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Propagator with friction in quantum mechanics
Volume 74A, number 1,2
PHYSICS LETTERS
29 October 1979
PROPAGATOR WITH FRICTION IN QUANTUM MECHANICS A.D. JANNUSSIS, G.N. BRODIMAS and A. STRECLAS Department of Theoretical Physics, University of Patras, Greece Received 8 August 1979
In this paper we calculate the propagator for quantum-mechanical systems with friction. For the case where the friction is a linear function of the velocity with a friction constant ~‘ we can calculate exact propagators of quadratic form.
Recently, Moreira [1] and Khandekar and Lawande [2,3] calculated the exact propagator for a quadratic lagrangian with the help of the Van Vieck—Pauli formula, which, in one dimension, reads 2xp[(j/h)S(q”,t”;q’,t’)] K(q”~t”;q’,t’)=(~ ~q3~~Sq,)lIe
,
(1)
where S is the classical action, which corresponds to the given lagrangian with friction, from the space—time point (q’, t’) to (q t”). Instead of eq. (1) the following definition of the propagator can be used [3] “,
K(q”, t”;q’, t’)
~
‘I’~(q’,t’)’I’~(q”,t”),
(2)
where the wave function ‘Tin (q, t) is defined in ref. [3]. For the above propagator the Hamilton operator depends on the time, and in addition it is assumed that there exists a hermitian invariant operator 1(t) which does not involve time differentiation. This is the case when the hamiltonian is of quadratic form with parameters which are a function of time and with friction terms. For the case ‘I’ 0(q,
t)
=
exp [_(i/fl)Ent] U~(q),
(3)
where lJ~(q) is the eigenfunction of the hamilton operator, eq. (2) yields the well-known definition of the propagator [4] K(q”,t”,q’,t’)
~U~(q”)U~(q’)exp[—(i/h)E0(t”—t’)]
.
(4)
In what follows the general eq. (2) will be used for the calculation of the propagator of all quadratic hamiltonians with friction and external fields. The case of the classical harmonic oscillator will be considered first. 1. Damped harmonic oscillator. 2I2m)e_7t+~mw2q2e7t. H=(p The solution of the SchrOdinger equation is given in refs. [5—7]: 6
(5)
PHYSICS LETTERS
Volume 74A, number 1,2
29 October 1979
!)~2exp1[~~y—i~(n+ ~)]t (m/2h)(~+ ~i7)e~tq2}9(~[(m~2/1l)h/2e7t/2q] (6) ‘I’~(q,t) = (m~Z/h)~4(2’~n where f12 = ~,2 ~~2 >~,~ is the friction constant and ~1C~(x) are Hermite polynomials. With the aid of eq. (2) and after a certainamount of algebra the propagator of the damped harmonic oscillator can be expressed as —
,
—
K(q
+t’ )/4 e [(2irifl/m~Z)sin ~2(t” r’)]
;q,t)
,t
~
—
[cot
~(t”
The above eq. (7) for y
=
—
t’) (e7t”q “2
+
1/2
e7t’q ~2)
—
]
2q~~qle’”t”+t’~2 ~
(e7~”q“2
—
ei~t’qp2))
(7)
.
0 gives the propagator of the harmonic oscillator [4].
2. Forced and damped oscillator. H= (p2/2m)e_7t
+
(~mw2q2 qF(t))e~t.
