This chapter aims to solve the problem of creating discrete, reversible systems of various equations of physics. The Schrodinger Equation involves both a first derivative with respect to time and a second derivative with respect to space. The chapter presents Ed Barton with the best starting point in what was to be his quest for the summer. He was to do a series of projects with the keystone being to find a beautiful, discrete and reversible version of the Schrodinger Equation. Ed Barton's immediate observation of the reversibility of that equation remains one of author's most striking memories of 18 years on the Massachusetts Institute of Technology faculty. Richard P. Feynman showed that by calculating the amplitude for the 0-Dimensional case in a perfectly time symmetric fashion, keeping to author's esthetic principle, then the values computed by the reversible difference equation exactly conserved the value of the amplitude.
This chapter aims to solve the problem of creating discrete, reversible systems of various equations of physics. The Schrodinger Equation involves both a first derivative with respect to time and a second derivative with respect to space. The chapter presents Ed Barton with the best starting point in what was to be his quest for the summer. He was to do a series of projects with the keystone being to find a beautiful, discrete and reversible version of the Schrodinger Equation. Ed Barton's immediate observation of the reversibility of that equation remains one of author's most striking memories of 18 years on the Massachusetts Institute of Technology faculty. Richard P. Feynman showed that by calculating the amplitude for the 0-Dimensional case in a perfectly time symmetric fashion, keeping to author's esthetic principle, then the values computed by the reversible difference equation exactly conserved the value of the amplitude.
The first question to ask about the phenomenon of relaxation is why it occurs at all. Both the Newton and the Schrödinger equations are symmetrical under time reversal: The Newton equation, dx/dt = v ; dv/dt = −∂V/∂x, implies that particles obeying this law of motion will retrace their trajectory back in time after changing the sign of both the time t and the particle velocities v. The Schrödinger equation, ∂ψ/∂t = −(i/.h) Ĥ ψ, implies that if (ψ (t) is a solution then ψ *(−t) is also one, so that observables which depend on |ψ|2 are symmetric in time. On the other hand, nature clearly evolves asymmetrically as asserted by the second law of thermodynamics. How does this asymmetry arise in a system that obeys temporal symmetry in its time evolution? Readers with background in thermodynamics and statistical mechanics have encountered the intuitive answer: Irreversibility in a system with many degrees of freedom is essentially a manifestation of the system “getting lost in phase space”:Asystem starts from a given state and evolves in time. If the number of accessible states is huge, the probability that the system will find its way back to the initial state in finite time is vanishingly small, so that an observer who monitors properties associated with the initial state will see an irreversible evolution. The question is how is this irreversible behavior manifested through the reversible equations of motion, and how does it show in the quantitative description of the time evolution. This chapter provides an introduction to this subject using the time-dependent Schrödinger equation as a starting point. Chapter 10 discusses more advanced aspects of this problem within the framework of the quantum Liouville equation and the density operator formalism.
The general problem of finding exactly soluble quantum systems is considered. It is argued that discrete space quantum mechanics emerges in a natural way as an avenue of approach. Discrete space quantum mechanics is formulated and applied to one-dimensional quantum systems with emphasis on single-channel models. It is found that a large variety of systems are exactly soluble in the sense that they only require the inversion of a finite-dimensional matrix. The interaction may in general be both nonlocal and non-time-reversal-invariant. The analytic structure of the resolvent is worked out in detail for a simple class of examples. It is shown that a slightly modified version of the usual continuous space Schrödinger equation may in principle be solved exactly for any finite range local potential by writing the solution in terms of corresponding discrete space solutions. It is also shown that from an algebraic viewpoint the models constructed are realizations of generalized versions of the Weyl relations.
This talk is about a method for solving the numerical initial value problem of the time-dependent Schrodinger equation with a Hamiltonian which may depend on time. It is only for time independent Hamiltonians that I was able to describe the accuracy and stability properties in the form of proven theorems. All practical experience shows, that the time-dependent case behaves equally well. In this method, the Hamiltonian comes in only as a hermitian linear operator and not as a mathematical expression that can be reordered or otherwise transformed in order to speed up the necessary calculations. In all cases where it is a feasible computational task to apply the Hamilton operator to a generic state vector, my suggestion is to try this method as a workhorse. It can be considered a ‘Simpson’s rule’ for quantum dynamics. There is no way for me to explain the matter from a technical point of view in 10 minutes, so I’ll try to convey the basic idea of the method. This idea is to treat quantum dynamics in closer analogy to classical dynamics as one usually does. One may argue, that there is a fundamental difference between classical and quantum dynamics since the Schrodinger equation is first order with respect to the time derivative, whereas classical mechanics is inherently second order, since forces determine accelerations and not velocities. This is not so clear a distinction as one might think, since also classical mechanics can be given the first-order form of canonical equations. I propose the opposite of such an approach, namely to give quantum dynamics a second-order form by simply differentiating the time-dependent Schrodinger equation with respect to time. To be sure, this is only a heuristic device for defining a computationally feasible approximation and not a proposal to change quantum mechanics. For the resulting equation for the second time derivative of the wave function, I then carry over and use the integration methods from the field of granular matter dynamics which I have been using in an industrial project concerning simulation of the toning process in electro-photographic copiers/printers over some years [4]. Such methods come in
