The spin-statistics connection is derived in a simple manner under the postulates that the original and the exchange wave functions are simply added, and that the azimuthal phase angle, which defines the orientation of the spin part of each single-particle spin-component eigenfunction in the plane normal to the spin-quantization axis, is exchanged along with the other parameters. The spin factor (-1)^2s belongs to the exchange wave function when this function is constructed so as to get the spinor ambiguity under control. This is achieved by effecting the exchange of the azimuthal angle by means of rotations and admitting only rotations in one sense. The procedure works in Galilean as well as in Lorentz-invariant quantum mechanics. Relativistic quantum field theory is not required.
The connection is established between two theories that have developed independently with the aim to describe quantum mechanics as a stochastic process, namely stochastic quantum mechanics (sqm) and stochastic electrodynamics (sed). Important commonalities and complementarities between the two theories are identified, notwithstanding their dissimilar origins and approaches. Further, the dynamical equation of sqm is completed with the radiation terms that are an integral element in sed. The central problem of the transition to the quantum dynamics is addressed, pointing to the key role of diffu…
By considering (non-relativistic) quantum mechanics as it is done in practice in particular in condensed-matter physics, it is argued that a deterministic, unitary time evolution within a chosen Hilbert space always has a limited scope, leaving a lot of room for embedding the quantum-classical transition into our current theories without recurring to difficult-to-accept interpretations of quantum mechanics. Nonunitary projections to initial and final states, the breaking of time-reversal symmetry, a change of Hilbert space, and the introduction of classical concepts such as external potentials…
Recently there has been much interest in deriving the quantum formalism and the set of quantum correlations from simple axioms. In this paper, we provide a step-by-step derivation of the quantum formalism that tackles both these problems and helps us to understand why this formalism is as it is. We begin with a structureless system that only includes real-valued observables, states and a (not specified) state update, and we gradually identify theory-independent conditions that make the algebraic structure of quantum mechanics be assimilated by it. In the first part of the paper, we derive esse…
The relationship that is widely presumed to hold between physical theories and their successors, in which the successors in some sense explain the success of the theories they replace, is known commonly as 'reduction.' I argue that one traditional approach to theory reduction in physics, founded on the notion that a superseded theory should simply be a mathematical limit of the theory that supersedes it, is misleading as a general picture of the relationship whereby one theory encompasses the domain of empirical validity of another. I defend an alternative account that builds upon a certain ge…
We investigate the relationship between information scrambling and work statistics after a quench for the paradigmatic example of short-range interacting particles in a one-dimensional harmonic trap, considering up to five particles numerically. In particular, we find that scrambling requires finite interactions, in the presence of which the long-time average of the squared commutator for the individual canonical operators is directly proportional to the variance of the work probability distribution. In addition to the numerical results, we outline the mathematical structure of the N-body syst…
Simple theorems relating a quantum mechanical system to the corresponding classical one at equilibrium and connecting the quantum eigenvalues to the frequencies of normal modes oscillations are presented. Corresponding to each quantum eigenfunction, a ` classical eigenfunction' is associated. Those belonging to `elementary excitations' play an important role.
Given are a first principles derivation and formulation of the probabilistic concepts that underly equilibrium quantum statistical mechanics. The transition to non-equilibrium probability is traversed briefly.
In this paper, we investigate DNA denaturation through Statistical Mechanics and show that exceptional polynomials lead to DNA mutation. We consider a DNA model with two chains connected by Morse potential representing the H bonds, and then we calculate the partition function for this model. The partition function is converted into a Schrödinger like an equation. We exploit the techniques of SUSY quantum mechanics to model DNA mutation. We also compute the thermal denaturation of DNA for each mutated state.
In this paper we will demonstrate the use of Feynman Diagrams for one dimensional scattering in quantum mechanics. We will evaluate the S-Matrix explicitly for the Dirac delta and finite wall potentials by summing the full series of Feynman diagrams, illustrating the spirit of perturbation theory. This technique may be useful in introductory quantum mechanics courses, and provides the student with intuition about conservation laws in the context of scattering problems by connecting Feynman diagrams, free propagation, and conservation of the corresponding observable. It also provides a toy mode…
A pedagogical derivation of statistical mechanics from quantum mechanics is provided, by means of open quantum systems. Besides, a new definition of Boltzmann entropy for a quantum closed system is also given to count microstates in a way consistent with the superposition principle. In particular, this new Boltzmann entropy is a constant that depends only on the dimension of the system's relevant Hilbert subspace. Finally, thermodynamics for quantum systems is investigated formally.
