29 research outputs found

### Time-dependent Schr\"odinger equations having isomorphic symmetry algebras. II. Symmetry algebras, coherent and squeezed states

Using the transformations from paper I, we show that the Schr\"odinger
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.Comment: LaTeX, 22 pages, new format, edited, with added discussion of the
classical motio

### 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 Schr\"odinger equation for
systems with potentials $V(x,\tau)=g^{(2)}(\tau)x^2+g^{(1)}(\tau)x
+g^{(0)}(\tau)$. We describe a set of number-operator eigenstates states,
$\{\Psi_n(x,\tau)\}$, that form a complete set of states but which, however,
are usually not energy eigenstates. From the extremal state,
$\Psi_0$, 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, $\tau$. 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
$V(x,\tau)$ and having isomorphic Lie algebras, will be dealt with in the
following paper (II).Comment: 23 pages, LaTe

### Time-dependent Schr\"odinger equations having isomorphic symmetry algebras. I. Classes of interrelated equations

In this paper, we focus on a general class of Schr\"odinger equations that
are time-dependent and quadratic in X and P. We transform Schr\"odinger
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.Comment: LaTeX, 18 pages, new format, edite

### Supercoherent states and physical systems

A method is developed for obtaining coherent states of a system admitting a supersymmetry. These states are called supercoherent states. The presented approach is based on an extension to supergroups of the usual group-theoretic approach. The example of the supersymmetric harmonic oscillator is discussed, thereby illustrating some of the attractive features of the method. Supercoherent states of an electron moving in a constant magnetic field are also described

### 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.Comment: 23 pages including 9 figures. To be published in Optics
Communication

### 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 Schr\"odinger
equations and obtain eigensolutions. The supercoherent states for this system
are constructed.Comment: 47 pages, submitted to Phys. Rev. A, LaTeX, IUHET 243 and
LA-UR-93-20

### 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.Comment: 8 pages, LAUR-93-1721, LaTe

### Algebraic approach in the study of time-dependent nonlinear integrable systems: Case of the singular oscillator

The classical and the quantal problem of a particle interacting in
one-dimension with an external time-dependent quadratic potential and a
constant inverse square potential is studied from the Lie-algebraic point of
view. The integrability of this system is established by evaluating the exact
invariant closely related to the Lewis and Riesenfeld invariant for the
time-dependent harmonic oscillator. We study extensively the special and
interesting case of a kicked quadratic potential from which we derive a new
integrable, nonlinear, area preserving, two-dimensional map which may, for
instance, be used in numerical algorithms that integrate the
Calogero-Sutherland-Moser Hamiltonian. The dynamics, both classical and
quantal, is studied via the time-evolution operator which we evaluate using a
recent method of integrating the quantum Liouville-Bloch equations \cite{rau}.
The results show the exact one-to-one correspondence between the classical and
the quantal dynamics. Our analysis also sheds light on the connection between
properties of the SU(1,1) algebra and that of simple dynamical systems.Comment: 17 pages, 4 figures, Accepted in PR