The space quantization induced by a Stern-Gerlach experiment is normally explained by invoking the ``collapse of the wave function.'' This is a rather mysterious idea; it would be better to explain the Stern-Gerlach results without using it. We re-analyze the Stern-Gerlach experiment using path integrals. We find if we model explicitly the finite width of the beam, coherent interference within the beam itself provides the space quantization -- without need to invoke the collapse. If we insist on employing only wave functions with the space and spin parts kept forcibly disentangled, we recreate the need to invoke the collapse. The collapse-free approach makes more specific predictions about the shape and position of the scattered beams; if the interaction region has finite length, these may be testable. Pending experimental disambiguation, the chief arguments in favor of the collapse-free approach are that it is simpler and less mysterious, has no adjustable parameters, and requires the invocation of no new forces.
It is shown that von Neumann-Landau equation for wave functions can present a mathematical formalism of motion of quantum mechanics. The wave functions of von Neumann-Landau equation for a single particle are `bipartite', in which the associated Schrödinger's wave functions correspond to those `bipartite' wave functions of product forms. This formalism establishes a mathematical expression of wave-particle duality and that von Neumann's entropy is a quantitative measure of complementarity between wave-like and particle-like behaviors. Furthermore, this extension of Schrödinger's form suggests …
It is shown that `bipartite' wave functions can present a mathematical formalism of quantum theory for a single particle, in which the associated Schrödinger's wave functions correspond to those `bipartite' wave functions of product forms. This extension of Schrödinger's form establishes a mathematical expression of wave-particle duality and that von Neumann's entropy is a quantitative measure of complementarity between wave-like and particle-like behaviors. In particular, this formalism suggests that collapses of Schrödinger's wave functions can be regarded as the simultaneous transition of t…
In quantum theory particles are represented as wave packets. Shock wave analysis of quantum equations of motion shows that wave function representation in general and wave packet description in particular contains discontinuities due to a non-zero quantum force. The quantum force causes wave packet dispersion which results in the intersection of characteristic curves developing a shock discontinuity. Since quantum force vanishes for localized quantum density waves [1], it is thus established that localized quantum density waves form the only class of wave function representation of particles i…
Photon wave function is a controversial concept. Controversies stem from the fact that photon wave functions can not have all the properties of the Schroedinger wave functions of nonrelativistic wave mechanics. Insistence on those properties that, owing to peculiarities of photon dynamics, cannot be rendered, led some physicists to the extreme opinion that the photon wave function does not exist. I reject such a fundamentalist point of view in favor of a more pragmatic approach. In my view, the photon wave function exists as long as it can be precisely defined and made useful.
We propose an optimized algorithm for the numerical simulation of two-time correlation functions by means of stochastic wave functions. As a first application, we investigate the two-time correlation function of a nonlinear optical parametric oscillator.
Canonical quantization of electromagnetic field is traditionally done using plane waves. It is possible to formulate the quantization using other complete set of basis functions. Wavelets are a special kind of functions which are localized in real as well as in Fourier space. In this paper we show how wavelets can be used as basis functions in canonical quantization. A countable set of mode functions are obtained. The general formalism of the change of basis is the same for all wavelets which satisfy a multiresolution analysis.
The monochromatic Dirac and polychromatic Titulaer-Glauber quantized field theories (QFTs) of electromagnetism are derived from a photon-energy wave function in much the same way that one derives QFT for electrons, that is, by quantization of a single-particle wave function. The photon wave function and its equation of motion are established from the Einstein energy-momentum-mass relation, assuming a local energy density. This yields a theory of photon wave mechanics (PWM). The proper Lorentz-invariant single-photon scalar product is found to be non-local in coordinate space, and is shown to c…
The quantum field of a single particle is expressed as the sum of the particle's ordinary wave function and the vacuum fluctuations. An exact quantum-field calculation shows that the squared amplitude of this field sums, at any time, to a delta function representing a discrete corpuscle at one point and zero everywhere else. The peak of the delta function is located at the point where the vacuum fluctuations interfere constructively with the ordinary wave function. Similarly, the collapsed wave function after a measurement of an observable is determined by interference between the initial wave…
An differential equation for wave functions is proposed, which is equivalent to Schrödinger's wave equation and can be used to determine energy-level gaps of quantum systems. Contrary to Schrödinger's wave equation, this equation is on `bipartite' wave functions. It is shown that those `bipartite' wave functions satisfy all the basic properties of Schrödinger's wave functions. Further, it is argued that `bipartite' wave functions can present a mathematical expression of wave-particle duality. This provides an alternative approach to the mathematical formalism of quantum mechanics.
