One and two photon wave functions are derived by projecting the quantum state vector onto simultaneous eigenvectors of the number operator and a recently constructed photon position operator [Phys. Rev A 59, 954 (1999)] that couples spin and orbital angular momentum. While only the Landau-Peierls wave function defines a positive definite photon density, a similarity transformation to a biorthogonal field-potential pair of positive frequency solutions of Maxwell's equations preserves eigenvalues and expectation values. We show that this real space description of photons is compatible with all of the usual rules of quantum mechanics and provides a framework for understanding the relationships amongst different forms of the photon wave function in the literature. It also gives a quantum picture of the optical angular momentum of beams that applies to both one photon and coherent states. According to the rules of qunatum mechanics, this wave function gives the probability to count a photon at any position in space.
We have recently constructed a photon position operator with commuting components. This was long thought to be impossible, but our position eigenvectors have a vortex structure like twisted light. Thus they are not spherically symmetric and the position operator does not transform as a vector, so that previous non-existence arguments do not apply. We find two classes of position eigenvectors and obtain photon wave functions by projection onto the bases of position eigenkets that they define, following the usual rules of quantum mechanics. The hermitian position operator, r0, leads to a Landau-…
We show that the cylindrical symmetry of the eigenvectors of the photon position operator with commuting components, x, reflects the E(2) symmetry of the photon little group. The eigenvectors of x form a basis of localized states that have definite angular momentum, J, parallel to their common axis of symmetry. This basis is well suited to the description of "twisted light" that has been the subject of many recent experiments and calculations. Rotation of the axis of symmetry of this basis results in the observed Berry phase displacement. We prove that {x1,x2,J3} is a realization of the two di…
What is called "orthodox" quantum mechanics, as presented in standard foundational discussions, relies on two substantive assumptions --- the projection postulate and the eigenvalue-eigenvector link --- that do not in fact play any part in practical applications of quantum mechanics. I argue for this conclusion on a number of grounds, but primarily on the grounds that the projection postulate fails correctly to account for repeated, continuous and unsharp measurements (all of which are standard in contemporary physics) and that the eigenvalue-eigenvector link implies that virtually all interes…
An inconsistency is pointed out within Quantum Mechanics as soon as successive joint measurements are involved on entangled states. The resolution of the inconsistency leads to a refutation of the use of entangled states as eigenvectors. Hence, the concept of quantum teleportation, which is based on the use of such entangled states--the Bell states--as eigenvectors, is demonstrated to be irrelevant to Quantum Mechanics.
In biorthogonal quantum mechanics, the eigenvectors of a quasi-Hermitian operator and those of its adjoint are biorthogonal and complete and the probability for a transition from a quantum state to any one of these eigenvectors is positive definite. We apply this formalism to the long standing problem of the position observable in quantum field theory. The dual bases are positive and negative frequency one-particle states created by the field operator and its conjugate and biorthogonality is a consequence of their commutation relations. In these biorthogonal bases the position operator is cova…
We present an efficient method for preparing the initial state required by the eigenvalue approximation quantum algorithm of Abrams and Lloyd. Our method can be applied when solving continuous Hermitian eigenproblems, e.g., the Schroedinger equation, on a discrete grid. We start with a classically obtained eigenvector for a problem discretized on a coarse grid, and we efficiently construct, quantum mechanically, an approximation of the same eigenvector on a fine grid. We use this approximation as the initial state for the eigenvalue estimation algorithm, and show the relationship between its s…
We find that biorthogonal quantum mechanics with a scalar product that counts both absorbed and emitted particles leads to covariant position operators with localized eigenvectors. In this manifestly covariant formulation the probability for a transition from a one-photon state to a position eigenvector is the first order Glauber correlation function, bridging the gap between photon counting and the sensitivity of light detectors to electromagnetic energy density. The position eigenvalues are identified as the spatial parameters in the canonical quantum field operators and the position basis d…
An analogue of Kolmogorov's superconvergent perturbation theory in classical mechanics is constructed for self adjoint operators. It is different from the usual Rayleigh--Schrödinger perturbation theory and yields expansions for eigenvalues and eigenvectors in terms of functions of the perturbation parameter.
