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Universal Probability Distribution for the Wave Function of a Quantum System Entangled with Its Environment
No full textA quantum system (with Hilbert space H1) entangled with its environment (with Hilbert space H2) is usually not attributed a wave function but only a reduced density matrix ρ1. Nevertheless, there is a precise way of attributing to it a random wave function ψ1, called its conditional wave function, whose probability distribution μ1 depends on the entangled wave function ψ∈H1⊗H2 in the Hilbert space of system and environment together. It also depends on a choice of orthonormal basis of H2 but in relevant cases, as we show, not very much. We prove several universality (or typicality) results about μ1, e.g., that if the environment is sufficiently large then for every orthonormal basis of H2, most entangled states ψ with given reduced density matrix ρ1 are such that μ1 is close to one of the so-called GAP (Gaussian adjusted projected) measures, GAP(ρ1). We also show that, for most entangled states ψ from a microcanonical subspace (spanned by the eigenvectors of the Hamiltonian with energies in a narrow interval [E,E+δE]) and most orthonormal bases of H2, μ1 is close to GAP(tr2ρmc) with ρmc the normalized projection to the microcanonical subspace. In particular, if the coupling between the system and the environment is weak, then μ1 is close to GAP(ρβ) with ρβ the canonical density matrix on H1 at inverse temperature β=β(E). This provides the mathematical justification of our claim in [J. Statist. Phys. 125:1193 (2006), http://arxiv.org/abs/quant-ph/0309021] that GAP measures describe the thermal equilibrium distribution of the wave function.Peer reviewe
Arrival Times Versus Detection Times
No full textHow to compute the probability distribution of a detection time, i.e., of the time which a detector registers as the arrival time of a quantum particle, is a long-debated problem. In this regard, Bohmian mechanics provides in a straightforward way the distribution of the time at which the particle actually does arrive at a given surface in 3-space in the absence of detectors. However, as we discuss here, since the presence of detectors can change the evolution of the wave function and thus the particle trajectories, it cannot be taken for granted that the arrival time of the Bohmian trajectories in the absence of detectors agrees with the one in the presence of detectors, and even less with the detection time. In particular, we explain why certain distributions that Das and Dürr (Sci. Rep. 9: 2242, 2019) presented as the distribution of the detection time in a case with spin, based on assuming that all three times mentioned coincide, are actually not what Bohmian mechanics predicts
Is the hypothesis about a low entropy initial state of the Universe necessary for explaining the arrow of time?
No full textAccording to statistical mechanics, microstates of an isolated physical system (say, a gas in a box) at time t0 in a given macrostate of less-than-maximal entropy typically evolve in such a way that the entropy at time t increases with |t-t0| in both time directions. In order to account for the observed entropy increase in only one time direction, the thermodynamic arrow of time, one usually appeals to the hypothesis that the initial state of the Universe was one of very low entropy. In certain recent models of cosmology, however, no hypothesis about the initial state of the Universe is invoked. We discuss how the emergence of a thermodynamic arrow of time in such models can nevertheless be compatible with the above-mentioned consequence of statistical mechanics, appearances to the contrary notwithstanding
On the spin dependence of detection times and the nonmeasurability of arrival times
No full textAccording to a well-known principle of quantum physics, the statistics of the outcomes of any quantum experiment are governed by a Positive-Operator-Valued Measure (POVM). In particular, for experiments designed to measure a specific physical quantity, like the time of a particle’s first arrival at a surface, this principle establishes that if the probability distribution of that quantity does not arise from a POVM, no such experiment exists. Such is the case with the arrival time distributions proposed by Das and Dürr, due to the nature of their spin dependence
Who’s Afraid of the Measurement Problem?
