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

Journal of Physics A Mathematical and General

The SU(2)(+)h(4) Hamiltonian

C M Cheng, P. C. W. Fung

Published Jan 7, 1990• 4 citations
Publisher

Abstract

In a previous paper the authors have employed a group theoretic method to find the evolution operator for a quantum system having a SU(2) Hamiltonian. In this paper, they consider a more complicated system whose Hamiltonian consists of SU(2) and h(4) group generators. A transformation method will be introduced, in conjunction with the above group theoretic method, to tackle this quantum problem. The result thus obtained will be applied to the problem of a mass-varying harmonic oscillator under an external force. Their result shows that the equation of motion of this oscillator is identical to a damped harmonic oscillator under an external force. In addition, it is shown that an initial coherent state will evolve as a squeezed state under the above Hamiltonian.

DOI
10.1088/0305-4470/23/1/014

Related papers

The evolution operator technique in solving the Schrodinger equation, and its application to disentangling exponential operators and solving the problem of a mass-varying harmonic oscillator

C M Cheng, P. C. W. Fung

The authors present a method for finding the evolution operator for the Schrodinger equation for the Hamiltonian expressible as H(t)=a 1 (t)J 4 +a 2 (t)J 0 +a 3 (t) J - where J + , J 0 and J - are the SU(2) group generators. Such a method is applied to the disentangling technique for exponential operators which are not necessarily unitary. As a demonstration of our general approach, they solved the problem of a harmonic oscillator with a varying mass.

Journal of Physics A Mathematical and General•Nov 21, 1988•97 citations

Squeezing in the Schrodinger picture: normal ordering technique to solve the mass-varying oscillator

A. L. De Brito, B. Baseia

Recent papers by Cheng and Fung (1988) and Lo (1990) have studied the generation of squeezed states out of coherent states in systems described by some time-dependent Hamiltonians. They have employed the evolution operator method for the Schrodinger equation with a Hamiltonian expressible as H(t)=a l (t)J + +a 2 (t)J 0 + +a 3 (t)J - , where J +- ,J 0 are the SU(2) group generators. Here, for the sake of comparison of methods and results, the present authors employ the normal ordered Hamiltonian H(t)= Sigma h l,m (t)(a) l a m and investigate squeezing through an alternative approach using the normal ordering technique to solve the evolution operator for the same system.

Quantum Optics Journal of the European Optical Society Part B•Aug 1, 1992•3 citations

Group Theoretical Approach to the Coherent and the Squeeze States of a Time-Dependent Harmonic Oscillator with a Singular Term

Jung Kon Kim, Sang Pyo Kim

For a time-dependent harmonic oscillator with an inverse squared singular term, we find the generalized invariant using the Lie algebra of $SU(2)$ and construct the number-type eigenstates and the coherent states using the spectrum-generating Lie algebra of $SU(1,1)$. We obtain the evolution operator in both of the Lie algebras. The number-type eigenstates and the coherent states are constructed group-theoretically for both the time-independent and the time-dependent harmonic oscillators with the singular term. It is shown that the squeeze operator transforms unitarily the time-dependent basis of the spectrum-generating Lie algebra of $SU(1,1)$ for the generalized invariant, and thereby evolves the initial vacuum into a final coherent vacuum.

arXiv:quant-ph/9511013•Journal of the Korean Physical Society•Nov 13, 1995•5 citations

Unitary approach to the quantum forced harmonic oscillator

D. Velasco-Martínez, Victor Guadalupe Ibarra‐Sierra, Juan Carlos Sandoval-Santana +2 more

In this paper we introduce an alternative approach to studying the evolution of a quantum harmonic oscillator subject to an arbitrary time dependent force. With the purpose of finding the evolution operator, certain unitary transformations are applied successively to Schrödinger's equation reducing it to its simplest form. Therefore, instead of solving the original Schrödinger's partial differential equation in time and space the problem is replaced by a system of ordinary differential equations. From the obtained evolution operator we workout the propagator. Even though we illustrate the use of unitary transformations on the solution of a forced harmonic oscillator, the method presented here might be used to solve more complex systems. The present work addresses many aspects regarding unitary transformations and the dynamics of a forced quantum harmonic oscillator that should be useful for students and tutors of the quantum mechanics courses at the senior undergraduate and graduate level.

arXiv:1409.0236•Aug 31, 2014•1 citation

Squeezing caused by a change of mass in a harmonic oscillator

Fan Hong-Yi, Haider R. Zaidi

It is shown that a mass change in a harmonic oscillator generates a squeezing transformation. Time-independent as well as time-dependent transformations are investigated. An expression for the interaction Hamiltonian responsible for squeezing and the equations of motion for the time evolution are derived.

