee of the classicality of it: we can have macroscopic quantum system (i.e., superconductors), which are macroscopic system that show a quantum behvior. So, “mascroscopicness” seems not to be the only requirement for the origin of the classical regime.
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