P
RAMANA °c Indian Academy of Sciences Vol. 65, No. 2—journal of August 2005
physics pp. 165–176
Quantum states with continuous spectrum for a general time-dependent oscillator
JEONG-RYEOL CHOI
Department of New Material Science, Division of Natural Sciences, Sun Moon University, Asan 336-708, Korea
E-mail: choiardor@hanmail.net
MS received 14 November 2004; accepted 22 March 2005
Abstract. We investigated quantum states with continuous spectrum for a general time-dependent oscillator using invariant operator and unitary transformation methods together. The form of the transformed invariant operator by a unitary operator is the same as the Hamiltonian of the simple harmonic oscillator: ˆI0= ˆp2/2 +ω2qˆ2/2. The fact that ω2 of the transformed invariant operator is constant enabled us to investigate the system separately for three cases, where ω2 >0, ω2 < 0, andω2 = 0. The eigenstates of the system are discrete for ω2 > 0. On the other hand, forω2 ≤ 0, the eigenstates are continuous. The time-dependent oscillators whose spectra of the wave function are continuous are not oscillatory. The wave function forω2 <0 is expressed in terms of the parabolic cylinder function. We applied our theory to the driven harmonic oscillator with strongly pulsating mass.
Keywords. Quantum states with continuous spectrum; time-dependent oscillator; in- variant operator; unitary operator; propagator.
PACS Nos 03.65.Ca; 03.65.-w
1. Introduction
Harmonic oscillators which have time-variable mass and/or frequency may be good examples of the time-dependent quadratic Hamiltonian systems. Although various dynamical systems have been investigated using approximation and perturbation methods [1,2], we confine our concern to the research of exact quantum state of the general time-dependent harmonic oscillator. There are a large number of pa- pers concerning the investigation of quantum states with discrete spectrum for time-dependent harmonic oscillator [3–8] and they can be applied to path-integral formulation of real-time finite-temperature field theory [9–11], dissipative quantum tunnelling effect in macroscopic system [12–15], scalar particle creation in cosmol- ogy [16], electrical behavior of RLC-circuit [17], and quantum motion of an ion in a Paul trap [3,18,19]. Even though the whole system is closed, it’s subsystem may implicitly depend on time via interaction with the remnant of the system. The
study of the properties for time-independent Hamiltonian systems whose eigen- states are continuous, such as energy dissipation, quantum and classical correspon- dence, decoherence and geometric/nongeometric effects are performed and found in [20–23]. The investigation of the eigenstates with continuous spectrum for the damped driven harmonic oscillator can be found in [24,25]. In this paper, we ex- tended this idea to the general time-dependent oscillator using invariant operator and unitary transformation methods together. The quantum state whose spectrum is continuous has plentiful applications in physical branches such as FRW universe model [26,27], surface diffusion [28,29], and the analysis of electron beam in a mag- netic field [30].
After the introduction of dynamical invariant operator by Lewis [31], the sys- tematic investigation of quantum-mechanical time-dependent quadratic oscillator has been facilitated. The key idea of the invariant operator method to obtain the solution of the Schr¨odinger equation for the time-dependent system is that the wave function corresponding to the Hamiltonian is different from the eigenstate of dynamical invariant operator by only some time-dependent phase factors [32].
2. Preliminary concepts
The Hamiltonian describing quantum general time-dependent quadratic oscillator is given by [7]
H(ˆˆ q,p, t) =ˆ A(t)ˆp2+B(t)(ˆqpˆ+ ˆpˆq) +C(t)ˆp+D(t)ˆq2+E(t)ˆq+F(t), (1) where A(t)–F(t) are time-dependent coefficients which are real and differentiable with respect to t. Note that A(t)6= 0. By applying Hamilton’s equation into eq.
(1), we can derive the classical equation of motion of the system as [33]
¨ˆ q−A˙
Aq˙ˆ+ Ã
2 ˙AB
A −2 ˙B−4B2+ 4AD
! ˆ q+AC˙
A −C˙ −2BC+ 2AE= 0.
