[Paper Review] On a Class of Quantum Canonical Transformations and the Time-Dependent Harmonic Oscillator

Alí Mostafazadeh|ArXiv.org|Dec 12, 1996

Quantum chaos and dynamical systems3 citations

TL;DR

This paper introduces a class of quantum canonical transformations generated by the unitary operator $ e^{i\epsilon(t)\sqrt{f(x)}p\sqrt{f(x)}} $, showing that for $ f(x) = x $, it rescales position and momentum operators by $ e^{\epsilon(t)} $ and $ e^{-\epsilon(t)} $, respectively. This leads to the identification of a new class of exactly solvable time-dependent harmonic oscillators, including the Caldirola-Kanai oscillator with mass $ m = m_0 e^{\gamma t} $, and maps free particles with position-dependent mass to those with constant mass via a change in the metric of space.

ABSTRACT

Quantum canonical transformations corresponding to the action of the unitary operator $e^{iε(t)\sqrt{f(x)}p\sqrt{f(x)}}$ is studied. It is shown that for $f(x)=x$, the effect of this transformation is to rescale the position and momentum operators by $e^{ε(t)}$ and $e^{-ε(t)}$, 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 $m=m_0 e^{γt}$, belongs to this class. It is also shown that for arbitrary $f(x)$, 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.

Motivation & Objective

• To identify a new class of exactly solvable time-dependent harmonic oscillators using quantum canonical transformations.
• To explore the physical implications of the unitary operator $ e^{i\epsilon(t)\sqrt{f(x)}p\sqrt{f(x)}} $ in quantum mechanics.
• To demonstrate that these transformations map free particles with position-dependent mass to those with constant mass, effectively changing the space's metric.
• To generalize the solution structure of time-dependent quantum systems beyond standard approaches.
• To establish a connection between canonical transformations and solvable models in quantum mechanics, particularly in time-dependent settings.

Proposed method

• The paper analyzes the unitary operator $ e^{i\epsilon(t)\sqrt{f(x)}p\sqrt{f(x)}} $ as a generator of quantum canonical transformations.
• For $ f(x) = x $, the transformation is shown to rescale position and momentum operators as $ x \to e^{\epsilon(t)}x $, $ p \to e^{-\epsilon(t)}p $.
• The method involves computing the action of the unitary operator on the canonical operators using operator identities and Baker-Campbell-Hausdorff techniques.
• The transformation is applied to the Hamiltonian of a free particle, showing that it induces a position-dependent mass in the effective dynamics.
• The paper derives the effective metric of the space under the transformation, showing that it corresponds to a change in the inner product structure.
• It connects the resulting dynamics to known solvable models, particularly the Caldirola-Kanai oscillator, by matching the time-dependent mass parameter.

Experimental results

Research questions

• RQ1 What are the physical effects of the canonical transformation generated by $ e^{i\epsilon(t)\sqrt{f(x)}p\sqrt{f(x)}} $ on quantum operators?
• RQ2 Can this class of transformations lead to new exactly solvable models in time-dependent quantum mechanics?
• RQ3 How does the transformation relate to the dynamics of free particles with position-dependent mass?
• RQ4 Does the transformation induce a change in the metric of the underlying Hilbert space?
• RQ5 Is the Caldirola-Kanai oscillator a special case of this broader class of solvable time-dependent harmonic oscillators?

Key findings

• The transformation with $ f(x) = x $ rescales position and momentum operators by $ e^{\epsilon(t)} $ and $ e^{-\epsilon(t)} $, respectively, leading to a time-dependent scaling of the canonical variables.
• This transformation identifies a new class of exactly solvable time-dependent harmonic oscillators, including the Caldirola-Kanai oscillator with mass $ m = m_0 e^{\gamma t} $.
• For arbitrary $ f(x) $, the transformation maps the dynamics of a free particle with constant mass to that of a free particle with position-dependent mass.
• The transformation induces a change in the metric of the space, effectively redefining the inner product structure of the Hilbert space.
• The resulting dynamics for position-dependent mass systems are equivalent to those of free particles in a transformed geometry.
• The method provides a systematic way to generate solvable time-dependent Hamiltonians from known free-particle systems.