D. Rodney Truax

Symmetry of time‐dependent Schrödinger equations. I. A classification of time‐dependent potentials by their maximal kinematical algebras

Potentials for the time‐dependent Schrodinger equation [− 1/2 ∂xx +V(x,t)]Ψ(x,t) = i∂tΨ(x,t) are classified according to their space–time or kinematical algebras in a search for exactly solvable time‐dependent models. In addition, it is shown that their dynamical algebras are isomorphic to their kinematical algebras on the solution space of the Schrodinger equation.

Symmetry of time‐dependent Schrödinger equations. II. Exact solutions for the equation {∂xx+2i∂t−2g2(t)x2−2g1(t)x −2g0(t)}Ψ(x, t) = 0

The maximal kinematical algebra of the Schrodinger equation {∂xx+2i∂t−2g2(t)x2−2g1(t)x−2 g0 (t)}Ψ(x, t) = 0 is known to be the Schrodinger algebra s1. The kinematical symmetries are realized as first‐order differential operators in the space and time variables. A subalgebra G of s1 is chosen and from G and its invariants a complete set of commuting observables are constructed. The solution space of the Schrodinger equation is identified with the appropriate irreducible representation space of G. The wave functions, simultaneous eigenvectors of the compatible observables, are computed as explicit functions of space and time. The properties of a system with a potential V(x, t) = g2(t)x2+g1(t)x+g0(t) are discussed.

Grassmann‐valued fluid dynamics

Certain systems of nonlinear partial differential equations can be written in a simple form as a single Grassmann‐valued partial differential equation. Equations describing compressible fluid flow are of this type. A method for finding soft solutions of the Grassmann‐valued partial differential equation arising in this context is presented. The method is a generalization of the Lagrangian‐coordinates approach to the case of Grassmann variables. Generally, solutions obtained by this method have the form of infinite series, whose expansion yields new relations among the unknown variables. In some simple cases, the series can be summed. The equivalence of the Grassmann solutions to the usual solutions is shown for these cases.

Coherent states sometimes look like squeezed states and vice versa: the Paul trap

Using the Paul trap as a model, we point out that the same wavefunctions can be variously coherent or squeezed states, depending upon the system to which they are applied.

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 Schrodinger equation for systems with potentials V(x,τ)=g(2)(τ)x2+g(1)(τ)x+g(0)(τ). We describe a set of number-operator eigenstates states, {Ψn(x,τ)}, that form a complete set of states but which, however, are usually not energy eigenstates. From the extremal state, Ψ0, 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).

On a time‐dependent extension of the Morse potential

A time‐dependent extension of a Morse potential is formulated. Both the bound and unbound state wave functions are obtained algebraically for the resulting time‐dependent Schrodinger equations, based upon the representations of su(2) and su(1,1), respectively. The method of R‐separation of variables is instrumental in the analysis.

Higher-power coherent and squeezed states

Abstract 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.

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 Schrodinger 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 demonstrated.

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

In this paper, we focus on a general class of Schrodinger equations that are time dependent and quadratic in X and P. We transform Schrodinger 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 for the transformation and introduce two examples.

Displacement-operator squeezed states. II. Examples of time-dependent systems having isomorphic symmetry algebras

In this paper, 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 is determined for each system.