(2) In fact, the Schr¨odinger equation for eq. (1) is too difficult to solve straightforward, since the separation of the coordinate and time is impossible.
The introduction of the dynamical invariant operator ˆI(t) may save us much time and labor in order to solve the quantum-mechanical solutions of the time-dependent Hamiltonian system because the wave function corresponding to the Hamiltonian is the same as the eigenstate of dynamical invariant operator except for only some time-dependent phase factors [32]. From d ˆI(t)/dt= 0, we obtain [7]
I(t) =ˆ α(t)[ˆp−pp(t)]2+β(t){[ˆq−qp(t)][ˆp−pp(t)]
+[ˆp−pp(t)][ˆq−qp(t)]}+γ(t)[ˆq−qp(t)]2. (3) In the above equation,qp(t) is the particular solution of classical equation (eq. (2)) of motion inq-space, andpp(t) is the corresponding particular solution inp-space.
Time-variable functionsα(t),β(t), andγ(t) are
α(t) =c1ρ21(t) +c2ρ1(t)ρ2(t) +c3ρ22(t), (4) β(t) = 1
4A{4[c1ρ21(t) +c2ρ1(t)ρ2(t) +c3ρ22(t)]B
−[2c1ρ1(t) ˙ρ1(t) +c2ρ˙1(t)ρ2(t)
+c2ρ˙2(t)ρ1(t) + 2c3ρ2(t) ˙ρ2(t)]}, (5) γ(t) = 1
2A2 (
1
2[c1ρ˙21(t) +c2ρ˙1(t) ˙ρ2(t) +c3ρ˙22(t)]
−B[2c1ρ1(t) ˙ρ1(t) +c2ρ˙1(t)ρ2(t) +c2ρ˙2(t)ρ1(t) + 2c3ρ2(t) ˙ρ2(t)]
+2B2[c1ρ21(t) +c2ρ1(t)ρ2(t) +c3ρ22(t)]
)
, (6)
where c1–c3 are arbitrary constants andρ1,2(t) are two independent homogeneous solutions of the following differential equation.
¨
ρ1,2(t)−A˙
Aρ˙1,2(t) + Ã2 ˙AB
A −2 ˙B−4B2+ 4AD
!
ρ1,2(t) = 0. (7) ForB(t) = 0, eqs (4)–(6) correspond to that of ref. [6].
We can convert the invariant operator into a simple form using unitary transfor- mation method. To do this, let us introduce the following unitary operator:
Uˆ = ˆU3Uˆ2Uˆ1, (8)
where
Uˆ1= exp µi
~qppˆ
¶ exp
µ
−i
~ppqˆ
¶
, (9)
Uˆ2= exp µ
i β 2α~qˆ2
¶
, (10)
Uˆ3= exp
· i
4~(ˆqˆp+ ˆpˆq) ln(2α)
¸
. (11)
Using eq. (8), we can transform invariant operator eq. (3):
Iˆ0= ˆUIˆUˆ†. (12)
Then, after performing some algebra, ˆI0 reduces to the following simple form:
Iˆ0=1 2pˆ2+1
2ω2qˆ2, (13)
where
ω2= 4(αγ−β2)
= 1
4A2(ρ1ρ˙2−ρ˙1ρ2)2(4c1c3−c22) = Constant. (14) Note that eq. (14) is constant. This can be checked by direct differentiation of ω2 with respect to time. The fact thatω2is constant enables us to investigate the system separately for three cases, whereω2>0,ω2<0, andω2= 0. The eigenstate of the system is discrete for ω2 > 0 since the transformed invariant operator eq.