(8)
—
The corresponding solution of the Schrodinger equation is given by Kerner [1]: 1/4
‘I’~(q,t)
=
(2’1n !)~‘2exp(_~_ [fi~t~it
(-~~)
+E~t+ ~mye7tq2
—
q(p 0(t)
X exp{—(m~2/2h)[q ~
/2
where ~(t)
=
e’Ytt2}
~m7qo(t)e7t)]}
,
(9)
E~= 1~2(n+ ~), p
t+ ~i’y, L0
[q —q0(t)]
+
+ ~ m740(t)q0(t)e~
0(t) = m~o(t)e7t,
~2
=
—
~ 72 >0 ,
(10)
and L0 represents the classical lagrangian for the damped but unforced motion as a function in time of the damped and forced position and velocity. The position coordinate q0(t) satisfies the classical equation: 2q mq0+yq0+mw 0F(t). (11) With the aid of the generalized propagator, given by eq. (2), and after a certain amount of algebra we finally find ,,
K(q ,
t
,,
r
, , mcz ,q , t ~ = [2~hsin fl(t”
2 e~t’cos ~(t” ÷ (q’
—
X exp
,l/2
—
—
t’)jI t’)
—
1112~2
exp 2flsin fl(t” 2(q”
q0(t’)) (t
+
u
i
t’) [(q —q0(t
))2 e ‘yt”
~
—q
0(t”)) (q’
(—i--i [~ “)dt”
—
—
q0(t’)) e7(t”+t’)/2]} 2e7t)
—
~
(t’)
dt’]
~
(q~l2e~Yt” — q‘
~m 7q0(t1’)e7t”) —q’~0(t’)+ ~m7qo(tl)e7t~)1}
(12)
.
For the case where there is no external force, that is,F(t)= 0, eq. (ll)gives q0 with formula (7).
=
0 and formula (12) coincides
3. Applied electric field and friction. The case of electric field and friction is the most interesting. The solution of the Schrodinger equation for an electric field has been given by Buch and Denman [8]. Accordingto Husimi [9] the wave function ‘I’~(q,t) with friction and for an applied electric field c~(t) has the following form: 7
Volume 74A, number 1, 2
~
t) = exp [ik[q
PHYSICS LETTERS
— ~f
e_~Tf~(v)e~1 dv]
+~
29 October 1979
dr]q}
k2 e~t + i ~[f~(r)e7T
(13)
Xexp[_~fe_~T[fr(V)e7I’dV]dT}.
According to eq. (2) the propagator becomes: K(q”, t”;q’, t’)
x
[f~
2exp(~
(r)e~Tdrq” _f~(r)e7T dTq’]
f expt~-(e_7t~— e_7t)k2 +ik[q”
—q’
_J~2J” e77[J~(v)e~ dV]
dl)
—~(J”e_1TJ~(v)e~1~’dv)]}dk;
(14)
the above integral is of Frensel type [10] and eq. (14) can finally be written as: K(q”, t”;q’, t’)= [2~ih(e~
exp
e_1t’)]
_f
x exp{~[f’~(r)e7Tdrqh1
~(r)e7~drq~]
L
2h(e7t”
~
The case where the function d(t) is constant, d(t) = ~
_
fe~
e_7~’)[q” _q9_~(fe_YTf~(~e~vdv)]2)
[J~(V)e7”dV]dr).
(15)
has been studied by Moreira [11
4. The forced harmonic oscillator. H
—(h2/2m)a2/8q2
+
~m w(t)2q2 —f(t)q
(16)
.
According to Husimi [9] the Schrodinger equation with a hamiltonian given by eq. (16) has a solution of gaussian type, namely ‘T’k (q, t) = exp {(i/2h) [a(t)q2 + 2~(t)q + ~‘(t)] — (ik2/2mh)A (t) + (ik/l1) [qB(t) where the function a (t) satisfies the following Ricatti equation: m~da/dt
=
—a2/m2
exp [_~fa(r)dr]ff(r)exp[~-f
~(t)
=
~!~~~fa(r)dr _-Lfdr exp
[_±j
a(r’) dr’]dr,
(19)
[—--~-f a(r’)dr] {ff(r’)exP [-~-f a(r”)dr”] drt}~
t a(r’)dr’]dr,
B(t)= exp [_~f
By means of eq. (2) the propagator can be expressed as:
(20)
(21,22)
a(r)dr],
r(t)=~fdrexP[_~fa(r’)dr]{ff(r’)exP[~fa(r”)dr”]dr’}
8
(17) (18)
=
exp
[‘(t)] } ,
— w2(t) ,
~(t)
A(t)f
+
.
