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

Michael Martin Nieto, D. Rodney Truax

Published Jan 1, 1997• 10 citations
OpenAlex

Abstract

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)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).

Related papers

Displacement-operator squeezed states. I. Time-dependent systems having isomorphic symmetry algebras

Michael Martin Nieto, D. Rodney Truax

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)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).

arXiv:quant-ph/9608008•Journal of Mathematical Physics•Jan 1, 1997•10 citations

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

Michael Martin Nieto, D. Rodney Truax

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.

arXiv:quant-ph/9608009•Journal of Mathematical Physics•Jan 1, 1997•6 citations

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

Michael Martin Nieto, D. Rodney Truax

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 demonstrated.

arXiv:quant-ph/9811076•Journal of Mathematical Physics•May 1, 2000•10 citations

The Schrödinger System H=−12eϒ(t−to)∂xx+12ω2e−ϒ(t−to)x2

Michael Martin Nieto, D. Rodney Truax

In this paper, we attack the specific time-dependent Hamiltonian problem H=−12eϒ(t−to)∂xx+12ω2e−ϒ(t−to)x2. This corresponds to a time-dependent mass (TM) Schrodinger equation. We give the specific transformations to (i) the more general quadratic (TQ) Schrodinger equation and to (ii) a different time-dependent oscillator (TO) equation. For each Schrodinger system, we give the Lie algebra of space-time symmetries, the number states, the coherent states, the squeezed-states, and the time-dependent 〈x〉, 〈p〉, (Δx)2, (Δp)2, and uncertainty product.

arXiv:quant-ph/9911094•Aug 25, 2001•1 citation

SCHRÖDINGER EQUATIONS WITH TIME-DEPENDENT P2 AND X2 TERMS

Michael Martin Nieto, D. Rodney Truax

We present some general results for the time-dependent mass Hamiltonian problem with H=-½e-2ν(t)∂xx+h(2)(t)e2ν(t)x2, where ν(t) is a continuous function of t. This Hamiltonian corresponds to a time-dependent mass (TM) Schrödinger equation with the restriction that there are only P2 and X2 terms. We give the specific transformations to a different quadratic 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 works as special cases.

International Journal of Modern Physics A•Apr 30, 2002•0 citations

The Schrödinger System H=−12(tot)a∂xx+12ω2(tto)bX2

Michael Martin Nieto, D. Rodney Truax

Abstract We attack the specific time-dependent Hamiltonian problem H=− 1 2 ( t o t ) a ∂ xx + 1 2 ω 2 ( t t o ) b x 2 . This corresponds to a time-dependent mass ( TM ) Schrodinger equation. We give the specific transformations to a different time-dependent quadratic Schrodinger equation ( TQ ) and to a different time-dependent oscillator ( TO ) equation. For each Schrodinger system, we give the Lie algebra of space-time symmetries, the number states, the squeezed-state 〈 x 〉 and 〈 p 〉 (with their classical motion), (Δ x ) 2 , (Δ p ) 2 , and the uncertainty product.

arXiv:quant-ph/9911095•Aug 25, 2001•2 citations
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)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).
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)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).
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.
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 demonstrated.
In this paper, we attack the specific time-dependent Hamiltonian problem H=−12eϒ(t−to)∂xx+12ω2e−ϒ(t−to)x2. This corresponds to a time-dependent mass (TM) Schrodinger equation. We give the specific transformations to (i) the more general quadratic (TQ) Schrodinger equation and to (ii) a different time-dependent oscillator (TO) equation. For each Schrodinger system, we give the Lie algebra of space-time symmetries, the number states, the coherent states, the squeezed-states, and the time-dependent 〈x〉, 〈p〉, (Δx)2, (Δp)2, and uncertainty product.
We present some general results for the time-dependent mass Hamiltonian problem with H=-½e-2ν(t)∂xx+h(2)(t)e2ν(t)x2, where ν(t) is a continuous function of t. This Hamiltonian corresponds to a time-dependent mass (TM) Schrödinger equation with the restriction that there are only P2 and X2 terms. We give the specific transformations to a different quadratic 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 works as special cases.
Abstract We attack the specific time-dependent Hamiltonian problem H=− 1 2 ( t o t ) a ∂ xx + 1 2 ω 2 ( t t o ) b x 2 . This corresponds to a time-dependent mass ( TM ) Schrodinger equation. We give the specific transformations to a different time-dependent quadratic Schrodinger equation ( TQ ) and to a different time-dependent oscillator ( TO ) equation. For each Schrodinger system, we give the Lie algebra of space-time symmetries, the number states, the squeezed-state 〈 x 〉 and 〈 p 〉 (with their classical motion), (Δ x ) 2 , (Δ p ) 2 , and the uncertainty product.