(13) corresponds to that of the oscillating system while that of the other two cases are continuous. The wave functions forω2>0 are [7]
hq|Ψn(t)i= exp[i²n(t)]hq|Φn(t)i, (15)
wherehq|Φn(t)iare the eigenstates of the invariant operator eq. (3), that are given by
hq|Φn(t)i=
³ ω 2α~π
´1/4 1
√2nn!Hn
µr ω
2α~(q−qp)
¶ exp
µi
~ppq
¶
×exp
·
− 1 2α~
³ω 2 +iβ
´
(q−qp)2
¸
, (16)
and²n(t) are phases of the form
²n(t) =−ωT(t) µ
n+1 2
¶
−1
~ Z t
0
·
Lp(qp(t0),q˙p(t0), t0)−C2(t0)
4A(t0)+F(t0)
¸
dt0, (17)
with
T(t) = Z t
0
A(t0)
α dt0, (18)
Lp(qp(t),q˙p(t), t) = 1
4A(t)q˙2p(t)−B(t)
A(t)qp(t) ˙qp(t)
− µ
D(t)−B2(t) A(t)
¶
q2p(t). (19)
The propagator which describes quantum mechanics in terms of the generalized path integral provides us the information about the evolution of the state for a particle. If an object placed originally q1 at t1 evolves toq2 at t2, we can express the propagator of the system whose spectrum of the eigenstate is discrete as [34]
K(q2, t2;q1, t1) = X∞
n=0
hq2|Ψn(t2)ihΨn(t1)|q1i. (20) When we perform the summation after inserting eq. (15) into the above equation with the aid of Mehler’s formula [35],
X∞
n=0
(z/2)n
n! Hn(x)Hn(y) = 1
(1−z2)1/2exp
·2xyz−(x2+y2)z2 1−z2
¸ , (21) eq. (20) becomes
K(q2, t2;q1, t1) = Ã
ω 4ip
α∗(t1)α(t2)π~sin[ω(T(t2)−T∗(t1))]
!1/2
×exp (
ω
4i~sin[ω(T(t2)−T∗(t1))]
× (
p 2
α∗(t1)α(t2)(q1−qp(t1))(q2−qp(t2))
−
·(q1−qp(t1))2
α∗(t1) +(q2−qp(t2))2 α(t2)
¸
cos[ω(T(t2)−T∗(t1))]
))
×exp
½
− i 2~
·β(t2)
α(t2)(q2−qp(t2))2−β∗(t1)
α∗(t1)(q1−qp(t1))2
¸¾
×exp
½
−i
~ Z t2
t1
·
Lp(qp(t0),q˙p(t0), t0)−C2(t0)
4A(t0)+F(t0)
¸ dt0
¾
×exp
·i
~(pp(t2)q2−pp(t1)q1)
¸
. (22)
This is the probability amplitude for the evolution of an oscillator fromq1 toq2 at time intervalt2−t1:
hq2|Ψn(t2)i= Z ∞
−∞
K(q2, t2;q1, t1)hq1|Ψn(t1)idq1. (23)
3. Quantum states with continuous spectrum
In this section, we investigate the quantum states with continuous spectrum for the time-dependent oscillator. In the case of ω2≤0, eq. (13) says that the system is not oscillatory and its eigenvalue becomes continuous.
We shall begin the investigation of the quantum state withω2 <0. For conve- nience, let us introduce a notation ˜ω2 that is given by
˜
ω2=−ω2>0. (24)
Then, eq. (13) can be rewritten as Iˆ0=1
2pˆ2−1
2ω˜2qˆ2. (25)
This is the same as the Hamiltonian of the harmonic parabola potential system.