We derive simple formulas connecting the generalized Wigner functions for $s$-ordering with the density matrix, and vice-versa. These formulas proved very useful for quantum mechanical applications, as, for example, for connecting master equations with Fokker-Planck equations, or for evaluating the quantum state from Monte Carlo simulations of Fokker-Planck equations, and finally for studying positivity of the generalized Wigner functions in the complex plane.
The derivation becomes possible when we find a new formalism which connects the relativistic mechanics with the quantum mechanics. In this paper, we explore the quantum wave nature from the Newtonian mechanics by using a concept: velocity field. At first, we rewrite the relativistic Newton's second law as a field equation in terms of the velocity field, which directly reveals a new relationship connecting to the quantum mechanics. Next, we show that the Dirac equation can be derived from the field equation in a rigorous and consistent manner.
These are the lecture notes for quantum and statistical mechanics courses that are given by DC at Ben-Gurion University. They are complementary to "Lecture Notes in Quantum Mechanics" [arXiv: quant-ph/0605180]. Some additional topics are covered, including: introduction to master equations; non-equilibrium processes; fluctuation theorems; linear response theory; adiabatic transport; the Kubo formalism; and the scattering approach to mesoscopics.
Measurement quantum mechanics, the theory of a quantum system which undergoes a measurement process, is introduced by a loop of mathematical equivalencies connecting previously proposed approaches. The unique phenomenological parameter of the theory is linked to the physical properties of an informational environment acting as a measurement apparatus which allows for an objective role of the observer. Comparison with a recently reported experiment suggests how to investigate novel interesting regimes for the quantum Zeno effect.
In this paper we discuss and analyse the idea of trying to see (non-relativistic) quantum mechanics as a ``space-time statistical mechanics'', by using the classical statistical mechanical method on objective microscopic space-time configurations. It is argued that this could perhaps be accomplished by giving up the assumption that the objective ``state'' of a system is independent of a future measurement performed on the system. This idea is then applied in an example of quantum state estimation on a qubit system.
We present here a canonical description for quantizing classical maps on a torus. We prove theorems analagous to classical theorems on mixing and ergodicity in terms of a quantum Koopman space $ L^2 (A_\hbar},τ_\hbar) $ obtained as the completion of the algebra of observables $ A_\hbar $ in the norm induced by the following inner product $(A,B) =τ_{\hbar}(A^{\dagger}B) $, where $τ_{\hbar}$ is a linear functional on the algebra analogous to the classical ``integral over phase space.'' We also derive explicit formulas connecting this formulation to the $θ$-torus decomposition of Bargmann space i…
Quantum mechanics can emerge from classical statistics. A typical quantum system describes an isolated subsystem of a classical statistical ensemble with infinitely many classical states. The state of this subsystem can be characterized by only a few probabilistic observables. Their expectation values define a density matrix if they obey a "purity constraint". Then all the usual laws of quantum mechanics follow, including Heisenberg's uncertainty relation, entanglement and a violation of Bell's inequalities. No concepts beyond classical statistics are needed for quantum physics - the differenc…
It is shown that the statistical conception of quantum mechanics is dynamical but not probabilistic, i.e. the statistical description in quantum mechanics is founded on dynamics. A use of the probability theory, when it takes place, is auxiliary. Attention is drawn to the fact that in the quantum mechanics there are two different objects: an individual object to be statistically described and a statistical average object, which is a result of the statistical description. Identification of the two different objects (a use of the same term for both) is an origin of many known quantum mechanics p…
We derive an exact one-component equation of motion for the probability amplitude of a target time-dependent state, and use the equation to reformulate quantum dynamics and control for both closed and open systems. Using the one-component equation, we show that an unexpected time-dependent leakage-free path can be induced and we capture the essential quantity in determining the effect of decoherence suppression. Our control protocol based on the nonperturbative leakage elimination operator provides a unified perspective connecting some subtle, popular and important concepts of quantum control,…
We emphasize that there is no spin-statistics connection in nonrelativistic quantum mechanics. In several recent papers, including Phys. Rev. A 67, 042102 (2003) [quant-ph/0207017], quantum mechanics is modified so as to force a spin-statistics connection, but the resulting theory is quite different from standard physics.