(23)
Volume 74A, number 1,2
PHYSICS LETTERS
29 October 1979
K(q”, t”;q’, t’)= exp~(i/2h)[(a(t”)q”2 + 2~(t”)q”+~(t”)) (a(t’)q’2 —
f
X~
If we set k given by
exp{—(ik2/2mh)[A(t”) —A(t’)] -~
+ (ik/h)
+ 2~(t’)q’+~(t’))]}
[q”B(t”) q’B(t’) + F(t”) —
—
F(t’)] } dk
(24)
.
hk in the integral of the r.hs. of eq. (24), this is again of Frensel type and the propagator is now
K(q”, t”, q’, t’)exp f(i/2h)[(a(t”)q”2 ÷2~(t”)q+ ~(t’)) ex {2irih[A(t”)—A(t’)]}1’2
—
(a(t’)q’2
+ 2~(t’)q’+
im [q”B(t”)—q’B(t’)+1’(t”)—r’(t’)]2 2h~ A(t”)A(t’)
~
(25)
In the same way, we can study the case of the damped and forced harmonic oscillator. The hamiltonian has now the form: H_(h2/2m)e_7ta2/aq2 +e7t [~mw2(t)q2 —f(t)q]
(26)
.
With the help of the following contact transformation [7] Wk(Qt)
q _~e_7ti’2Q, ~Il(q,t)exp[~7t+(im7/4fl)e7tq2]
(27,28)
the wave function satisfies the Schrodinger equation =
—(fl2/2m) a2”4’/aQ’2
+
[~m cl(t)2Q’2
—
F(t)Q’] XI’,
(29)
where =
w(t)2 _~72,
F(t)= e7t/2f(t).
(30)
So we can easily find the propagator: ~(q”, t”, q’, t’) = exp [h(t” + t’) ÷ (im7/4h) (e7t q”2
—
e7t q’~2)jK(Q”, t”; Q’, t’) ,
(31)
where K(Q”, t”; Q’, t’) = K (elt ‘2q”, t”, e7t’t2qt, t’) is the propagator (25). The evaluation of the propagator for the hamiltonian (26) has been carried out by Khandekar and Lawande [2, 3] by means of the known method of path integrals [11] , which has been used for the same purpose by other authors [12,13]. 5. Damped harmonic oscillator in a uniform magnetic field. Another interesting case is that of the damped harmonic oscillator in a uniform magnetic field, which we will study now. Accoding to Jannussis et al. [14] , the Hamilton operator is given by the relation H(t) = (l/2m) [p + (e/c)H(t)X q] 2 e_7t
+
~e7tmw2(q~
+ q~+
q~),
(32)
where H(t) = He7t and His the constant intensity of the magnetic field. Using the contact transformation q eltI’2Q, the solution of the time-dependent SchrOdinger equation has the following form: ‘P(Q, t) exp[~’yt (im’y/411)Q2] F 1(Q1 , Q2, t)F3(Q3, t), =
—
where 2
F1(Q1, Q2, t) = (m~l)~~’
exp ~
[(mfZ/h)r2] e~e~t~’n1,
(33)
9
Volume 74A, number 1,2
PHYSICS LETTERS
29 October 1979
m~23 mm !)_~2 exp(_~-~_ Q~)~m[(m~l
1/2
F3(Q3, t) = (~~) (2
3/1’1)Q3] e_(1~tEm
and L~(x)are the generalized Laguerre polynomials. The above solutions have been expressed in polar coordinates, that is Q1 = rcos ~, Q2 = rsin s~,Q3 = Q3. The energy eigenvalues are as follows: En =h~l(2n+l+l)±flwLl,
~2 ~
_~72,
eH/2mc,
WL
(34)
2—~y2. Em=h~l3(m+~), ~2~=w The calculation of the propagator proceeds now easily through relation (2). After a certain amount of algebra we obtain m~.2 K(q”, r”;q’, t’)
+~ +
~3(t
2n~t”
— 2cos
WL(t
3 [2i~hsin ~3(t”
— t’) — t’)
1/2 i’)]
—
3~y(t”+t’)/4
2i~hsin~(t”
—
t’) exp (_~7(e7t~12 — e7t”q?P2)
2+ e7t’q~2)— 2et+t)I2q~~~2q2]
[cos ~3(t”
— 1) (e7t”q~’
{cos &~(t” t’) [e~t’(qj’2 + q~2)+ e7t’ (q~2+ q~)]
_tP)e7(t”+t~2(qjIqj + q~q~)—
