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arXiv:quant-ph/9807002 · Journal of Physics A Mathematical and General

Time-dependent diffeomorphisms as quantum canonical transformations and the time-dependent harmonic oscillator

Ali Mostafazadeh

Published Jul 31, 1998• 37 citations
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Abstract

Quantum canonical transformations corresponding to time-dependent diffeomorphisms of the configuration space are studied. A special class of these transformations which correspond to time-dependent dilatations is used to identify a previously unknown class of exactly solvable time-dependent harmonic oscillators. The Caldirola-Kanai oscillator belongs to this class. For a general time-dependent harmonic oscillator, it is shown that choosing the dilatation parameter to satisfy the classical equation of motion, one obtains the solution of the Schrödinger equation. A simple generalization of this result leads to the reduction of the Schrödinger equation to a second-order ordinary differential equation whose special case is the auxiliary equation of the Lewis-Riesenfeld invariant theory. The time-evolution operator is expressed in terms of a positive real solution of this equation in a closed form, and the time-dependent position and momentum operators are calculated.

DOI
10.1088/0305-4470/31/30/014

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Quantum canonical transformations corresponding to time-dependent diffeomorphisms of the configuration space are studied. A special class of these transformations which correspond to time-dependent dilatations is used to identify a previously unknown class of exactly solvable time-dependent harmonic oscillators. The Caldirola-Kanai oscillator belongs to this class. For a general time-dependent harmonic oscillator, it is shown that choosing the dilatation parameter to satisfy the classical equation of motion, one obtains the solution of the Schrödinger equation. A simple generalization of this result leads to the reduction of the Schrödinger equation to a second-order ordinary differential equation whose special case is the auxiliary equation of the Lewis-Riesenfeld invariant theory. The time-evolution operator is expressed in terms of a positive real solution of this equation in a closed form, and the time-dependent position and momentum operators are calculated.
Quantum canonical transformations corresponding to the action of the unitary operator eiε(t)f(x)pf(x)e^{iε(t)\sqrt{f(x)}p\sqrt{f(x)}}eiε(t)f(x)​pf(x)​ is studied. It is shown that for f(x)=xf(x)=xf(x)=x, the effect of this transformation is to rescale the position and momentum operators by eε(t)e^{ε(t)}eε(t) and e−ε(t)e^{-ε(t)}e−ε(t), respectively. This transformation is shown to lead to the identification of a previously unknown class of exactly solvable time-dependent harmonic oscillators. It turns out that the Caldirola-Kanai oscillator whose mass is given by m=m0eγtm=m_0 e^{γt}m=m0​eγt, belongs to this class. It is also shown that for arbitrary f(x)f(x)f(x), this canonical transformations map the dynamics of a free particle with constant mass to that of free particle with a position-dependent mass. In other words, they lead to a change of the metric of the space.
In this work we present the classical and quantum solutions of time-dependent coupled harmonic oscillators. In these systems the masses, frequencies and coupling parameter (k) are functions of time. Four systems are investigated. To obtain the classical solutions we use a coordinate and momentum transformations along with a canonical transformation to write the original Hamiltonian as the sum of two Hamiltonians of uncoupled harmonic oscillators with modified time-dependent frequencies and unitary masses. We find the analytical expression for position and velocity of each oscillator of the systems. To obtain the exact quantum solutions we use a unitary transformation and the Lewis and Riesenfeld invariant method. The wave functions obtained are written in terms of a c-number quantity () which is solution of the Milne-Pinney equation. For each system we solve the respective Milne-Pinney equation and discuss how the quantum fluctuations and the uncertainty product evolve with time.
A time-dependent unitary (canonical) transformation is found that maps the Hamiltonian for a harmonic oscillator with time-dependent real mass and real frequency to that of a generalized harmonic oscillator with time-dependent real mass and imaginary frequency. The latter may be reduced to an ordinary harmonic oscillator by means of another unitary (canonical) transformation. A simple analysis of the resulting system leads to the identification of a previously unknown class of exactly solvable time-dependent oscillators. Furthermore, it is shown how one can apply these results to establish a canonical equivalence between some real and imaginary frequency oscillators. In particular it is shown that a harmonic oscillator whose frequency is constant and whose mass grows linearly in time is canonically equivalent with an oscillator whose frequency changes from being real to imaginary and vice versa repeatedly.
In this work we present the classical and quantum solutions of two classes of time-dependent harmonic oscillators, namely: (a) the log-periodic and (b) the Caldirola-Kanai-type oscillators. For class (a) we study the following oscillators: (I) m(t)=m0fractt0m(t)=m_0frac{t}{t_0}m(t)=m0​fractt0​, (II) m(t)=m0m(t)=m_0m(t)=m0​ and (III) m(t)=m0ajustfractt02m(t)=m_0ajust{frac{t}{t_0}}^2m(t)=m0​ajustfractt0​2. In all three cases omega(t)=omega0fract0tomega(t)=omega_0frac{t_0}{t}omega(t)=omega0​fract0​t. For class (b) we study the Caldirola-Kanai oscillator (IV)where omega(t)=omega0omega(t)=omega_0omega(t)=omega0​ and m(t)=m0extexpajustgammatm(t)=m_0 ext{exp}ajust{gamma t}m(t)=m0​extexpajustgammat and the oscillator with omega(t)=omega0omega(t)=omega_0omega(t)=omega0​ and m(t)=m0ajust1+fractt0alpham(t)=m_0ajust{1+frac{t}{t_0}}^alpham(t)=m0​ajust1+fractt0​alpha, for alpha=2alpha=2alpha=2 (V) and alpha=4alpha=4alpha=4 (VI). To obtain the classical solution for each oscillator we solve the respective equation of motion and analyze the behavior of q(t)q(t)q(t), p(t)p(t)p(t) as well as the phase diagram q(t)q(t)q(t) vs p(t)p(t)p(t). To obtain the quantum solutions we use a unitary transformation and the Lewis and Riesenfeld quantum invariant method. The wave functions obtained are written in terms of a function ($ ho$) which is solution of the Milne-Pinney equation. Futhermore, for each system we solve the respective Milne-Pinney equation and discuss how the uncertainty product evolves with time.