No full textScientific realists usually claim that quantum mechanics can be made compatible with scientific realism by solving the measurement problem, even if there is disagreement about which solution is best. In this paper I argue this is due to having different views about what it means to make quantum theory compatible with scientific realism: ‘relaxed’ realists think it is enough to solve the adequacy problem, ‘modest’ realists believe that there is also a precision problem, while ‘robust’ realists insist that quantum theory still needs to be suitably completed. These attitudes are connected with the type of explanation one favors: while relaxed realists favor principle theories, robust realists prefer constructive theories, and modest realists provide non-constructive dynamical hybrids as long as they preserve locality and separability
Any orthonormal basis in high dimension is uniformly distributed over the sphere
No full textLet Xd be a real or complex Hilbert space of finite but large dimension d, let S(Xd ) denote the unit sphere of Xd, and let u denote the normalized uniform measure on S(Xd ). For a finite subset B of S(Xd ), we may test whether it is approximately uniformly distributed over the sphere by choosing a partition A1, . . . , Am of S(Xd ) and checking whether the fraction of points in B that lie in Ak is close to u(Ak) for each k = 1, . . . , m. We show that if B is any orthonormal basis of Xd and m is not too large, then, if we randomize the test by applying a random rotation to the sets A1, . . . , Am, B will pass the random test with probability close to 1. This statement is related to, but not entailed by, the law of large numbers. An application of this fact in quantum statistical mechanics is briefly described
Universal Probability Distribution for the Wave Function of a Quantum System Entangled with its Environment
No full textA quantum system (with Hilbert space (Formula presented.)) entangled with its environment (with Hilbert space (Formula presented.)) is usually not attributed to a wave function but only to a reduced density matrix (Formula presented.). Nevertheless, there is a precise way of attributing to it a random wave function (Formula presented.) , called its conditional wave function, whose probability distribution (Formula presented.) depends on the entangled wave function (Formula presented.) in the Hilbert space of system and environment together. It also depends on a choice of orthonormal basis of (Formula presented.) but in relevant cases, as we show, not very much. We prove several universality (or typicality) results about (Formula presented.) , e.g., that if the environment is sufficiently large then for every orthonormal basis of (Formula presented.) , most entangled states (Formula presented.) with given reduced density matrix (Formula presented.) are such that (Formula presented.) is close to one of the so-called GAP (Gaussian adjusted projected) measures, (Formula presented.). We also show that, for most entangled states (Formula presented.) from a microcanonical subspace (spanned by the eigenvectors of the Hamiltonian with energies in a narrow interval (Formula presented.)) and most orthonormal bases of (Formula presented.) , (Formula presented.) is close to (Formula presented.) with (Formula presented.) the normalized projection to the microcanonical subspace. In particular, if the coupling between the system and the environment is weak, then (Formula presented.) is close to (Formula presented.) with (Formula presented.) the canonical density matrix on (Formula presented.) at inverse temperature (Formula presented.). This provides the mathematical justification of our claim in Goldstein et al. (J Stat Phys 125: 1193–1221, 2006) that GAP measures describe the thermal equilibrium distribution of the wave function
Gibbs and Boltzmann Entropy in Classical and Quantum Mechanics
No full textThe Gibbs entropy of a macroscopic classical system is a function of a probability distribution over phase space, i.e., of an ensemble. In contrast, the Boltzmann entropy is a function on phase space, and is thus defined for an individual system. Our aim is to discuss and compare these two notions of entropy, along with the associated ensemblist and individualist views of thermal equilibrium. Using the Gibbsian ensembles for the computation of the Gibbs entropy, the two notions yield the same (leading order) values for the entropy of a macroscopic system in thermal equilibrium. The two approaches do not, however, necessarily agree for non-equilibrium systems. For those, we argue that the Boltzmann entropy is the one that corresponds to thermodynamic entropy, in particular, in connection with the second law of thermodynamics. Moreover, we describe the quantum analog of the Boltzmann entropy, and we argue that the individualist (Boltzmannian) concept of equilibrium is supported by the recent works on thermalization of closed quantum systems
Physical Meaning of Neumann and Robin Boundary Conditions for the Schrödinger Equation
Full text linkThe non-relativistic Schrödinger equation on a domain with boundary is often considered with homogeneous Dirichlet boundary conditions ( for on the boundary), homogeneous Neumann boundary conditions ( for on the boundary and the normal derivative), or Robin boundary conditions ( for on the boundary and a real parameter). Physically, the Dirichlet condition applies if the potential is much higher outside than inside the domain (``potential well\u27\u27). We ask, when does the Neumann or Robin condition apply physically? Our answer is, when the potential is much lower (at the appropriate level) in a thin layer along the surface of a potential well, or when a negative delta potential of the appropriate strength is added at a surface close to the surface of the potential well.14 pages LaTeX, no figure files; v3: major revision, a gap in the argument has been closed, and more detailed explanations have been adde
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