2sin
WL(t”
—
(35)
t’) e7(t’ ft’)/2 (qjq~’— q~qj’)}~
It is clear from the result of the present work that the generalized propagator defined by eq. (2) considerably simplifies the evaluation of the propagator for quadratic forms with friction; moreover, it can be used for time-dependent hamiltonians of a more general form. References [1] [2] [3] [4]
[5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
10
IC. Moreira, Lett. Nuovo Cimento 23 (1978) 294. D.C. Khandekar and S.V. Lawande, Phys. Lett. 67A (1975) 175. D.C. Khandekar and S.V. Lawande, J. Math. Phys. 16 (1975) 384. B. Kursunoglou, Modern quantum theory (Freedman, London, 1962). V.V. Dodonov and VI. Man’co, Nuovo Cimento 44B (1978) 265. F. Bopp, Z. Angew. Phys. 14 (1962) 699. E.H. Kerner, Can. J. Phys. 36 (1958) 371. L.H. Buch and H.H. Denman, Am. J. Phys. 42 (1974) 304. K. Husimi, Prog. Theor. Phys. 9 (1953) 381. W. Magnus, F. Oberbettinger and R. Soni, Formulae and theorems for the special functions of Mathematical Physics (Springer, 1966) § 9.2. R.P. Feynman and A. Hibbs, Quantum mechanics and path integrals (New York, 1965). G.J. Papadopoulos, J. Phys. A7 (1974) 209; Al (1968) 593. A.V. Jones and G. Papadopoulos, J. Phys. A4 (1971) 86. A. Jannussis, G. Brodimas and A. Streclas, Lett. Nuovo Cimento 25 (1979) no. 9.
PHYSICS LETTERS
29 October 1979
PROPAGATOR WITH FRICTION IN QUANTUM MECHANICS A.D. JANNUSSIS, G.N. BRODIMAS and A. STRECLAS Department of Theoretical Physics, University of Patras, Greece Received 8 August 1979
In this paper we calculate the propagator for quantum-mechanical systems with friction. For the case where the friction is a linear function of the velocity with a friction constant ~‘ we can calculate exact propagators of quadratic form.
Recently, Moreira [1] and Khandekar and Lawande [2,3] calculated the exact propagator for a quadratic lagrangian with the help of the Van Vieck—Pauli formula, which, in one dimension, reads 2xp[(j/h)S(q”,t”;q’,t’)] K(q”~t”;q’,t’)=(~ ~q3~~Sq,)lIe
,
(1)
where S is the classical action, which corresponds to the given lagrangian with friction, from the space—time point (q’, t’) to (q t”). Instead of eq. (1) the following definition of the propagator can be used [3] “,
K(q”, t”;q’, t’)
~
‘I’~(q’,t’)’I’~(q”,t”),
(2)
where the wave function ‘Tin (q, t) is defined in ref. [3]. For the above propagator the Hamilton operator depends on the time, and in addition it is assumed that there exists a hermitian invariant operator 1(t) which does not involve time differentiation. This is the case when the hamiltonian is of quadratic form with parameters which are a function of time and with friction terms. For the case ‘I’ 0(q,
t)
=
exp [_(i/fl)Ent] U~(q),
(3)
where lJ~(q) is the eigenfunction of the hamilton operator, eq. (2) yields the well-known definition of the propagator [4] K(q”,t”,q’,t’)
~U~(q”)U~(q’)exp[—(i/h)E0(t”—t’)]
.