Recently, we have shown how the interpretation of quantum mechanics due to Lande' can be used to derive from first principles generalized formulas for the operators and some eigenvectors for spin 1/2 Though we gave the operators for all the components of the spin, we did not give the eigenvectors of the operators for the x and y components of the spin. We now give these vectors. In addition, we present a new and simple method of deriving the operators for the x and y components of the spin as well as their vectors from those for the z component. We give a general proof that the operator for th…
We extend classical Maxwell field theory to a first quantized theory of the photon by deriving a conserved Lorentz four-current whose zero component is a positive definite number density. Fields are real and their positive (negative) frequency parts are interpreted as absorption (emission) of a positive energy photon. With invariant plane wave normalization, the photon position operator is Hermitian with instantaneously localized eigenvectors that transform as Lorentz four-vectors. Reality of the fields and wave function ensure causal propagation and zero net absorption of energy in the absenc…
We show that the quantization of a simple damped system leads to a self-adjoint Hamiltonian with a family of complex generalized eigenvalues. It turns out that they correspond to the poles of energy eigenvectors when continued to the complex energy plane. Therefore, the corresponding generalized eigenvectors may be interpreted as resonant states. We show that resonant states are responsible for the irreversible quantum dynamics of our simple model.
The formalism of subdynamics is extended to the functional approach of quantum systems, and used for the Friedrichs model, in which diagonal singularities in states and observables are included. We compute in this approach the generalized eigenvectors and eigenvalues of the Liouvulle-Von Newmann operator, using an iterative scheme. As complex generalized eigenvalues are obtained, the decay rates of unstable modes are included in the spectral decomposition.
It is demonstrated that, making minimal changes in ordinary quantum mechanics, a reasonable irreversible quantum mechanics can be obtained. This theory has a more general spectral decompositions, with eigenvectors corresponding to unstable states that vanish when $t \to \infty .$ These ''Gamov vectors'' have zero norm, in such a way that the norm and the energy of the physical states remain constant. The evolution operator has no inverse, showing that we are really dealing with a time-asymmetric theory. Using Friedrichs model reasonable physical results are obtained, e. g. : the remaining of a…
Remarkably we find that for a ring with linear boundary conditions such that the eigenvector and its derivative are continuous, there does not seem to be a way for the well-known de Broglie relation to be gauge invariant. Certain nonlinear boundary conditions assure gauge invariance, and lead to eigenfunctions with a discontinuous but differentiable phase and a continuous spectrum. A discrete subset of this spectrum forms a Hilbert space, while another subset is excluded by the nonlinear boundaries. We conclude that discontinuous momentum eigenfunctions are tenable, and that it is possible tha…
In Bohmian mechanics particles follow continuous trajectories, hence 2-time position correlations are well defined. Nevertheless, Bohmian mechanics predicts the violation of Bell inequalities. Motivated by this fact we investigate position measurements in Bohmian mechanics by coupling the particles to macroscopic pointers. This explains the violation of Bell inequalities despite 2-time position correlations. We relate this fact to so-called surrealistic trajectories that, in our model, correspond to slowly moving pointers. Next, we emphasize that the nice feature of Bohmian mechanics, which do…
Quantum mechanics postulates the existence of states determined by a particle position at a single time. This very concept, in conjunction with superposition, induces much of the quantum-mechanical structure. In particular, it implies the time evolution to obey the Schroedinger equation, and it can be used to complete a truely basic derivation of the statistical axiom as recently proposed by Deutsch.
