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D. Rodney Truax

Publications and source records attributed to D. Rodney Truax.

16 recordsLinked to original sources
arXiv2002

Schrödinger equations with time-dependent P^2 and X^2 terms

We present some general results for the time-dependent mass Hamiltonian problem with H=-{1/2}e^{-2ν}\partial_{xx} +h^{(2)}(t)e^{2ν}x^2. This Hamiltonian corresponds to a time-dependent mass (TM) Schrödinger equation with the restriction that there are only P^2 and X^2 terms. We give the specific transformations to a different quantum Schrödinger(TQ) equation and to a different time-dependent oscillator (TO) equation. For each Schrödinger system, we give the Lie algebra of space-time symmetries and (x,t) representations for number states, coherent states, and squeezed states. These general results include earlier work as special cases.

quant-ph↗
arXiv2002

The Schrödinger system H=-{1/2} (t_o/t)^a \partial_{xx} + (1/2) ω^2 (t/t_o)^b x^2

We attack the specific time-dependent Hamiltonian problem H=-{1/2} (t_o/t)^a \partial_{xx} + (1/2) ω^2 (t/t_o)^b x^2. This corresponds to a time-dependent mass (TM) Schrödinger equation. We give the specific transformations to a different time-dependent quadratic Schrödinger equations (TQ) and to a different time-dependent oscillator (TO) equation. For each Schrödinger system, we give the Lie algebra of space-time symmetries, the number states, the squeezed-state and (with their classical motion), (Δx)^2, (Δp)^2, and the uncertainty product.

quant-ph↗
arXiv2000

Symmetries and solutions of the three-dimensional Paul trap

Using the symmetries of the three-dimensional Paul trap, we derive the solutions of the time-dependent Schrödinger equation for this system, in both Cartesian and cylindrical coordinates. Our symmetry calculations provide insights that are not always obvious from the conventional viewpoint.

quant-ph↗
arXiv2000

The Schrödinger system H=-{1/2}e^{Υ(t-t_o)}\partial_{xx} +\lfrac{1}{2}ω^2e^{-Υ(t-t_o)}x^2

In this paper, we attack the specific time-dependent Hamiltonian problem H=-{1/2}e^{Υ(t-t_o)}\partial_{xx} +\lfrac{1}{2}ω^2e^{-Υ(t-t_o)}x^2. This corresponds to a time-dependent mass (TM) Schrödinger equation. We give the specific transformations to i) the more general quadratic (TQ) Schrödinger equation and to ii) a different time-dependent oscillator (TO) equation. For each Schrödinger system, we give the Lie algebra of space-time symmetries, the number states, the coherent states, the squeezed-states and the time-dependent , , (Δx)^2, (Δp)^2, and uncertainty product.

quant-ph↗
arXiv1999

Time-dependent Schrödinger equations having isomorphic symmetry algebras. I. Classes of interrelated equations

In this paper, we focus on a general class of Schrödinger equations that are time-dependent and quadratic in X and P. We transform Schrödinger equations in this class, via a class of time-dependent mass equations, to a class of solvable time-dependent oscillator equations. This transformation consists of a unitary transformation and a change in the ``time'' variable. We derive mathematical constraints forthe transformation and introduce two examples.

quant-ph↗
arXiv1999

Time-dependent Schrödinger equations having isomorphic symmetry algebras. II. Symmetry algebras, coherent and squeezed states

Using the transformations from paper I, we show that the Schrödinger equations for: (1)systems described by quadratic Hamiltonians, (2) systems with time-varying mass, and (3) time-dependent oscillators, all have isomorphic Lie space-time symmetry algebras. The generators of the symmetry algebras are obtained explicitly for each case and sets of number-operator states are constructed. The algebras and the states are used to compute displacement-operator coherent and squeezed states. Some properties of the coherent and squeezed states are calculated. The classical motion of these states is deomonstrated.

quant-ph↗
arXiv1999

Higher-Power Coherent and Squeezed States

A closed form expression for the higher-power coherent states (eigenstates of $a^{j}$) is given. The cases j=3,4 are discussed in detail, including the time-evolution of the probability densities. These are compared to the case j=2, the even- and odd-coherent states. We give the extensions to the "effective" displacement-operator, higher-power squeezed states and to the ladder-operator/minimum-uncertainty, higher-power squeezed states. The properties of all these states are discussed.

