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Time evolution operator for a class of miltidimensional driven oscillators
Volume 125, number 2,3
PHYSICSLETTERSA
2 November 1987
T I M E E V O L U T I O N OPERATOR FOR A CLASS OF M U L T I D I M E N S I O N A L DRIVEN OSCILLATORS Francisco M. FERNANDEZ and Eduardo A. CASTRO l Instituto de InvestigacionesFisicoquimicasTerricasy Aplicadas (INIFTA),Divisirn Qulmica Terrica, Sucursal 4, Casillade Correo16, 1900La Plata,Argentina Received 13 January 1987; revised manuscript received26 August 1987; accepted for publication 31 Augustus 1987 Communicated by D.D. Holm
The Schr6dinger equation for multidimensional time-dependent, second-order polynomial oscillators is exactly solved. The time evolution operator is obtained from the Heisenbergequations of motion. The problem is reduced to integrating Hamilton's equations of motion for the classicalanalog.
The driven oscillator has a large number of applications in theoretical physics and chemistry. Among them we mention the quantum theory of the laser and other questions of quantum optics [ 1 ], the motion of a charged particle in a time-dependent electromagnetic field [2], and the vibrationaltranslational and vibrational-vibrational energy transfer in molecular collisions (see refs. [ 3-5 ] and references therein). In addition to this, the exactly solvable quantum-mechanical models are frequently helpful in testing the approximate methods such as the time-dependent perturbation theory of the basis set expansion. It is well known that the Schrrdinger equation for some one-dimensional cases can be exactly solved (see refs. [4,6] and references therein). On the other hand, no attempt has been made, as fas as we know, at the multidimensional problem, except for some approximate treatments of two- and three-dimensional driven oscillators [3,5 ]. Since coupled timedependent harmonic oscillators are suitable for molecular collisions [ 3 ] the solution of the SchrSdinger equation in such cases is of utmost interest. In this paper we obtain the time evolution operator exactly for a time-dependent oscillator with linear and bilinear terms in coordinates and momenta. The procedure is based on the Heisenberg equations To whom correspondenceshould be addressed.
of motion that are shown to determine, except for a phase factor, the form of the time evolution operator which is written as a product of unitary operators. We consider the following quite general hamiltonian operator
H= ½PVA( t)p+ ½qVB( t )q+ ½[pV C( t)q + qV CX ( t)p](1) + D-r ( t)p+ FV ( t)q ,
(l)
where the elements of the matrices A, B (AX=A, BX=B), and C, and column vectors D and F are assumed to be real continuous functions of the time. The components of the column vectors p and q are the quantum-mechanical momenta and coordinate operators, respectively, that obey the usual commutation relation ( qj, Pk)=ih~jk 0", k= 1, 2, ..., n). The Schrrdinger equation for the time evolution operator U( t, to) reads d ih dt U(t, to) =HU(t, to),
(2)
where U(to, to) = T (the identity operator). The Heisenberg operators pj and #j,
~j=U+psU,
(tj=U+qjU, j = l , 2 ..... n
(3)
satisfy the quantum-mechanical equations of motion (matrix notation is used throughout)
0375-9601/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
77
Volume 125, number 2,3
p=-Cp-BO-F,
PHYSICSLETTERSA
~I=Ap+CXO+D,
4)
where the dot means time differentiation. A solution to eq. (4) can be written /5= Po(t) + P l ( t ) p + P 2 ( t ) q ,
O=Qo( t) + Q,( t)p+ Q2( t)q ,
5)
where the matrices Pj and Qi (J= 1, 2) and the column vectors P0 and Qo obey
t'o = - CPo - BQo - F ,
(6a)
Qo =APo + CXQo + D ,
(6b)
['j = - CP, - BQI ,
(7a)
Qj=APj+CWQj,
j=l,2.
(7b)
The initial conditions are Po(to)=Qo(to)=O. QI (t0) =P2(to) = 0, Q2(to) =PI (to) = L where I is the identity matrix. Only 2n2+n of the 4n 2 elements of the matrices Pj and Qj ( j = 1, 2) are independent since the Heisenberg coordinate and momenta obey
(p,,pk)=(q~,qk)=O,
(q,,pk)=ihO,~ .
