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K. Zelaya and V. Hussin
required to provide an accurate description. Physical applications are found in
electromagnetic traps of particles [1–3], in which external time-dependent electric
and magnetic fields allow the confinement of particles [4]. In such a case, we
describe the respective Hamiltonian through a parametric oscillator potential, also
known as nonstationary oscillator, which consists of an oscillator-like interaction
with a frequency that varies in time. Exact solutions were studied in detail by
Lewis and Riesenfeld for the classical and quantum cases [5, 6]. Given that the
Hamiltonian has an explicit dependence on time, an eigenvalue equation associated
with Hamiltonian is no longer feasible and the existence of an orthonormal set of
solutions cannot be taken for granted, as it is customary for the stationary quantum
oscillator. Nevertheless, Lewis and Riesenfeld introduced an approach in which a
nonstationary eigenvalue equation can be still found once the appropriate constant of
motion is determined [6]. With the latter, solutions to the Schrödinger equation are
found by adding the appropriate time-dependent complex-phase to the nonstationary
eigenfunctions. The constant of motion, or invariant operator ˆ
I (t), is usually
imposed as an ansatz and determined from the condition [i∂/∂t − ˆ
H (t), ˆ
I (t)] = 0.
Such approach has been applied successfully to other time-dependent models as
well [7].
In this note, we address the solutions of the nonstationary oscillator with timedependent mass. To this end, we consider the method of point transformations,
which has been used in the context of quadratic time-dependent potentials [8] and
nonstationary Darboux transformations [9]. In particular, we use the construction
introduced in [10]. The latter allows deforming the well-known Schrödinger
equation of the stationary oscillator into the one of the time-dependent model.
This method leads to a straightforward way to obtain the solutions of the timedependent model as deformations of the stationary oscillator. Remarkably, the point
transformation preserves the first integrals, this means that the constants of motion
and the spectral properties for the time-dependent model are inherited from the
stationary oscillator, without requiring to impose any anstaz [6].
2 Nonstationary Oscillator with Time-Dependent Mass
Let us first consider the quantum harmonic oscillator, defined through the Hamiltonian
ˆ
H osc =
ˆ
p 2
y
2
+
ˆ
y 2
2
,
(1)
where ˆ
y and ˆ
p y stand for the canonical position and momentum operators, [ ˆ
y, ˆ
p y ] =
i. With the latter, the Schrödinger equation, represented in the spatial coordinate ‘y’,
reads as
K. Zelaya and V. Hussin
required to provide an accurate description. Physical applications are found in
electromagnetic traps of particles [1–3], in which external time-dependent electric
and magnetic fields allow the confinement of particles [4]. In such a case, we
describe the respective Hamiltonian through a parametric oscillator potential, also
known as nonstationary oscillator, which consists of an oscillator-like interaction
with a frequency that varies in time. Exact solutions were studied in detail by
Lewis and Riesenfeld for the classical and quantum cases [5, 6]. Given that the
Hamiltonian has an explicit dependence on time, an eigenvalue equation associated
with Hamiltonian is no longer feasible and the existence of an orthonormal set of
solutions cannot be taken for granted, as it is customary for the stationary quantum
oscillator. Nevertheless, Lewis and Riesenfeld introduced an approach in which a
nonstationary eigenvalue equation can be still found once the appropriate constant of
motion is determined [6]. With the latter, solutions to the Schrödinger equation are
found by adding the appropriate time-dependent complex-phase to the nonstationary
eigenfunctions. The constant of motion, or invariant operator ˆ
I (t), is usually
imposed as an ansatz and determined from the condition [i∂/∂t − ˆ
H (t), ˆ
I (t)] = 0.
Such approach has been applied successfully to other time-dependent models as
well [7].
In this note, we address the solutions of the nonstationary oscillator with timedependent mass. To this end, we consider the method of point transformations,
which has been used in the context of quadratic time-dependent potentials [8] and
nonstationary Darboux transformations [9]. In particular, we use the construction
introduced in [10]. The latter allows deforming the well-known Schrödinger
equation of the stationary oscillator into the one of the time-dependent model.
This method leads to a straightforward way to obtain the solutions of the timedependent model as deformations of the stationary oscillator. Remarkably, the point
transformation preserves the first integrals, this means that the constants of motion
and the spectral properties for the time-dependent model are inherited from the
stationary oscillator, without requiring to impose any anstaz [6].
2 Nonstationary Oscillator with Time-Dependent Mass
Let us first consider the quantum harmonic oscillator, defined through the Hamiltonian
ˆ
H osc =
ˆ
p 2
y
2
+
ˆ
y 2
2
,
(1)
where ˆ
y and ˆ
p y stand for the canonical position and momentum operators, [ ˆ
y, ˆ
p y ] =
i. With the latter, the Schrödinger equation, represented in the spatial coordinate ‘y’,
reads as