Point Transformations: Exact Solutions
of the Quantum Time-Dependent Mass
Nonstationary Oscillator
Kevin Zelaya and Véronique Hussin
Abstract In this note, we address the exact solutions of a time-dependent Hamiltonian composed by an oscillator-like interaction with both frequency and mass terms
that depend on time. The latter is achieved by constructing the appropriate point
transformation that deforms the Schrödinger equation of a stationary oscillator into
the one of the time-dependent model. In this form, the solutions of the latter can
be seen as deformations of the well-known stationary oscillator solutions, and thus
an orthogonal set of solutions can be determined straightforwardly. The latter is
possible since the inner product structure is preserved by the point transformation.
Moreover, any invariant operator of the stationary oscillator is transformed into an
invariant of the time-dependent model. This property leads to a straightforward way
to determine constants of motion without using any ansatz.
Keywords Nonstationary oscillator · Point transformations · Time-dependent
Hamiltonians · Exact solutions · Quantum invariants
1 Introduction
The dynamics of non-relativistic quantum systems is determined through the
solutions of the Schrödinger equation. In the latter, the information of the system
under consideration is coded in the Hamiltonian operator. Most of the physical
systems of interest are stationary and described by time-independent Hamiltonians. On the other hand, for open systems, time-dependent Hamiltonians are
K. Zelaya ()
Centre de Recherches Mathématiques, Université de Montréal, Montréal, QC, Canada
e-mail: zelayame@crm.umontreal.ca
V. Hussin
Centre de Recherches Mathématiques, Université de Montréal, Montréal, QC, Canada
Départment de Mathématiques et de Statistique, Université de Montréal, Montréal, QC, Canada
e-mail: hussin@dms.umontreal.ca
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_28
295
of the Quantum Time-Dependent Mass
Nonstationary Oscillator
Kevin Zelaya and Véronique Hussin
Abstract In this note, we address the exact solutions of a time-dependent Hamiltonian composed by an oscillator-like interaction with both frequency and mass terms
that depend on time. The latter is achieved by constructing the appropriate point
transformation that deforms the Schrödinger equation of a stationary oscillator into
the one of the time-dependent model. In this form, the solutions of the latter can
be seen as deformations of the well-known stationary oscillator solutions, and thus
an orthogonal set of solutions can be determined straightforwardly. The latter is
possible since the inner product structure is preserved by the point transformation.
Moreover, any invariant operator of the stationary oscillator is transformed into an
invariant of the time-dependent model. This property leads to a straightforward way
to determine constants of motion without using any ansatz.
Keywords Nonstationary oscillator · Point transformations · Time-dependent
Hamiltonians · Exact solutions · Quantum invariants
1 Introduction
The dynamics of non-relativistic quantum systems is determined through the
solutions of the Schrödinger equation. In the latter, the information of the system
under consideration is coded in the Hamiltonian operator. Most of the physical
systems of interest are stationary and described by time-independent Hamiltonians. On the other hand, for open systems, time-dependent Hamiltonians are
K. Zelaya ()
Centre de Recherches Mathématiques, Université de Montréal, Montréal, QC, Canada
e-mail: zelayame@crm.umontreal.ca
V. Hussin
Centre de Recherches Mathématiques, Université de Montréal, Montréal, QC, Canada
Départment de Mathématiques et de Statistique, Université de Montréal, Montréal, QC, Canada
e-mail: hussin@dms.umontreal.ca
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_28
295