quant-ph↗
arXiv1996

Displacement-Operator Squeezed States. I. Time-Dependent Systems Having Isomorphic Symmetry Algebras

In this paper we use the Lie algebra of space-time symmetries to construct states which are solutions to the time-dependent Schrödinger equation for systems with potentials $V(x,τ)=g^{(2)}(τ)x^2+g^{(1)}(τ)x +g^{(0)}(τ)$. We describe a set of number-operator eigenstates states, $\{Ψ_n(x,τ)\}$, that form a complete set of states but which, however, are usually not energy eigenstates. From the extremal state, $Ψ_0$, and a displacement squeeze operator derived using the Lie symmetries, we construct squeezed states and compute expectation values for position and momentum as a function of time, $τ$. We prove a general expression for the uncertainty relation for position and momentum in terms of the squeezing parameters. Specific examples, all corresponding to choices of $V(x,τ)$ and having isomorphic Lie algebras, will be dealt with in the following paper (II).

quant-ph↗
arXiv1996

Displacement-Operator Squeezed States. II. Examples of Time-Dependent Systems Having Isomorphic Symmetry Algebras

In this article, results from the previous paper (I) are applied to calculations of squeezed states for such well-known systems as the harmonic oscillator, free particle, linear potential, oscillator with a uniform driving force, and repulsive oscillator. For each example, expressions for the expectation values of position and momentum are derived in terms of the initial position and momentum, as well as in the $(α,z)$- and in the $(z,α)$-representations described in I. The dependence of the squeezed-state uncertainty products on the time and on the squeezing parameters are determined for each system.

quant-ph↗
arXiv1996

Holstein-Primakoff/Bogoliubov Transformations and the Multiboson System

As an aid to understanding the {\it displacement operator} definition of squeezed states for arbitrary systems, we investigate the properties of systems where there is a Holstein-Primakoff or Bogoliubov transformation. In these cases the {\it ladder-operator or minimum-uncertainty} definitions of squeezed states are equivalent to an extent displacement-operator definition. We exemplify this in a setting where there are operators satisfying $[A, A^{\dagger}] = 1$, but the $A$'s are not necessarily the Fock space $a$'s; the multiboson system. It has been previously observed that the ground state of a system often can be shown to to be a coherent state. We demonstrate why this must be so. We close with a discussion of an alternative, effective definition of displacement-operator squeezed states.

quant-ph↗
arXiv1995

ARBITRARY-ORDER HERMITE GENERATING FUNCTIONS FOR COHERENT AND SQUEEZED STATES

For use in calculating higher-order coherent- and squeezed- state quantities, we derive generalized generating functions for the Hermite polynomials. They are given by $\sum_{n=0}^{\infty}z^{jn+k}H_{jn+k}(x)/(jn+k)!$, for arbitrary integers $j\geq 1$ and $k\geq 0$. Along the way, the sums with the Hermite polynomials replaced by unity are also obtained. We also evaluate the action of the operators $\exp[a^j(d/dx)^j]$ on well-behaved functions and apply them to obtain other sums.

quant-ph↗
arXiv1993

Squeezed States for General Systems

We propose a ladder-operator method for obtaining the squeezed states of general symmetry systems. It is a generalization of the annihilation-operator technique for obtaining the coherent states of symmetry systems. We connect this method with the minimum-uncertainty method for obtaining the squeezed and coherent states of general potential systems, and comment on the distinctions between these two methods and the displacement-operator method.

hep-th↗
arXiv1993

Supersymmetry and a Time-Dependent Landau System

A general technique is outlined for investigating supersymmetry properties of a charged spin-$\half$ quantum particle in time-varying electromagnetic fields. The case of a time-varying uniform magnetic induction is examined and shown to provide a physical realization of a supersymmetric quantum-mechanical system. Group-theoretic methods are used to factorize the relevant Schrödinger equations and obtain eigensolutions. The supercoherent states for this system are constructed.

hep-th↗
arXiv1992

Supersqueezed States

We derive the supersqueeze operator for the supersymmetric harmonic oscillator, using Baker-Campbell-Hausdorff relations for the supergroup OSP(2/2). Combining this with the previously obtained superdisplacement operator, we derive the supersqueezed states. These are the supersymmetric generalization of the squeezed states of the harmonic oscillator.

hep-ph↗