The present note aims to provide a clear and explicit derivation of the orthonormality condition, and the completeness property of the Volkov wave function. Thus, none of the results are new.
In this letter we investigate the common procedure in which any wave function is expanded into a series of eigenfunctions. It is shown that as far as dynamical systems are concerned the expanding procedure involves various mathematical and physical difficulties. With or without introducing phase factors, such expansions do not represent dynamical wave functions.
Instability-induced random branching of deterministic dynamics is discussed as a possible mechanism of random wave function collapse. In the case of two level systems, the Born probability rule emerges as the simplest linear solution to the equation for measurement probabilities.
Schmidt modes of non-collinear biphoton angular wave functions are found analytically. The experimentally realizable procedure is described for their separation. Parameters of the Schmidt decomposition are used for evaluation of the degree of biphoton's angular entanglement.
My view on the meaning of the quantum wave function and its connection to protective measurements is described. The wave function and only the wave function is the ontology of the quantum theory. Protective measurements support this view although they do not provide a decisive proof. A brief review of the discovery and the criticism of protective measurements is presented. Protective measurements with postselection are discussed.
We study the optimal Slater-determinant approximation of an $N$-fermion wave function analytically. That is, we seek the Slater-determinant (constructed out of $N$ orthonormal single-particle orbitals) wave function having largest overlap with a given $N$-fermion wave function. Some simple lemmas have been established and their usefulness is demonstrated on some structured states, such as the Greenberger-Horne-Zeilinger state. In the simplest nontrivial case of three fermions in six orbitals, which the celebrated Borland-Dennis discovery is about, the optimal Slater approximation wave function…
The most puzzling issue in the foundations of quantum mechanics is perhaps that of the status of the wave function of a system in a quantum universe. Is the wave function objective or subjective? Does it represent the physical state of the system or merely our information about the system? And if the former, does it provide a complete description of the system or only a partial description? We shall address these questions here mainly from a Bohmian perspective, and shall argue that part of the difficulty in ascertaining the status of the wave function in quantum mechanics arises from the fact…
After summarizing three versions of trajectory-based quantum mechanics, it is argued that only the original formulation due to Bohm, which uses the Schrödinger wave function to guide the particles, can be readily extended to particles with spin. To extend the two wave function-free formulations, it is argued that necessarily particle trajectories not only determine location, but also spin. Since spin values are discrete, it is natural to revert to a variation of Bohm's pilot wave formulation due originally to Bell. It is shown that within this formulation with stochastic quantum trajectories, …
The probabilistic prediction of quantum theory is mystery. I solved the mystery by a geometrical interpretation of a wave function. This suggests the unification between quantum theory and the theory of relativity. This suggests Many-Worlds Interpretation is true, too.
The internal phase dynamics of a quantum system is revealed in details. Theoretical and experimental evidences of existence of a causal relation of the phase of the wave function with the dynamics of the quantum system are presented sistematically for the first time. A new, dynamics-statistical interpretation of the quantum mechanics is introduced. A particle-wave duality picture incorporated in the wave function through its phase and amplitude is considered.
We use a factorization technique and representation of canonical transformations to construct globally valid closed form expressions without singularities of semi-classical wave functions for arbitrary smooth potentials over a one-dimensional position space.
The perturbation method in supersymmetric quantum mechanics (SUSYQM) is used to study the spheroidal wave functions' recurrence relations, which are revealed by the shape-invariance property of the super-potential. The super-potential is expanded by the parameter alpha and could be gotten by approximation method. Up to the first order, it has the shape-invariance property and the excited spheroidal wave functions are gotten. Also, all the first term eigenfunctions obtained are in closed form. They are advantageous to investigating for involved physical problems of spheroidal wave function.
A new approach to the geometrization of the electron theory is proposed. The particle wave function is represented by a geometric entity, i.e., Clifford number, with the translation rules possessing the structure of Dirac equation for any manifold. A solution of this equation is obtained in terms of geometric treatment. Interference of electrons whose wave functions are represented by geometric entities is considered. New experiments concerning the geometric nature of electrons are proposed.