PeP~-P,P[=O,
ilton's equations of motion of its classical counterpart. It is worth noticing that the Heisenberg operator 17= U+HU can be expressed as a sum of a classical hamiltonian HcL and a quantum-mechanical one H•, where
H('I . =½t~rAPo+ ~i Qox BQo + P~ CQc) + DT Po + Fr Qo ,
(9)
+DDp+F~q,
.4<>=P~AP~ +Q~BQ, +P~ CQ~ + Q~ c r p I , B o = P~AP2 + Q] BQ2 + P~ CQ~ + Q~ C v P2 . CO = PI AP: + Q~ BQ2 + P~ CQ2 + Q~ c~ P. , DQ = P~ APo + Q~ BQo + P~ CQo +Q~ Cr po + P ~ D + Q ] F , I;~ = P~APo + Q] BQo + P~ CQo +Q~CT Po + P [ D + QT F .
(14)
Except for a trivial phase factor that can be obtained, if necessary', from the Schrtidinger equation. the operator U(t. to) can be witten
(10)
U= U 1 U2 U3 ~74U5 ,
Q : P ~ - Q I P ~ =1.
(11)
where
78
(13)
and
Q2Q~ - Q , Q ~ = 0 ,
It can be easily verified that the solution to the equations of motion for the classical case can be obtained by substituting the classical momenta and coordinate p(t), q(t), p(to), and q(to) for the operators/~, 0, p, and q, respectively, in eq. (5). This fact is due to the formal resemblance between the Heisenberg and Hamilton's equations of motion for quadratic hamiltonians and was noted before by other authors [4,6-9]. Unitary and linear canonical transformations such as the one discussed above prove to be useful, for instance, in obtaining quantum-mechanical and classical invariants (see refs. [7-9] and references therein). It is shown below that the time evolution operator U can be exactly written in terms of the matrices P and Q. In other words, the dynamics of the quantum-mechanical model is determined by Ham-
(12)
Ho = ~PXAoP+ ½q, B o q + ½(pr Coq+ qX C~>p)
(8)
In fact, it follows immediately from eqs. (8) that
2 November 1987
~rl
(15)
=exp[ - (i/2h)p 1 a(t)p] .
(72 =exp[ - (i/2h)qVb(t)q] , U3 =exp{ - (i/2h) [p 1c(t)q+qXc(t)p] }, U4 =exp[ - ( i / h ) d S ( t ) p l , U5 =exp[ - (i/h)f~(t)q] .
(16)
The elements of the real matrices a, b ( b r = a , b T= b) and c, and column vectors d and f can be shown to be continuous functions of the time. Notice that there are as many functions in U as there are independent elements in the matrices and vectors P and Q. In order to obtain the Heisenberg operators we make use of the well-known formula eWVe " = V o + V I + ....
(17)
Volume 125, number 2,3
PHYSICS LETTERS A
where V0= V and ~ = ( W, ~_ 1 ). A straightforward calculation leads to U~pUz = p - b q ,
U~ qU~ =q+ap ,
Uf qU3 = e : q ,
U2 qU4=q+d,
(18)
from which we conclude that Po = - e - of_ be : d , Pt = - e - C , P2 = - b e : , Qo=(I-ab)e: d-ae-~f , Q t = a e -~,
Q2=(I-ab)e : .
(19)
After solving the classical equations of motion (6) and (7) we obtain the time evolution operator from a=QiP? L,
b=-PIP~,
f=Q~Po-Q~P2,
e-~=Pl,
d=Q~P~-QTPo.