Classical statistical average values are generally generalized to average values of quantum mechanics, it is discovered that quantum mechanics is direct generalization of classical statistical mechanics, and we generally deduce both a new general continuous eigenvalue equation and a general discrete eigenvalue equation in quantum mechanics, and discover that a eigenvalue of quantum mechanics is just an extreme value of an operator in possibility distribution, the eigenvalue f is just classical observable quantity. A general classical statistical uncertain relation is further given, the general…
We discuss the propagation dynamics of nonspreading wave packets. We decompose the Hamiltonian into two parts. The first part is such that wave packets is its instantaneous eigenstate and is therefore irrelevant to the propagation of the packet. The second part is shown to be the effective Hamiltonian governing the motion of the packet both classically and quantum mechanically. Thus, analogous to Ehrenfest's theorem, nonspreading wave packets offer another view point directly connecting quantum mechanics and classical mechanics. This analysis also works for non-square-integrable packets, such …
Understanding the demarcation line between classical and quantum is an important issue in modern physics. The development of such an understanding requires a clear picture of the various concurrent notions of `classicality' in quantum theory presently in use. Here, we focus on the relationship between Kolmogorov consistency of measurement statistics -- the foundational footing of classical stochastic processes in standard probability theory -- and the commutativity (or absence thereof) of measurement operators -- a concept at the core of quantum theory. Kolmogorov consistency implies that the …
It is proved from stated assumptions of nonrelativistic quantum mechanics based on the Schroedinger equation that identical spin-zero particles must obey symmetric statistics.
We present a general method to derive the classical mechanics of a system of identical particles in a way that retains information about quantum statistics. The resulting statistical mechanics can be interpreted as a classical version of Haldane's exclusion statistics.
In this work we present a general formalism to treat non-Hermitian and noncommutative Hamiltonians. This is done employing the phase-space formalism of quantum mechanics, which allows to write a set of robust maps connecting the Hamitonians and the associated Wigner functions to the different Hilbert space structures, namely, those describing the non-Hermitian and noncommutative, Hermitian and noncommutative, and Hermitian and commutative systems. A general recipe is provided to obtain the expected values of the more general Hamiltonian. Finally, we apply our method to the harmonic oscillator …
The classical-statistical limit of quantum mechanics is studied. It is proved that the limit $\hbar \to 0$ is the good limit for the operators algebra but it si not so for the state compact set. In the last case decoherence must be invoked to obtain the classical-statistical limit.
We study the use of methods based on the real symplectic groups $Sp(2n,\mathcal{R})$ in the analysis of the Arthurs-Kelly model of proposed simultaneous measurements of position and momentum in quantum mechanics. Consistent with the fact that such measurements are in fact not possible, we show that the observable consequences of the Arthurs-Kelly interaction term are contained in the symplectic transformation law connecting the system plus apparatus variance matrices at an initial and a final time. The individual variance matrices are made up of averages and spreads or uncertainties for single…
The spin-statistics connection has been proved for nonrelativistic quantum mechanics (Jabs, A., 2010: Found. Phys., {\bf 40}, 776-792). The proof is extended here to the relativistic regime using the parametrized Dirac equation. A causality condition is not required.
Based on the concept of ensemble, it is proved in the manuscript that the probability amplitude function can also been used to describe the classical statistical system. The motion equations of probability amplitude functions of classical statistical system are obtained, which are similar to that in quantum mechanics. It is also proved in the meantime that microsystems that obey quantum mechanics can also be described in the forms of classical statistical mechanics and deterministic classical mechanics. The quantum revised formula of classical interaction force and the revised Liouville's equa…
We implement Feynman's suggestion that the only missing notion needed for the puzzle of Quantum Measurement is the statistical mechanics of amplifying apparatus. We define a thermodynamic limit of quantum amplifiers which is a classically describable system in the sense of Bohr, and define macroscopic pointer variables for the limit system. Then we derive the probabilities of Quantum Measurement from the deterministic Schroedinger equation by the usual techniques of Classical Statistical Mechanics.
