Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

International Journal of Modern Physics A

SCHRÖDINGER EQUATIONS WITH TIME-DEPENDENT P2 AND X2 TERMS

Michael Martin Nieto, D. Rodney Truax

Published Apr 30, 2002• 0 citations
Publisher

Abstract

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.

DOI
10.1142/s0217751x02009667

Related papers

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

Michael Martin Nieto, D. Rodney Truax

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.

arXiv:quant-ph/9911093•Nov 20, 1999•1 citation

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

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

Time-dependent mass Schr\"odinger equations. II. Example 1

Michael Martin Nieto, D. Rodney Truax

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

•Nov 20, 1999•0 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

EQUIVALENCE OF TIME-DEPENDENT SCHRÖDINGER EQUATIONS WITH CONSTANT AND TIME-DEPENDENT MASS

Axel Schulze‐Halberg

We show that the time-dependent Schrödinger equation (TDSE) for a potential of the form V(x,t)=A(t)x2+B(t)x+C(t) and time-dependent mass can be transformed into the same TDSE with constant mass. We obtain an explicit formula relating solutions of the TDSE for time-dependent mass and for constant mass to each other.

Modern Physics Letters A•Dec 4, 2003•2 citations
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.
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.
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.
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.
In this paper, we attack the specific time-dependent Hamiltonian problem ${H}=-\lfrac{1}{2}e^{\Upsilon(t-t_o)}\partial_{xx} +\lfrac{1}{2}\omega^2e^{-\Upsilon(t-t_o)}x^2$. This corresponds to a time-dependent mass (TM) Schr\odinger equation. We give the specific transformations to i) the more general quadratic (TQ) Schr\odinger equation and to ii) a different time-dependent oscillator (TO) equation. For each Schr\odinger 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(\Delta x)^2(Δx)2, (Δp)2(\Delta p)^2(Δp)2, and uncertainty product.
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.