The quantum mechanical expression relating two commuting operators is reformulated such that the power method (also called method of moments) for iteratively calculating eigenvalues and eigenvectors becomes applicable. The new iterative scheme thus obtained allows to pre-select a quantum number of one of the two commuting operators and then calculate a corresponding eigenvalue of the other operator. The result is the common eigenvector of the eigenvalue pair. Among others the method may be used to calculate excited states, cope with degeneracy and/or accelerate convergence. Small example calcu…
We provide a squeeze-like transformation that allows one to remove a position dependent mass from the Hamiltonian. Methods to solve the Schrödinger equation may then be applied to find the respective eigenvalues and eigenfunctions. As an example, we consider a position-dependent-mass that leads to the integrable Morse potential and therefore to well-known solutions.
The problem of the position and spin in relativistic quantum mechanics is analyzed in detail. It is definitively shown that the position and spin operators in the Foldy-Wouthuysen representation (but not in the Dirac one) are quantum-mechanical counterparts of the classical position and spin variables. The probabilistic interpretation is valid only for Foldy-Wouthuysen wave functions. The relativistic spin operators are discussed. The spin-orbit interaction does not exist for a free particle if the conventional operators of the orbital angular momentum and the rest-frame spin are used. Alterna…
The de Broglie - Bohm Interpretation of Quantum Mechanics assigns positions and trajectories to particles. We analyze the validity of a formula for the velocities of Bohmian particles which makes the analysis of these trajectories particularly simple. We apply it to four different types of particle detectors and show that three types of the detectors lead to "surrealistic trajectories", i.e., leave a trace where the Bohmian particle was not present.
Quantizing the transfer of energy and momentum between interacting particles, we obtain a quantum impulse equation and relations that the corresponding mechanical power, force and torque satisfy. In addition to the energy-frequency and momentum-wavelength relations, we introduce the power-frequency and force-wavelength analogs, respectively. Further, we obtain an operator representation for the mechanical power and impact force in the position space and discuss their correspondence with the relevant momentum operator. The position representation of the torque operator and its relation to the o…
A challenge of modern physics is to investigate the quantum behavior of a bulk material object, for instance a mechanical oscillator. We have earlier demonstrated that by coupling a mechanical oscillator to the energy levels of embedded rare-earth ion dopants, it is possible to prepare such a resonator in a low phonon number state. Here, we describe how to extend this protocol in order to prepare momentum- and position squeezed states, and we analyze how the obtainable degree of squeezing depends on the initial conditions and on the coupling of the oscillator to its thermal environment.
A supersymmetric technique for the solution of the effective mass Schrö% dinger equation is proposed. Exact solutions of the Schroedinger equation corresponding to a number of potentials are obtained. The potentials are fully isospectral with the original potentials. The conditions for the shape invariance of the potentials are discussed.
The Dirac delta function can be defined by the limitation of the rectangular function covering a unit area with decrease of the width of the rectangle to zero, and in quantum mechanics the eigenvectors of the position operator take the form of the delta function. When discussing the position measurement in quantum mechanics, one is prompted by the mathematical convention that uses the rectangular wave function of sufficiently narrow width to approximate the delta function in order to making the state of the position physical. We argue that such an approximation is improper in physics, because …
In a recent article, Kiukas and Werner claim to have shown that Bohmian Mechanics does not make the same empirical predictions as ordinary Quantum Mechanics. More precisely, they have shown that ordinary Quantum Mechanics predicts maximal violations of the CHSH-Bell inequality for a certain experiment in which only position measurements are performed on two noninteracting entangled free non-relativistic particles. Kiukas and Werner claim that Bohmian Mechanics doesn't predict a violation of the CHSH-Bell inequality for that experiment. We explain that such claim is wrong and that the argument …
We recycle Cruz et al.'s (Phys. Lett. A 369 (2007) 400) work on the classical and quantum position-dependent mass (PDM) oscillators. To elaborate on the ordering ambiguity, we properly amend some of the results reported in their work and discuss the classical and quantum mechanical correspondence for the PDM harmonic oscillators. We use a point canonical transformation and show that one unique quantum PDM oscillator Hamiltonian (consequently, one unique ordering-ambiguity parametric set j=l=-1/4 and k=-1/2) is obtained. To show that such a parametric set is not just a manifestation of the quan…
In the context of a two-parameter $(α, β)$ deformation of the canonical commutation relation leading to nonzero minimal uncertainties in both position and momentum, the harmonic oscillator spectrum and eigenvectors are determined by using techniques of supersymmetric quantum mechanics combined with shape invariance under parameter scaling. The resulting supersymmetric partner Hamiltonians correspond to different masses and frequencies. The exponential spectrum is proved to reduce to a previously found quadratic spectrum whenever one of the parameters $α$, $β$ vanishes, in which case shape inva…
Bohm Mechanics and Nelson Stochastic Mechanics are confronted with Quantum Mechanics in presence of non-interacting subsystems. In both cases, it is shown that correlations at different times of compatible position observables on stationary states agree with Quantum Mechanics only in the case of product wave functions. By appropriate Bell-like inequalities it is shown that no classical theory, in particular no stochastic process, can reproduce the quantum mechanical correlations of position variables of non interacting systems at different times.