A generalized Hamilton-Jacobi representation describes microstates of the Schrödinger wave function for bound states. At the very points that boundary values are applied to the bound state Schrödinger wave function, the generalized Hamilton-Jacobi equation for quantum mechanics exhibits a nodal singularity. For initial value problems, the two representations are equivalent.
The notion of a physical collapse of the wave function is embodied in dynamical collapse models. These involve a modification of the unitary evolution of the wave function such as to give a dynamical account of collapse. The resulting dynamics is at first sight time asymmetric for the simple reason that the wave function depends on those collapse events in the past but not those in the future. Here we show that dynamical wave function collapse models admit a general description that has no inbuilt direction of time. Given some simple constraints, we show that there exist empirically equivalent…
The perturbation method in supersymmetric quantum mechanics (SUSYQM) is used to study the spheroidal wave functions' eigenvalue problem. Expanding the super-potential in series of the parameter alpha, the first order term of ground eigen-value and the eigen-function are gotten. In the paper, the very excellent results are that all the first two terms approximation on eigenfunctions obtained are in closed form. They give useful information for the involved physical problems in application of spheroidal wave functions.
We prove that the bound-state momentum-space wave function $ϕ(p)$ for the quasi-one-dimensional hydrogen atom as used by Olendski in two recent publications (Eur. J. Phys.38, 038001(2017), arXiv : 1703.10042v1, 26 Mar2017) is incorrect. Further, we reconfirm that the wave function used by us in Eur. J.. Phys. 38, 025103 is correct.
The finite difference time domain (FDTD) method has been successfully applied to obtain energies and wave functions for two electrons in a quantum dot modeled by a three dimensional harmonic potential. The FDTD method uses the time-dependent Schrödinger equation (TDSE) in imaginary time. The TDSE is numerically solved with an initial random wave function and after enough simulation time, the wave function converges to the ground state wave function. The excited states are determined by using the same procedure for the ground state with additional constraints that the wave function must be orth…
The ontology of Bohmian mechanics includes both the universal wave function (living in 3N-dimensional configuration space) and particles (living in ordinary 3-dimensional physical space). Proposals for understanding the physical significance of the wave function in this theory have included the idea of regarding it as a physically-real field in its 3N-dimensional space, as well as the idea of regarding it as a law of nature. Here we introduce and explore a third possibility in which the configuration space wave function is simply eliminated -- replaced by a set of single-particle pilot-wave fi…
Radial wave functions for power-law potentials are approximated with the help of power-law substitution and explicit summation of the leading constituent WKB series. Our approach reproduces the correct behavior of the wave functions at the origin, at the turning points and far away from the turning points
All integrals needed to evaluate the correlated wave functions with polynomial terms of inter-electronic distance are included. For this form of the wave function, the integrals needed can be expressed as a product of integrals involving at most four electrons.
We study the inverse problem of constructing an appropriate Hamiltonian from a physically reasonable set of orthogonal wave functions for a quantum spin system. Usually, we are given a local Hamiltonian and try to characterize the relevant wave functions and energies (the spectrum) of the system. Here, we take the opposite approach; starting from a collection of orthogonal wave functions, our goal is to characterize the associated parent Hamiltonian, to see how the wave functions and the energy values determine the structure of the parent Hamiltonian. Specifically, we obtain (quasi) local Hami…
We present and motivate an efficient way to include orbital dependent many--body correlations in trial wave function of real--space Quantum Monte Carlo methods for use in electronic structure calculations. We apply our new orbital--dependent backflow wave function to calculate ground state energies of the first row atoms using variational and diffusion Monte Carlo methods. The systematic overall gain of correlation energy with respect to single determinant Jastrow-Slater wave functions is competitive with the best single determinant trial wave functions currently available. The computational c…
Wave functions of a new functional kind have been proposed for Helium-like atoms in this work . These functions explicitly depend on interelectronic and hyperspherical coordinates. The best ground state energy for the Helium atom $ -2.903724376677 a.u.$ has been calculated with variational method with basis set of simple functions with a single exponential parameter. To the author's knowledge, this is the best result with use of hyperspherical coordinates so far. Comparable result has been obtained for the hydrogen anion. For Helium atom, our best wave functions matched the Kato cusp condition…
