300
K. Zelaya and V. Hussin
¨
σ +
Ω
2
−
¨
μ
μ
σ =
1
σ 3 , ¨
γ +
Ω
2
−
¨
μ
μ
γ =
F
μ
,
μ
σ
η = ξ −
i
2
ln
σ
μ
, (19)
where the real-valued function ξ(t) is given by
ξ(t) =
γ W γ
2σ
−
1
2
t
dt
F (t )γ (t )
μ(t )
.
(20)
From (19) it follows that σ (t) satisfies the Ermakov equation [13], whose solutions
are well-known in the literature [13–15]. In general, for a set of nonnegative
parameters {a, b, c} we have [13]
σ (t) =
aq
2
1 (t) + bq 1 (t)q 2 (t) + cq
2
2 (t)
1/2
, b
2
− 4ac = −
4
W 2
0
,
(21)
where q 1 and q 2 are two linearly independent real solutions of the linear equation
¨
q 1,2 +
Ω
2
−
¨
μ
μ
q 1,2 = 0 ,
(22)
and the Wronskian W (q 1 , q 2 ) = W 0 is a constant. The constraint in the constants
a, b, c ensures that σ > 0 at any time [14, 15]. Thus, the transformed coordinate
y(x, t) and time parameter τ (t) are free of singularities at any time. Notice that (21)
corresponds to the classical equation of motion of the parametric oscillator [4, 5]. On
the other hand, γ (t) is a solution to the classical parametric oscillator subjected to a
driving force F (t). In general, γ can be expressed as the sum of the homogeneous
solution γ h = γ 1 q 1 +γ 2 q 2 and the particular solution γ p (t), where the real constants
γ 1,2 are fixed according to the initial conditions and the function γ p (t) is determined
once the driving force F (t) has been specified. Moreover, the function τ introduced
in (15) can be rewritten in terms of q 1 and q 2 as well, leading to [15]
τ (t) =
t dt
σ 2 (t )
= arctan
W 0
2
b + 2c
q 2
q 1
.
(23)
From (10) and with the functions σ , γ and τ already identified, the solutions
to the Schrödinger equation (7) are simply given in terms of the solutions of the
stationary oscillator Ψ (y, τ ) as
ψ(x, t) = exp
−i
μ
σ
W μ
2
x
2
+ W γ x
− iξ
μ
σ
Ψ (y(x, t), τ (t)) .
(24)
That is, the solutions of the nonstationary oscillator with time-dependent mass ˆ
H (t)
can be seen as a mere deformation of the solutions of the stationary oscillator,
provided by the appropriate point transformation. From the latter, it is natural to
ask whether the structure of the inner product defined in terms of the new solutions
K. Zelaya and V. Hussin
¨
σ +
Ω
2
−
¨
μ
μ
σ =
1
σ 3 , ¨
γ +
Ω
2
−
¨
μ
μ
γ =
F
μ
,
μ
σ
η = ξ −
i
2
ln
σ
μ
, (19)
where the real-valued function ξ(t) is given by
ξ(t) =
γ W γ
2σ
−
1
2
t
dt
F (t )γ (t )
μ(t )
.
(20)
From (19) it follows that σ (t) satisfies the Ermakov equation [13], whose solutions
are well-known in the literature [13–15]. In general, for a set of nonnegative
parameters {a, b, c} we have [13]
σ (t) =
aq
2
1 (t) + bq 1 (t)q 2 (t) + cq
2
2 (t)
1/2
, b
2
− 4ac = −
4
W 2
0
,
(21)
where q 1 and q 2 are two linearly independent real solutions of the linear equation
¨
q 1,2 +
Ω
2
−
¨
μ
μ
q 1,2 = 0 ,
(22)
and the Wronskian W (q 1 , q 2 ) = W 0 is a constant. The constraint in the constants
a, b, c ensures that σ > 0 at any time [14, 15]. Thus, the transformed coordinate
y(x, t) and time parameter τ (t) are free of singularities at any time. Notice that (21)
corresponds to the classical equation of motion of the parametric oscillator [4, 5]. On
the other hand, γ (t) is a solution to the classical parametric oscillator subjected to a
driving force F (t). In general, γ can be expressed as the sum of the homogeneous
solution γ h = γ 1 q 1 +γ 2 q 2 and the particular solution γ p (t), where the real constants
γ 1,2 are fixed according to the initial conditions and the function γ p (t) is determined
once the driving force F (t) has been specified. Moreover, the function τ introduced
in (15) can be rewritten in terms of q 1 and q 2 as well, leading to [15]
τ (t) =
t dt
σ 2 (t )
= arctan
W 0
2
b + 2c
q 2
q 1
.
(23)
From (10) and with the functions σ , γ and τ already identified, the solutions
to the Schrödinger equation (7) are simply given in terms of the solutions of the
stationary oscillator Ψ (y, τ ) as
ψ(x, t) = exp
−i
μ
σ
W μ
2
x
2
+ W γ x
− iξ
μ
σ
Ψ (y(x, t), τ (t)) .
(24)
That is, the solutions of the nonstationary oscillator with time-dependent mass ˆ
H (t)
can be seen as a mere deformation of the solutions of the stationary oscillator,
provided by the appropriate point transformation. From the latter, it is natural to
ask whether the structure of the inner product defined in terms of the new solutions