The well known Poisson Summation Formula is analysed from the perspective of the coherent state systems associated with the Heisenberg--Weyl group. In particular, it is shown that the Poisson summation formula may be viewed abstractly as a relation between two sets of bases (Zak bases) arising as simultaneous eigenvectors of two commuting unitary operators in which geometric phase plays a key role. The Zak bases are shown to be interpretable as generalised coherent state systems of the Heisenberg--Weyl group and this, in turn, prompts analysis of the sampling theorem (an important and useful c…
We study the classical and quantum oscillator in the context of a non-additive (deformed) displacement operator, associated with a position-dependent effective mass, by means of the supersymmetric formalism. From the supersymmetric partner Hamiltonians and the shape invariance technique we obtain the eigenstates and the eigenvalues along with the ladders operators, thus showing a preservation of the supersymmetric structure in terms of the deformed counterpartners. The deformed space in supersymmetry allows to characterize position-dependent effective mass, uniform field interactions and to ob…
Within the context of non-Hermitian quantum mechanics, we use the generators of eigenvectors of the Hamiltonian to construct a unitary inner product space. Such models have been of interest in recent years, for instance, in the context of ${\cal PT}$ symmetry, although our construction extends to the larger class of so-called pseudo-Hermitian Operators. We provide a detailed example to illustrate the concept and compare with known results.
We extend the formulation of pseudo-Hermitian quantum mechanics to eta-pseudo-Hermitian Hamiltonian operators H with an unbounded metric operator eta. In particular, we give the details of the construction of the physical Hilbert space, observables, and equivalent Hermitian Hamiltonian for the case that H has a real and discrete spectrum and its eigenvectors belong to the domain of eta and consequently its positive square root.
The characterization of observables, expressed via Hermitian operators, is a crucial task in quantum mechanics. For this reason, an eigensolver is a fundamental algorithm for any quantum technology. In this work, we implement a semi-autonomous algorithm to obtain an approximation of the eigenvectors of an arbitrary Hermitian operator using the IBM quantum computer. To this end, we only use single-shot measurements and pseudo-random changes handled by a feedback loop, reducing the number of measures in the system. Due to the classical feedback loop, this algorithm can be cast into the reinforce…
The most peculiar, specifically quantum, features of quantum mechanics --- quantum nonlocality, indeterminism, interference of probabilities, quantization, wave function collapse during measurement --- are explained on a logical-geometrical basis. It is shown that truths of logical statements about numerical values of quantum observables are quantum observables themselves and are represented in quantum mechanics by density matrices of pure states. Structurally, quantum mechanics is a result of applying non-Abelian symmetries to truth operators and their eigenvectors --- wave functions. Wave fu…
The Bohmian interpretation of quantum mechanics adds particle trajectories to the wave function and ensures that the probability distribution of the particle positions agrees with quantum mechanics at any time. This is not sufficient to avoid contradictions with quantum mechanics. There are correlations between particle positions at different times which cannot be reproduced with real particle trajectories. A simple rearrangement of an experimental test of the Bell-CHSH inequality demonstrates this.