The eigenvalue equation for eq. (25) can be represented as
Iˆ0|φλ(t)i=λ|φλ(t)i. (26) By substitution of eq. (25) into eq. (26), we can obtain the following differential equation in q-space:
∂2hq|φλ(t)i
∂Q2 + µ
Λ +1 4Q2
¶
hq|φλ(t)i= 0, (27) where
Q= r2˜ω
~ q, (28)
Λ = λ
~˜ω. (29)
By solving eq. (27), we can derive the eigenstate of transformed invariant operator as
hq|φλ(t)i= µ ω˜
8π2~
¶1/4
×
·
eiπ/4D−iΛ−1/2 µ1 +i
√2 Q
¶
+e−iπ/4D−iΛ−1/2
µ
−1 +i
√2 Q
¶¸
, (30)
whereDν(x) is the parabolic cylinder function which is defined as [35,36]
Dν(x) = 2(ν−1)/2exp(−x2/4)xΨ(1/2¯ −ν/2,3/2;x2/2), (31) with ¯Ψ given by [36]
Ψ(a, c;¯ y) = 1 Γ(a)
Z ∞
0
e−ytta−1(1 +t)c−a−1dt. (32) The eigenstate of the untransformed invariant operator hq|Φλ(t)i can be derived from [37]
hq|Φλ(t)i= ˆU†hq|φλ(t)i. (33)
Using eq. (8), the above equation can be evaluated as hq|Φλ(t)i= 1
(2α)1/4 µ ω˜
8π2~
¶1/4 exp
·
− iβ
2α~(q−qp(t))2
¸
eipp(t)q/~
×
·
eiπ/4D−iΛ−1/2 µ1 +i
2√ αQ0
¶
+ e−iπ/4D−iΛ−1/2 µ
−1 +i 2√
αQ0
¶¸
, (34)
where
Q0= r2˜ω
~ (q−qp(t)). (35)
It is known that the wave function of the system is different from the eigenstate of the invariant operator by some time-dependent phase factor:
hq|Ψλ(t)i= exp[i²(t)]hq|Φλ(t)i. (36)
By substituting eq. (36) into Schr¨odinger equation, we can obtain that
~²(t) =˙ hΦλ(t)|
µ i~∂
∂t−Hˆ
¶
|Φλ(t)i. (37)
The phase can be derived from eq. (37) with eqs (1) and (34) as
²(t) =−λ
~t− 1
~ Z t
0
·
Lp(qp(t0),q˙p(t0), t0)−C2(t0)
4A(t0)+F(t0)
¸
dt0. (38) Then, substitution of eqs (34) and (38) into eq. (36), we can express the full wave function of the system as
hq|Ψλ(t)i= 1 (2α)1/4
µ ω˜ 8π2~
¶1/4 exp
·
− iβ
2α~(q−qp(t))2
¸
eipp(t)q/~
×
·
eiπ/4D−iΛ−1/2 µ1 +i
2√ α Q0
¶
+ e−iπ/4D−iΛ−1/2 µ
−1 +i 2√
α Q0
¶¸
×exp
½
−iλ
~t− i
~ Z t
0
·
Lp(qp(t0),q˙p(t0), t0)
−C2(t0)
4A(t0)+F(t0)
¸ dt0
¾
, (39)
where the eigenvalueλis continuous.