X21 = - e - C 2 2 c 1 2
(20)
The unitary operator U3 can be written in many different ways. We may, for instance, try U3 = ( j _ ~ 1 ~ i ) ( jI~IkTjk),
(21a)
~k =exp[ - (i/h) CjkSjk],
(21b)
,
X22 = e - C : : ( 1
YII =eClI(l +cl2c21) , Yel=eC22c21,
Uf pU3 = e - ¢ p ,
U~pU5 = p - f ,
2 November 1987 "+Cl2C21 ) ,
Yl2=eCllcl2 ,
Y22~--e c22 ,
(23)
and the elements of c can be easily expressed in terms of those of X. We have shown how to solve the time-dependent Schr/Sdinger equation for the hamiltonian operator (1) exactly. The method is remarkably simple and holds for any number of degrees of freedom. It allows one to avoid usual approximations such as perturbation theory, basis set expansions [ 5 ] or neglect of two-quantum transitions [ 3 ] that proves to be the cause of a relatively large error for scattering of a heavy projectile by a light target [5]. After integrating the classical equations of motion and building the time evolution operator we may calculate, for example, the transition probabilities ~=
lim
t~oo t0 ~ --oo
I(k[ U(t, To)[J) [2 ,
(24)
where IJ) and [k) are the initial and final states, respectively. In this case it is most convenient to consider the time evolution operator in the interaction picture [ 4,5 ].
s : = ½(pjqk + q k P A •
References
In general the transformations read U;pUz=Xp,
Uf qU3=Yq,
(22)
where X= ( Y- l )T. When using this form of Us the matrices and vectors P and Q are obtained from eq. (19) where e-~ and e - : are replaced by X and Y, respectively. The resulting expressions are free from exponentials of matrices. For example, if n = 2 and U3= Tt t T22TI2T21, the matrices X and Y are found to be gll=e
-°l ,
gl2=--e-Cllc21
,
[1] V.S. Popov and A.M. Perelomov, Zh Eksp. Teor. Fiz. 57 (1969) 1684 [Soy. Phys. JETP 30 0970) 910]. [2] H.R. Lewis Jr. and W.B. Riesenfeld, J. Math. Phys. l0 (1969) 1458. [3] G.D. Billing, Chem. Phys. 33 (1978) 227. [4] B. Gazdy and D.A. Micha, J. Chem. Phys. 82 (1985) 4926. [5] J. Recamier, D.A. Micha and B. Gazdy, Chem. Phys. Lett. l l 9 (1985) 383. [6] P. Pechukas and J.C. Light, J. Chem. Phys. 44 (1966) 3897. [7] P.G.L. Leach, J. Math. Phys. 18 (1977) 1608. [8] P.G.U Leach, J. Math. Phys. 18 (1977) 1902. [9] P.G.L. Leach, J. Math. Phys. 19 (1978) 446.
79
PHYSICSLETTERSA
2 November 1987
T I M E E V O L U T I O N OPERATOR FOR A CLASS OF M U L T I D I M E N S I O N A L DRIVEN OSCILLATORS Francisco M. FERNANDEZ and Eduardo A. CASTRO l Instituto de InvestigacionesFisicoquimicasTerricasy Aplicadas (INIFTA),Divisirn Qulmica Terrica, Sucursal 4, Casillade Correo16, 1900La Plata,Argentina Received 13 January 1987; revised manuscript received26 August 1987; accepted for publication 31 Augustus 1987 Communicated by D.D. Holm
The Schr6dinger equation for multidimensional time-dependent, second-order polynomial oscillators is exactly solved. The time evolution operator is obtained from the Heisenbergequations of motion. The problem is reduced to integrating Hamilton's equations of motion for the classicalanalog.
The driven oscillator has a large number of applications in theoretical physics and chemistry. Among them we mention the quantum theory of the laser and other questions of quantum optics [ 1 ], the motion of a charged particle in a time-dependent electromagnetic field [2], and the vibrationaltranslational and vibrational-vibrational energy transfer in molecular collisions (see refs. [ 3-5 ] and references therein). In addition to this, the exactly solvable quantum-mechanical models are frequently helpful in testing the approximate methods such as the time-dependent perturbation theory of the basis set expansion. It is well known that the Schrrdinger equation for some one-dimensional cases can be exactly solved (see refs. [4,6] and references therein). On the other hand, no attempt has been made, as fas as we know, at the multidimensional problem, except for some approximate treatments of two- and three-dimensional driven oscillators [3,5 ]. Since coupled timedependent harmonic oscillators are suitable for molecular collisions [ 3 ] the solution of the SchrSdinger equation in such cases is of utmost interest. In this paper we obtain the time evolution operator exactly for a time-dependent oscillator with linear and bilinear terms in coordinates and momenta. The procedure is based on the Heisenberg equations To whom correspondenceshould be addressed.
