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American Journal of Physics

Free particle nonenergy eigenfunctions of a transformed momentum operator using classical solution variables

David Milton Fradkin

Published Feb 1, 1994• 1 citation
Publisher

Abstract

The classical solution is used to provide appropriate variables for the quantum-mechanical free particle time-dependent Schrödinger wave equation, which then becomes separable giving nonenergy eigenfunctions of closed form. Solutions derived in this fashion may fail to satisfy the homogeneous Schrödinger wave equation at one point in space/time; if this should occur, an inhomogeneous term is produced and a corresponding restricted solution reduces to the free particle propagator. In general, it is shown that the nonenergy eigenfunctions may also be interpreted as eigenfunctions of a time-independent transformed momentum operator. The similarity transformation, by means of which is constructed the transformed momentum operator, the transformed position operator, and the transformed Hamiltonian, are explicitly determined. The relationship of the nonenergy eigenfunctions to the energy eigenfunctions is also exhibited.

DOI
10.1119/1.17640

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The classical solution is used to provide appropriate variables for the quantum-mechanical free particle time-dependent Schrödinger wave equation, which then becomes separable giving nonenergy eigenfunctions of closed form. Solutions derived in this fashion may fail to satisfy the homogeneous Schrödinger wave equation at one point in space/time; if this should occur, an inhomogeneous term is produced and a corresponding restricted solution reduces to the free particle propagator. In general, it is shown that the nonenergy eigenfunctions may also be interpreted as eigenfunctions of a time-independent transformed momentum operator. The similarity transformation, by means of which is constructed the transformed momentum operator, the transformed position operator, and the transformed Hamiltonian, are explicitly determined. The relationship of the nonenergy eigenfunctions to the energy eigenfunctions is also exhibited.
Abstract In many texts, the transition from classical mechanics to quantum mechanics is achieved by substituting the action for the phase angle. The paper presents a different approach to show some connections between classical and quantum mechanics for a single particle for an audience at graduate and postgraduate levels. Firstly, it is shown that a wave equation of action can be derived under the free particle condition and the Legendre transform. The wave-like solutions of the action, Hamiltonian and momentum of the free particle are presented. Using the discrete approximation, the equation of motion of a single particle, in scalar potential field, is obtained in a similar form to Schrödinger’s equation. The rest of the paper discusses the propagation, superposition of the wave-like dynamic variables and their connections to quantum mechanics. The superposition of the variables of a particle is generally distinct from the superposition of classical waves (e.g. acoustics). The quantum superposition provides a self-consistent interpretation of the wave-like solutions of the variables. Connections between the classical and quantum relations for corresponding variables are observed from the one-to-one comparisons.
The classical solution is used as a variable for the one-dimensional quantum-mechanical oscillator, and this leads to a separable differential equation solvable in closed form which yields a set of new quantum mechanical solutions, not energy eigenfunctions, that have the property of being localized. These new solutions are shown to be eigenfunctions of a time-independent position operator. The operator algebra of that operator and an analogous momentum operator is given, as well as the relationship of the eigenfunctions to the energy eigenfunction set. The new eigenfunctions, which are not square integrable but possess a δ-function normalization, themselves represent wavepackets that at times corresponding to when the classical position is at an extremum collapse to Dirac δ functions.
Solutions of the Schrödinger equation with an exact time dependence are derived as eigenfunctions of dynamical invariants which are constructed from time-independent operators using time-dependent unitary transformations. Exact solutions and a closed form expression for the corresponding time evolution operator are found for a wide range of time-dependent Hamiltonians in d dimensions, including non-Hermitean -symmetric Hamiltonians. Hamiltonians are constructed using time-dependent unitary spatial transformations comprising dilatations, translations and rotations and solutions are found in several forms: as eigenfunctions of a quadratic invariant, as coherent state eigenfunctions of boson operators, as plane wave solutions from which the general solution is obtained as an integral transform by means of the Fourier transform, and as distributional solutions for which the initial wavefunction is the Dirac δ-function. For the isotropic harmonic oscillator in d dimensions radial solutions are found which extend known results for d = 1, including Barut–Girardello and Perelomov coherent states (i.e., vector coherent states), which are shown to be related to eigenfunctions of the quadratic invariant by the ζ-transformation. This transformation, which leaves the Ermakov equation invariant, implements SU(1, 1) transformations on linear dynamical invariants. coherent states are derived also for the time-dependent linear potential. Exact solutions are found for Hamiltonians with electromagnetic interactions in which the time-dependent magnetic and electric fields are not necessarily spatially uniform. As an example, it is shown how to find exact solutions of the time-dependent Schrödinger equation for the Dirac magnetic monopole in the presence of time-dependent magnetic and electric fields of a specified form.
We present a simple approach for finding the general solution of the time-dependent Schrödinger equation for a particle moving under the action of a uniform force. From such a general solution we derive the eigenfunctions of the energy as well as those of the momentum. Next we obtain the propagator and show that the time evolution of an arbitrarily chosen initial wavefunction is described by a Fresnel transformation. This establishes a useful link with optical phenomena and allows us to link the case of a particle moving under a uniform force to that of the free particle.
We reexamine the general solution of a Schr\"odinger equation in the presence of a time-dependent linear potential in configuration space based on the Lewis-Riesenfeld framework. For comparison, we also solve the problem in momentum space and then Fourier transform the solution to get the general wave function. Appropriately choosing the weight function in the latter method, we can obtain the same wave function as the former method. It is found that a non-Hermitian time-dependent linear invariant can be used to obtain Gaussian-type wave-packet solutions of the time-dependent system. This operator is a specific linear combination of the initial momentum and initial position operators. This fact indicates that the constants of integration such as the initial position and initial momentum that determine the classical motion play important roles in the time-dependent quantum system. The eigenfunction of the linear invariant is interpreted as a wave packet with a ``center of mass'' moving along the classical trajectory, while the ratio between the coefficients of the initial position and initial momentum determines the width of the wave packet.
The time-dependent Schrödinger equation (TDSE) contains two variables: position and time. Therefore, the solutions are functions of both space and time (x and t, if we limit ourselves to particles moving in one dimension). It turns out that the easiest way to "solve" such an equation of two variables is to perform a separation of variables. We show that if the potential does not change with time, then certain special solutions (eigenfunctions) can be written as a product of a space function and a time function: Ψ(x,t) = ψ(x) f(t). By assuming a "separable" solution of this form, and plugging into the TDSE, the TDSE itself can be separated into two equations: the time-independent Schrödinger equation (TISE) with solution ψ(x), and the space-independent Schrödinger equation (SISE) with solution f(t). Furthermore, f(t) always equals e -iωt , while ψ(x) depends on the potential. So, for potentials that do not vary with time, we need only solve the much simpler TISE in order to investigate the energy, momentum, and so on of the particle. This is why, for the remainder of this book, we only use the TISE.