The propagator of the system whose spectrum of the eigenstate is continuous can be derived from [24]
K(q2, t2;q1, t1) = 1
~˜ω Z ∞
−∞
hq2|Ψλ(t2)ihΨλ(t1)|q1idλ. (40) After substituting eq. (39) into the above equation, the integration can be performed using the following integral formula [35],
Z c+i∞
c−i∞
[Dν(x)D−ν−1(iy) +Dν(−x)D−ν−1(−iy)]t−ν−1dν sin(−νπ)
= 2√
√ 2πi
1 +t2exp
· 1−t2
4(1 +t2)(x2+y2) +i xyt 1 +t2
¸
−1< c <0, |argt|< π/2. (41)
Then, eq. (40) becomes K(q2, t2;q1, t1) =
à ω˜
4ip
α∗(t1)α(t2)π~sinh[˜ω(t2−t1)]
!1/2
×exp
·i
~(pp(t2)q2−pp(t1)q1)
¸ exp
( ω˜
4i~sinh[˜ω(t2−t1)]
×
( 2
pα∗(t1)α(t2)(q1−qp(t1))(q2−qp(t2))
−
·(q1−qp(t1))2
α∗(t1) +(q2−qp(t2))2 α(t2)
¸
cosh[˜ω(t2−t1)]
))
×exp
½
− i 2~
·β(t2)
α(t2)(q2−qp(t2))2−β∗(t1)
α∗(t1)(q1−qp(t1))2
¸¾
×exp
½
−i
~ Z t2
t1
·
Lp(qp(t0),q˙p(t0), t0)−C2(t0) 4A(t0)+F(t0)
¸ dt0
¾ . (42) Now, let us investigate the system forω2= 0, which is another case whose spec- trum of quantum state is continuous. In this case eq. (13) can be more simplified to
Iˆ0=1
2pˆ2. (43)
Note that eq. (43) is the same as the Hamiltonian of the free particle. We can represent the eigenvalue equation of eq. (43) as
Iˆ0|φλ0(t)i=λ0|φλ0(t)i. (44)
By substitution of eq. (43) into eq. (44) and after some algebra, we can obtain the eigenstate of transformed invariant operator inq-space:
hq|φλ0(t)i= µ λ0
8~2π2
¶1/4
×
·
eiπ/4exp µi
~
√2λ0 q
¶
+ e−iπ/4exp µ
−i
~
√2λ0 q
¶¸
, (45) where λ0 is some constant with dimension of energy. The eigenstate of untrans- formed invariant operator can be calculated using eqs (8) and (45) to be
hq|Φλ0(t)i= ˆU†hq|φλ0(t)i
= 1
(2α)1/4 µ λ0
8~2π2
¶1/4 exp
µ
− iβ
2α~(q−qp)2
¶ exp
µi
~ppq
¶
×
"
eiπ/4exp Ãi
~ rλ0
α(q−qp)
!
+ e−iπ/4exp Ã
−i
~ rλ0
α(q−qp)
!#
. (46)
The relation between eigenstate of the invariant operator and wave function is also the same as the previous one, eq. (36) and we can easily derive the corresponding phase as
²(t) =−λ0
~t−1
~ Z t
0
·
Lp(qp(t0),q˙p(t0), t0)−C2(t0)
4A(t0)+F(t0)
¸
dt0. (47) Then, substitution of eqs (46) and (47) into eq. (36), we can express the full wave function as
hq|Ψλ0(t)i= 1 (2α)1/4
µ λ0
8~2π2
¶1/4 exp
µ
− iβ
2α~(q−qp)2
¶ exp
µi
~ppq
¶
×
"
eiπ/4exp Ãi
~ rλ0
α (q−qp)
!
+ e−iπ/4exp Ã
−i
~ rλ0
α (q−qp)
!#
×exp
½
−iλ0
~t− i
~ Z t
0
·
Lp(qp(t0),q˙p(t0), t0)
−C2(t0)
4A(t0)+F(t0)
¸ dt0
¾
. (48)
In eq. (48), the eigenvalueλ0 is also continuous asλin Eq. (39). The propagator of the system can be derived from [24]
K(q2, t2;q1, t1) = 1
√λ0
Z ∞
−∞
hq2|Ψλ0(t2)ihΨλ0(t1)|q1id√
λ0. (49)
By substituting eq. (48) into the above equation we can obtain that K(q2, t2;q1, t1) =
à 1
4ip
α∗(t1)α(t2)π~(t2−t1)
!1/2
×exp
·i
~(pp(t2)q2−pp(t1)q1)
¸
×exp
i 4~(t2−t1)
Ãq1−qp(t1)
pα∗(t1) −q2−qp(t2) pα(t2)
!2
×exp
½
− i 2~
·β(t2)
α(t2)(q2−qp(t2))2
−β∗(t1)
α∗(t1)(q1−qp(t1))2
¸¾
×exp
½
−i
~ Z t2
t1
[Lp(qp(t0),q˙p(t0), t0)
−C2(t0)
4A(t0)+F(t0)
¸ dt0
¾
. (50)
4. Application to the driven oscillator with strongly pulsating mass The discussions in the previous sections may be applied to various kinds of time- dependent oscillators. As an example, we apply them to the driven oscillator with strongly pulsating mass [38,39]. In this case the Hamiltonian is given by
H(t) =ˆ pˆ2 2M(t)+1
2M(t)ω02qˆ2−M(t)f(t)ˆq, (51) where
M(t) =M0cos2ωMt, (52)
f(t) =f0cos(ωft+θ), (53)
where M0 is the mass at t = 0,ωM and ωf are arbitrary constant frequencies, f0
is the amplitude of the driving force, andθis the initial phase of the driving force.