The authors of the recent paper [1] boldly claim to discover a new fully quantum approach to foundation of statistical mechanics: "Our conceptually novel approach is free of mathematically ambiguous notions such as probability, ensemble, randomness, etc." The aim of this note is to show that this approach is neither specific for quantum systems nor really conceptually different from the standard textbook arguments supporting microcanonical or canonical ensembles in statistical mechanics.
We generalize classical statistical mechanics to describe the kinematics and the dynamics of systems whose variables are constrained by a single quantum postulate (discreteness of the spectrum of values of at least one variable of the theory). This is possible provided we adopt Feynman's suggestion of dropping the assumption that the probability for an event must always be a positive number. This approach has the advantage of allowing a reformulation of quantum theory in phase space without introducing the unphysical concept of probability amplitudes, together with all the problems concerning …
Where does quantum mechanics part ways with classical mechanics? How does quantum randomness differ fundamentally from classical randomness? We cannot fully explain how the theories differ until we can derive them within a single axiomatic framework, allowing an unambiguous account of how one theory is the limit of the other. Here we derive nonrelativistic quantum mechanics and classical statistical mechanics within a common framework. The common axioms include conservation of average energy and conservation of probability current. But two axioms distinguish quantum mechanics from classical st…
The classical limit of the Schrodinger equation implies the orthodox statistical interpretation for degrees of freedom in finite-dimensional subspaces of the full Hilbert space, but the argument presented does not imply the Born interpretation for degrees of freedom that have a classical counterpart.
We point out that Fratini and Hayrapetyan ignored the randomness of mixing, a basic request to prepare a statistical ensemble rather than an ensemble in general. Their analysis is irrelevant for standard statistical ensembles, their conclusions about the limitations of standard theory become unjustified.
We describe quantum entropic functionals and outline a research program dealing with entropic fluctuations in non-equilibrium quantum statistical mechanics.
We show two fascinating topics lying between quantum information processing and statistical mechanics. First, we introduce an elaborated technique, the surface code, to prepare the particular quantum state with robustness against decoherence. Second, we show another interesting technique to employ quantum nature, quantum annealing. Through both of the topics, we would shed light on the birth of the interdisciplinary field between quantum mechanics and statistical mechanics.
Starting from the quantum theory of identical particles, we show how to define a classical mechanics that retains information about the quantum statistics. We consider two examples of relevance for the quantum Hall effect: identical particles in the lowest Landau level, and vortices in the Chern-Simons Ginzburg-Landau model. In both cases the resulting {\em classical} statistical mechanics is shown to be a nontrivial classical limit of Haldane's exclusion statistics.
Quantum mechanics appears to contain ghosts from both classical statistical mechanics and special relativity. On one hand, both the Dirac and Schrödinger equations have classical analogs that emerge directly from classical statistical mechanics, unimpeded by major problems of interpretation. On the other hand, the formal analytic continuation that takes these classical equations to the quantum version introduces a velocity dependent phase. However, among classical theories, only in relativistic mechanics does one find path-dependent phase in the form of relativistic time dilation. This paper e…
Recently a sufficient and necessary condition for Pauli's spin- statistics connection in nonrelativistic quantum mechanics has been established [quant-ph/0208151]. The two-dimensional part of this result is extended to n-particle systems and reformulated and further simplified in a more geometric language.
This review covers applications of quantum Monte Carlo methods to quantum mechanical problems in the study of electronic and atomic structure, as well as applications to statistical mechanical problems both of static and dynamic nature. The common thread in all these applications is optimization of many-parameter trial states, which is done by minimization of the variance of the local or, more generally for arbitrary eigenvalue problems, minimization of the variance of the configurational eigenvalue.
We extend the mathematical theory of quantum hypothesis testing to the general $W^*$-algebraic setting and explore its relation with recent developments in non-equilibrium quantum statistical mechanics. In particular, we relate the large deviation principle for the full counting statistics of entropy flow to quantum hypothesis testing of the arrow of time.