Canadian Journal of Physics•Feb 1, 1989•9 citations

Evolution of SU(1,1) coherent states in harmonic oscillators with time-dependent masses

Christopher C. Gerry, M F Plumb

The authors study the evolution of SU(1,1) coherent states, which may represent squeezed vacuum states, in harmonic oscillators with time-dependent masses. Such oscillators may model a quantised electromagnetic field in a cavity where the field changes with time under the action of some reservoir. They consider three types of variable mass: a decaying mass, a pulsating mass, and a mass with damping and pulsation. They have found that in the most interesting cases involving pulsation, a high degree of squeezing (enhanced squeezing) may be obtained in the vicinity of the resonance where the pulsation frequency is twice the natural oscillator frequency.

Journal of Physics A Mathematical and General•Sep 7, 1990•15 citations
In a previous paper the authors have employed a group theoretic method to find the evolution operator for a quantum system having a SU(2) Hamiltonian. In this paper, they consider a more complicated system whose Hamiltonian consists of SU(2) and h(4) group generators. A transformation method will be introduced, in conjunction with the above group theoretic method, to tackle this quantum problem. The result thus obtained will be applied to the problem of a mass-varying harmonic oscillator under an external force. Their result shows that the equation of motion of this oscillator is identical to a damped harmonic oscillator under an external force. In addition, it is shown that an initial coherent state will evolve as a squeezed state under the above Hamiltonian.
Recent papers by Cheng and Fung (1988) and Lo (1990) have studied the generation of squeezed states out of coherent states in systems described by some time-dependent Hamiltonians. They have employed the evolution operator method for the Schrodinger equation with a Hamiltonian expressible as H(t)=a l (t)J + +a 2 (t)J 0 + +a 3 (t)J - , where J +- ,J 0 are the SU(2) group generators. Here, for the sake of comparison of methods and results, the present authors employ the normal ordered Hamiltonian H(t)= Sigma h l,m (t)(a) l a m and investigate squeezing through an alternative approach using the normal ordering technique to solve the evolution operator for the same system.
For a time-dependent harmonic oscillator with an inverse squared singular term, we find the generalized invariant using the Lie algebra of SU(2)SU(2)SU(2) and construct the number-type eigenstates and the coherent states using the spectrum-generating Lie algebra of SU(1,1)SU(1,1)SU(1,1). We obtain the evolution operator in both of the Lie algebras. The number-type eigenstates and the coherent states are constructed group-theoretically for both the time-independent and the time-dependent harmonic oscillators with the singular term. It is shown that the squeeze operator transforms unitarily the time-dependent basis of the spectrum-generating Lie algebra of SU(1,1)SU(1,1)SU(1,1) for the generalized invariant, and thereby evolves the initial vacuum into a final coherent vacuum.
In this paper we introduce an alternative approach to studying the evolution of a quantum harmonic oscillator subject to an arbitrary time dependent force. With the purpose of finding the evolution operator, certain unitary transformations are applied successively to Schrödinger's equation reducing it to its simplest form. Therefore, instead of solving the original Schrödinger's partial differential equation in time and space the problem is replaced by a system of ordinary differential equations. From the obtained evolution operator we workout the propagator. Even though we illustrate the use of unitary transformations on the solution of a forced harmonic oscillator, the method presented here might be used to solve more complex systems. The present work addresses many aspects regarding unitary transformations and the dynamics of a forced quantum harmonic oscillator that should be useful for students and tutors of the quantum mechanics courses at the senior undergraduate and graduate level.
The authors study the evolution of SU(1,1) coherent states, which may represent squeezed vacuum states, in harmonic oscillators with time-dependent masses. Such oscillators may model a quantised electromagnetic field in a cavity where the field changes with time under the action of some reservoir. They consider three types of variable mass: a decaying mass, a pulsating mass, and a mass with damping and pulsation. They have found that in the most interesting cases involving pulsation, a high degree of squeezing (enhanced squeezing) may be obtained in the vicinity of the resonance where the pulsation frequency is twice the natural oscillator frequency.