If we apply Hamilton’s equation of motion into eq. (51), we can obtain that
¨ˆ q+M˙
Mq˙ˆ+ω20qˆ=f(t), (54)
¨ˆ p−M˙
Mp˙ˆ+ω02pˆ=Mf˙(t). (55)
By applying eqs (52) and (53), the above two equations become
¨ˆ
q−2ωMtan(ωMt)˙ˆq+ω20qˆ=f0cos(ωft+θ), (56)
¨ˆ
p+ 2ωMtan(ωMt) ˙ˆp+ω02pˆ=−f0M0ωfcos2(ωMt) sin(ωft+θ), (57) and, eq. (7) becomes
¨
ρ1,2(t) +M˙
Mρ˙1,2(t) +ω02ρ1,2(t) = 0. (58) The twoc-number solutions of the above equation are
ρ1(t) =ρ1(0) sec(ωMt)eiΩt, (59)
ρ2(t) =ρ2(0) sec(ωMt)e−iΩt, (60)
where
Ω = q
ω20+ω2M. (61)
If we consider eqs (54) and (55), the two particular solutionsqpandppfollow that
¨ qp+M˙
Mq˙p+ω02qp=f(t), (62)
¨ pp−M˙
Mp˙p+ω02pp=Mf˙(t). (63)
The particular solutions satisfying the above two equations are given by [38]
qp=1
2f0sec(ωMt)
"
cos[(ωf+ωM)t+θ]−cos(Ωt) cosθ Ω2−(ωf+ωM)2
+cos[(ωf−ωM)t+θ]−cos(Ωt) cosθ Ω2−(ωf−ωM)2
#
, (64)
pp= [M0M(t)]1/2
·d
dt(qpcosωMt) +ωMqpsinωMt
¸
. (65)
In terms of eqs (59), (60), (64), and (65) the quantum solution of the system can be completely described.
5. Summary
We used both invariant operator and unitary transformation methods in order to investigate the quantum eigenstates with continuous spectrum for the general time- dependent oscillator. If we choose unitary operator as eq. (8) with eqs (9)–(11), the invariant operator can be transformed to eq. (13) which is the same as the Hamiltonian of the simple harmonic oscillator. The fact that ω2 in eq. (14) is constant enabled us to investigate the system separately for three cases, where ω2>0,ω2<0, and ω2= 0. The eigenvalue of the system forω2≤0 is continuous while that forω2>0 is discrete. For the latter case, the system is oscillatory and quantized. On the other hand, for the former case, the system is not oscillatory.
We obtained exact solutions of the Schr¨odinger equation for the system having continuous eigenvalue. The wave function forω2 <0 is expressed in terms of the parabolic cylinder function while that forω2>0 is expressed in terms of the well- known Hermite polynomial. We derived propagator of the system whose quantum states are both discrete and continuous by using the wave function, which is the probability amplitude for the evolution of an oscillator.
Our results can be applied to various time-dependent quadratic Hamiltonian systems beyond damped driven harmonic oscillator. As an example, we applied our theory to the driven oscillator with strongly pulsating mass. For damped driven harmonic oscillator, our results reduces to that of ref. [25].
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