of motion that are shown to determine, except for a phase factor, the form of the time evolution operator which is written as a product of unitary operators. We consider the following quite general hamiltonian operator
H= ½PVA( t)p+ ½qVB( t )q+ ½[pV C( t)q + qV CX ( t)p](1) + D-r ( t)p+ FV ( t)q ,
(l)
where the elements of the matrices A, B (AX=A, BX=B), and C, and column vectors D and F are assumed to be real continuous functions of the time. The components of the column vectors p and q are the quantum-mechanical momenta and coordinate operators, respectively, that obey the usual commutation relation ( qj, Pk)=ih~jk 0", k= 1, 2, ..., n). The Schrrdinger equation for the time evolution operator U( t, to) reads d ih dt U(t, to) =HU(t, to),
(2)
where U(to, to) = T (the identity operator). The Heisenberg operators pj and #j,
~j=U+psU,
(tj=U+qjU, j = l , 2 ..... n
(3)
satisfy the quantum-mechanical equations of motion (matrix notation is used throughout)
0375-9601/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
77
Volume 125, number 2,3
p=-Cp-BO-F,
PHYSICSLETTERSA
~I=Ap+CXO+D,
4)
where the dot means time differentiation. A solution to eq. (4) can be written /5= Po(t) + P l ( t ) p + P 2 ( t ) q ,
O=Qo( t) + Q,( t)p+ Q2( t)q ,
5)
where the matrices Pj and Qi (J= 1, 2) and the column vectors P0 and Qo obey
t'o = - CPo - BQo - F ,
(6a)
Qo =APo + CXQo + D ,
(6b)
['j = - CP, - BQI ,
(7a)
Qj=APj+CWQj,
j=l,2.
(7b)
The initial conditions are Po(to)=Qo(to)=O. QI (t0) =P2(to) = 0, Q2(to) =PI (to) = L where I is the identity matrix. Only 2n2+n of the 4n 2 elements of the matrices Pj and Qj ( j = 1, 2) are independent since the Heisenberg coordinate and momenta obey
(p,,pk)=(q~,qk)=O,
(q,,pk)=ihO,~ .
PeP~-P,P[=O,
ilton's equations of motion of its classical counterpart. It is worth noticing that the Heisenberg operator 17= U+HU can be expressed as a sum of a classical hamiltonian HcL and a quantum-mechanical one H•, where
H('I . =½t~rAPo+ ~i Qox BQo + P~ CQc) + DT Po + Fr Qo ,
(9)
+DDp+F~q,
.4<>=P~AP~ +Q~BQ, +P~ CQ~ + Q~ c r p I , B o = P~AP2 + Q] BQ2 + P~ CQ~ + Q~ C v P2 . CO = PI AP: + Q~ BQ2 + P~ CQ2 + Q~ c~ P. , DQ = P~ APo + Q~ BQo + P~ CQo +Q~ Cr po + P ~ D + Q ] F , I;~ = P~APo + Q] BQo + P~ CQo +Q~CT Po + P [ D + QT F .
(14)
Except for a trivial phase factor that can be obtained, if necessary', from the Schrtidinger equation. the operator U(t. to) can be witten
(10)
U= U 1 U2 U3 ~74U5 ,
Q : P ~ - Q I P ~ =1.
(11)
where
78
(13)
and
Q2Q~ - Q , Q ~ = 0 ,
It can be easily verified that the solution to the equations of motion for the classical case can be obtained by substituting the classical momenta and coordinate p(t), q(t), p(to), and q(to) for the operators/~, 0, p, and q, respectively, in eq. (5). This fact is due to the formal resemblance between the Heisenberg and Hamilton's equations of motion for quadratic hamiltonians and was noted before by other authors [4,6-9]. Unitary and linear canonical transformations such as the one discussed above prove to be useful, for instance, in obtaining quantum-mechanical and classical invariants (see refs. [7-9] and references therein). It is shown below that the time evolution operator U can be exactly written in terms of the matrices P and Q. In other words, the dynamics of the quantum-mechanical model is determined by Ham-
(12)
Ho = ~PXAoP+ ½q, B o q + ½(pr Coq+ qX C~>p)
(8)
In fact, it follows immediately from eqs. (8) that
2 November 1987
~rl
(15)
=exp[ - (i/2h)p 1 a(t)p] .