The Born postulate can be reduced to its deterministic content that only applies to eigenvectors of observables: the standard probabilistic interpretation of generic states then follows from algebraic properties of repeated measurements and states. Extending this reasoning suggests an interpretation of quantum mechanics generalized with indefinite quantum norm.
We derive the classical limit of quantum mechanics by describing the center of mass of a system constituted by a large number of particles. We will show that in that limit the commutator between the position and velocity of the center of mass is infinitesimal, which allows both to be known with great precision. We then show how the infinitesimal commutator allows for the definition of functions of position and velocity, and how the commutator reduces to a Poisson bracket.
We present a new causal quantum mechanics in one and two dimensions developed recently at TIFR by this author and V. Singh. In this theory both position and momentum for a system point have Hamiltonian evolution in such a way that the ensemble of system points leads to position and momentum probability densities agreeing exactly with ordinary quantum mechanics.
It is shown that the Fourier transformation that relates position and momentum representations of quantum mechanics can be understood as a consequence of a symmetry principle that establishes the equivalence of being and becoming in the description of reality. There are however other transformation compatible with the same principle that could lead to different formalisms of quantum mechanics.
The usual Heisenberg uncertainty relation for position and momentum may be replaced by an exact equality, for suitably chosen measures of position and momentum uncertainty. This "exact" uncertainty relation is valid for_all_ pure states, and is sufficiently strong to provide an axiomatic basis for moving from classical mechanics to quantum mechanics. In particular, the assumption of a nonclassical momentum fluctuation, having a strength which scales inversely with uncertainty in position, leads from the classical equations of motion to the Schroedinger equation.
Within the traditional Hilbert space formalism of quantum mechanics, it is not possible to describe a particle as possessing, simultaneously, a sharp position value and a sharp momentum value. Is it possible, though, to describe a particle as possessing just a sharp position value (or just a sharp momentum value)? Some, such as Teller (Journal of Philosophy, 1979), have thought that the answer to this question is No -- that the status of individual continuous quantities is very different in quantum mechanics than in classical mechanics. On the contrary, I shall show that the same subtle issues…
Trying to interpret B. Zilber's project on model theory of quantum mechanics we study a way of building limit models from finite-dimensional approximations. Our point of view is that of metric model theory, and we develop a method of taking ultraproducts of unbounded operators. We first calculate the Feynman propagator for the free particle as defined by physicists as an inner product $\langle x_{0}| K^{t}| x_{1}\rangle $ of the eigenvector $| x_{0}\rangle $ of the position operator with eigenvalue $x_{0}$ and $K^{t}(| x_{1}\rangle )$, where $K^{t}$ is the time evolution operator. However, due…
In an article published in 2010, Kiukas and Werner claim to have shown that Bohmian Mechanics does not make the same empirical predictions as ordinary Quantum Mechanics. We explain that such claim is wrong.