In 1948, Richard Feynman published the first paper on his new approach to non-relativistic quantum mechanics. Before Feynman's work there were two mathematical formulations of quantum mechanics. Schrödinger's formulation was based on PDE (the Schrödinger equation) and states representation by wave functions, so it was in the framework of analysis and differential equations. The other formulation was Heisenberg's matrix algebra. Initially, they were thought to be competing. The proponents of one claimed that the other was “ wrong. ” Within a couple of years, John von Neumann had proved that they are equivalent.\nAlthough Feynman's theory was not fundamentally new, it nonetheless offered an entirely fresh and different perspective: via a precise formulation of Bohr's correspondence principle, it made quantum mechanics similar to classical mechanics in a precise sense. In addition, Feynman's approach made it possible to explain physical experiments, and, via diagrams, link them directly to computations. What resulted was a very powerful device for computing energies and scattering amplitudes - the famous Feynman's diagrams.\nIn his formulation, Feynman aimed at representing the solution to the non-relativistic Schrödinger equation in the form of an “ average ” over histories or paths of a particle. This solution is commonly known as the Feynman path integral. It plays an important role in the theory but appears as a postulate based on intuition coming from physics rather than a justified mathematical object. This is why Feynman's vision has caught the attention of many mathematicians as well as physicists.\nThe papers of Gelfand, Cameron, and Nelson are among the first, and more substantial, attempts to supply Feynman's theory with a rigorous mathematical foundation. These attempts were followed by many others, but unfortunately all of them were not quite satisfactory. The difficulty comes from a need to define a measure on an infinite-dimensional space of continuous functions that represent all possible paths of a particle. This Feynman's measure has to produce an integral with the properties requested by Feynman. In particular, the expression for the Feynman measure has to involve the non-absolutely integrable Fresnel integrands. The non-absolute integrability of the Fresnel integrands makes the measure fail to be positive and to have the countably additive property. Thus, a well-defined measure in the case of the Feynman path integral does not exist.\nExtensive research has been done on the methods of relating the Feynman path integral to the integral with respect to the Wiener measure. The method of analytic continuation in mass defines the Feynman path integral as a certain limit of the Wiener integrals. Unfortunately, this method can be used as definition for only almost all values of the mass parameter in the Schrödinger equation. For physicists, this is not a satisfactory result and needs to be improved.\nIn this work we examine those questions which originally led to the Feynman path integral. By now we know that Feynman's “ dream ” cannot be realized as a positive and countably additive measure on the path-space. Here, we offer a new way out by modifying Feynman's question, and thereby achieving a solution to the Schrödinger equation via a different kind of averages in the path-space.\nWe give our version of the question that Feynman “ should have asked ” in order to realize the elusive path integral. In our formulation, we get a Feynman path integral as a limit of linear functionals, as opposed to the more familiar inductive limits of positive measures, traditionally used for constructing the Wiener measure, and related Gaussian families. We adapt here an approach pioneered by Patrick Muldowney. In it, Muldowney suggested a Henstock integration technique in order to deal with the non-absolute integrability of the kind of Fresnel integrals which we need in our solution to Feynman's question. By applying Henstock's theory to Fresnel integrals, we construct a complex-valued “ probability distribution functions ” on the path-space. Then we use this “ probability ” distribution function to define the Feynman path integral as an inductive limit. This establishes a mathematically rigorous Feynman limit, and at the same time, preserves Feynman's intuitive idea in resulting functional. In addition, our definition, and our solution, do not place any restrictions on any of the parameters in the Schrödinger equation, and have a potential to offer useful computational experiments, and other theoretical insights.
Abstract This chapter introduces Schrödinger’s equation and the action integral from which the differential equation is derived by the principle of stationary action. This principle is central to almost all of physics. The discretization of the integral with finite elements and interpolation with polynomials scaled by interpolation coefficients at the ends of the elements (nodes) allows us to integrate out the spatial dependence leaving the action as a binomial expression in the coefficients. Now the action is varied with respect to these coefficients to obtain a discretized matrix version of the Schrödinger’s equation. This typically leads to matrix equation or a generalized eigenvalue problem to solve. In simple terms, this is the essence of the finite element method. The boundary conditions for the wavefunction and the derivative of the wavefunction can be very easily imposed at the nodes prior to the solution of the matrix equations. The probability current has to be continuous at interfaces in the interior of the physical domain. The form of the current is obtained by a gauge-variational method in an unambiguous manner. Going from the differential equation to the action integral, and vice versa, are treated in great detail so that the reader can gain confidence in working with variational principles.
Lee's discretization Time is a continuous parameter in standard Schrödinger wave mechanics, and quantum wavefunctions are differentiable functions of that parameter. We shall refer to this as temporal differentiability . Temporal differentiability is assumed in Dirac's more abstract formulation of quantum mechanics, where state vectors in some abstract Hilbert space and Hermitian operators over that space depend on continuous time (CT) (Dirac, 1958). The same is assumed in relativistic quantum field theories, where quantum field operators in the Heisenberg picture are differentiable over time and space. Temporal differentiability is a necessary condition for the existence of the quantum wave equations and operator field equations found in such theories. Unfortunately, these differentiable equations are usually impossible to solve exactly, so various techniques such as perturbation theory and computer simulation are employed to make suitable approximations. Numerical techniques on their own, however, are insufficient to answer questions of principle, such as the meaning of quantized spacetime. A more principled approach to the solution of quantum differential equations, namely the path integral (PI), was developed, exploited and popularized principally by Feynman (Feynman and Hibbs, 1965). In some situations, such as quantum gravity, the PI was found to be virtually the only technology available to discuss the quantum physics. We discuss this approach in greater detail in Chapter 18.