(72 =exp[ - (i/2h)qVb(t)q] , U3 =exp{ - (i/2h) [p 1c(t)q+qXc(t)p] }, U4 =exp[ - ( i / h ) d S ( t ) p l , U5 =exp[ - (i/h)f~(t)q] .
(16)
The elements of the real matrices a, b ( b r = a , b T= b) and c, and column vectors d and f can be shown to be continuous functions of the time. Notice that there are as many functions in U as there are independent elements in the matrices and vectors P and Q. In order to obtain the Heisenberg operators we make use of the well-known formula eWVe " = V o + V I + ....
(17)
Volume 125, number 2,3
PHYSICS LETTERS A
where V0= V and ~ = ( W, ~_ 1 ). A straightforward calculation leads to U~pUz = p - b q ,
U~ qU~ =q+ap ,
Uf qU3 = e : q ,
U2 qU4=q+d,
(18)
from which we conclude that Po = - e - of_ be : d , Pt = - e - C , P2 = - b e : , Qo=(I-ab)e: d-ae-~f , Q t = a e -~,
Q2=(I-ab)e : .
(19)
After solving the classical equations of motion (6) and (7) we obtain the time evolution operator from a=QiP? L,
b=-PIP~,
f=Q~Po-Q~P2,
e-~=Pl,
d=Q~P~-QTPo.
X21 = - e - C 2 2 c 1 2
(20)
The unitary operator U3 can be written in many different ways. We may, for instance, try U3 = ( j _ ~ 1 ~ i ) ( jI~IkTjk),
(21a)
~k =exp[ - (i/h) CjkSjk],
(21b)
,
X22 = e - C : : ( 1
YII =eClI(l +cl2c21) , Yel=eC22c21,
Uf pU3 = e - ¢ p ,
U~pU5 = p - f ,
2 November 1987 "+Cl2C21 ) ,
Yl2=eCllcl2 ,
Y22~--e c22 ,
(23)
and the elements of c can be easily expressed in terms of those of X. We have shown how to solve the time-dependent Schr/Sdinger equation for the hamiltonian operator (1) exactly. The method is remarkably simple and holds for any number of degrees of freedom. It allows one to avoid usual approximations such as perturbation theory, basis set expansions [ 5 ] or neglect of two-quantum transitions [ 3 ] that proves to be the cause of a relatively large error for scattering of a heavy projectile by a light target [5]. After integrating the classical equations of motion and building the time evolution operator we may calculate, for example, the transition probabilities ~=
lim
t~oo t0 ~ --oo
I(k[ U(t, To)[J) [2 ,
(24)
where IJ) and [k) are the initial and final states, respectively. In this case it is most convenient to consider the time evolution operator in the interaction picture [ 4,5 ].
s : = ½(pjqk + q k P A •
References
In general the transformations read U;pUz=Xp,
Uf qU3=Yq,
(22)
where X= ( Y- l )T. When using this form of Us the matrices and vectors P and Q are obtained from eq. (19) where e-~ and e - : are replaced by X and Y, respectively. The resulting expressions are free from exponentials of matrices. For example, if n = 2 and U3= Tt t T22TI2T21, the matrices X and Y are found to be gll=e
-°l ,
gl2=--e-Cllc21
,
[1] V.S. Popov and A.M. Perelomov, Zh Eksp. Teor. Fiz. 57 (1969) 1684 [Soy. Phys. JETP 30 0970) 910]. [2] H.R. Lewis Jr. and W.B. Riesenfeld, J. Math. Phys. l0 (1969) 1458. [3] G.D. Billing, Chem. Phys. 33 (1978) 227. [4] B. Gazdy and D.A. Micha, J. Chem. Phys. 82 (1985) 4926. [5] J. Recamier, D.A. Micha and B. Gazdy, Chem. Phys. Lett. l l 9 (1985) 383. [6] P. Pechukas and J.C. Light, J. Chem. Phys. 44 (1966) 3897. [7] P.G.L. Leach, J. Math. Phys. 18 (1977) 1608. [8] P.G.U Leach, J. Math. Phys. 18 (1977) 1902. [9] P.G.L. Leach, J. Math. Phys. 19 (1978) 446.
79