We construct the algebra of operators acting on the Hilbert spaces of Quantum Mechanics for systems of $N$ identical particles from the field operators acting in the Fock space of Quantum Field Theory by providing the explicit relation between the position and momentum operators acting in the former spaces and the field operators acting on the latter. This is done in the context of the non-interacting Klein-Gordon field. It may not be possible to extend the procedure to interacting field theories since it relies crucially on particle number conservation. We find it nevertheless important that …
Consider a statistical model with an epistemic restriction such that, unlike in classical mechanics, the allowed distribution of positions is fundamentally restricted by the form of an underlying momentum field. Assume an agent (observer) who wishes to estimate the momentum field given information on the conjugate positions. We discuss a classically consistent, weakly unbiased, best estimation of the momentum field minimizing the mean squared error, based on which the abstract mathematical rules of quantum mechanics can be derived. The results suggest that quantum wave function is not an objec…
The interpretation of quantum mechanics due to Lande' is applied to the connection between wave mechanics and matrix mechanics. The connection between the differential eigenvalue equation and the matrix eigenvalue equation for an operator is elucidated. In particular, we show that the elements of a matrix vector state are probability amplitudes with a structure rather than being mere constants. We obtain the most general expressions for the probability amplitudes for the description of spin-1/2 measurements. As a result, we derive spin-1/2 operators and vectors from first principles. The proce…
We extend a procedure for construction of the photon position operators with transverse eigenvectors and commuting components [Phys. Rev. A 59, 954 (1999)] to body rotations described by three Euler angles. The axial angle can be made a function of the two polar angles, and different choices of the functional dependence are analogous to different gauges of a magnetic field. Symmetries broken by a choice of gauge are re-established by transformations within the gauge group. The approach allows several previous proposals to be related. Because of the coupling of the photon momentum and spin, our…
We propose a technique to couple the position operator of a nano mechanical resonator to a SQUID device by modulating its magnetic flux bias. By tuning the magnetic field properly, either linear or quadratic couplings can be realized, with a discretely adjustable coupling strength. This provides a way to realize coherent nonlinear effects in a nano mechanical resonator by coupling it to a Josephson quantum circuit. As an example, we show how squeezing of the nano mechanical resonator state can be realized with this technique. We also propose a simple method to measure the uncertainty in the po…
Exactly solvable potentials of nonrelativistic quantum mechanics are known to be shape invariant. For these potentials, eigenvalues and eigenvectors can be derived using well known methods of supersymmetric quantum mechanics. The majority of these potentials have also been shown to possess a potential algebra, and hence are also solvable by group theoretical techniques. In this paper, for a subset of solvable problems, we establish a connection between the two methods and show that they are indeed equivalent.
Bohmian mechanics is a theory that provides a consistent explanation of quantum phenomena in terms of point particles whose motion is guided by the wave function. In this theory, the state of a system of particles is defined by the actual positions of the particles and the wave function of the system; and the state of the system evolves deterministically. Thus, the Bohmian state can be compared with the state in classical mechanics, which is given by the positions and momenta of all the particles, and which also evolves deterministically. However, while in classical mechanics it is usually tak…
A relativistic spin operator is to be the difference between the total and orbital angular momentum. As the unique position operator for a localized state, the remarkable Newton-Wigner position operator, which has all desirable commutation relations as a position operator, can give a proper spin operator. Historically important three spin operators respectively proposed by Bogolubov et al., Pryce, and Foldy-Woutheysen are investigated to manifest a corresponding spin operator to the Newton-Wigner position operator. We clarify a unique spin operator in relativistic quantum mechanics described b…
In order to arrive at Bohmian mechanics from standard nonrelativistic quantum mechanics one need do almost nothing! One need only complete the usual quantum description in what is really the most obvious way: by simply including the positions of the particles of a quantum system as part of the state description of that system, allowing these positions to evolve in the most natural way. The entire quantum formalism, including the uncertainty principle and quantum randomness, emerges from an analysis of this evolution. This can be expressed succinctly---though in fact not succinctly enough---by …
It has been predicted and experimentally demonstrated that by injecting squeezed light into an optomechanical device it is possible to enhance the precision of a position measurement. Here, we present a fundamentally different approach where the squeezing is created directly inside the cavity by a nonlinear medium. Counterintuitively, the enhancement of the signal to noise ratio works by de-amplifying precisely the quadrature that is sensitive to the mechanical motion without losing quantum information. This enhancement works for systems with a weak optomechanical coupling and/or strong mechan…
The problem of spontaneous pair creation in static external fields is reconsidered. A weak version of the conjecture proposed by G Nenciu (1980) is seated and proved. The method reduces the proof of the general conjecture to the study of the evolution, associated with a time dependent Hamiltonian, of a vector which is eigenvector of this Hamiltonian at some given time. Possible ways of proving the general conjecture are discussed.