(4)
In what follows the general eq. (2) will be used for the calculation of the propagator of all quadratic hamiltonians with friction and external fields. The case of the classical harmonic oscillator will be considered first. 1. Damped harmonic oscillator. 2I2m)e_7t+~mw2q2e7t. H=(p The solution of the SchrOdinger equation is given in refs. [5—7]: 6
(5)
PHYSICS LETTERS
Volume 74A, number 1,2
29 October 1979
!)~2exp1[~~y—i~(n+ ~)]t (m/2h)(~+ ~i7)e~tq2}9(~[(m~2/1l)h/2e7t/2q] (6) ‘I’~(q,t) = (m~Z/h)~4(2’~n where f12 = ~,2 ~~2 >~,~ is the friction constant and ~1C~(x) are Hermite polynomials. With the aid of eq. (2) and after a certainamount of algebra the propagator of the damped harmonic oscillator can be expressed as —
,
—
K(q
+t’ )/4 e [(2irifl/m~Z)sin ~2(t” r’)]
;q,t)
,t
~
—
[cot
~(t”
The above eq. (7) for y
=
—
t’) (e7t”q “2
+
1/2
e7t’q ~2)
—
]
2q~~qle’”t”+t’~2 ~
(e7~”q“2
—
ei~t’qp2))
(7)
.
0 gives the propagator of the harmonic oscillator [4].
2. Forced and damped oscillator. H= (p2/2m)e_7t
+
(~mw2q2 qF(t))e~t.
(8)
—
The corresponding solution of the Schrodinger equation is given by Kerner [1]: 1/4
‘I’~(q,t)
=
(2’1n !)~‘2exp(_~_ [fi~t~it
(-~~)
+E~t+ ~mye7tq2
—
q(p 0(t)
X exp{—(m~2/2h)[q ~
/2
where ~(t)
=
e’Ytt2}
~m7qo(t)e7t)]}
,
(9)
E~= 1~2(n+ ~), p
t+ ~i’y, L0
[q —q0(t)]
+
+ ~ m740(t)q0(t)e~
0(t) = m~o(t)e7t,
~2
=
—
~ 72 >0 ,
(10)
and L0 represents the classical lagrangian for the damped but unforced motion as a function in time of the damped and forced position and velocity. The position coordinate q0(t) satisfies the classical equation: 2q mq0+yq0+mw 0F(t). (11) With the aid of the generalized propagator, given by eq. (2), and after a certain amount of algebra we finally find ,,
K(q ,
t
,,
r
, , mcz ,q , t ~ = [2~hsin fl(t”
2 e~t’cos ~(t” ÷ (q’
—
X exp
,l/2
—
—
t’)jI t’)
—
1112~2
exp 2flsin fl(t” 2(q”
q0(t’)) (t
+
u
i
t’) [(q —q0(t
))2 e ‘yt”
~
—q
0(t”)) (q’
(—i--i [~ “)dt”
—
—
q0(t’)) e7(t”+t’)/2]} 2e7t)
—
~
(t’)
dt’]
~
(q~l2e~Yt” — q‘
~m 7q0(t1’)e7t”) —q’~0(t’)+ ~m7qo(tl)e7t~)1}
(12)
.
For the case where there is no external force, that is,F(t)= 0, eq. (ll)gives q0 with formula (7).
=
0 and formula (12) coincides
3. Applied electric field and friction. The case of electric field and friction is the most interesting. The solution of the Schrodinger equation for an electric field has been given by Buch and Denman [8]. Accordingto Husimi [9] the wave function ‘I’~(q,t) with friction and for an applied electric field c~(t) has the following form: 7
Volume 74A, number 1, 2
~
t) = exp [ik[q
PHYSICS LETTERS
— ~f
e_~Tf~(v)e~1 dv]
+~
29 October 1979
dr]q}
k2 e~t + i ~[f~(r)e7T
(13)
Xexp[_~fe_~T[fr(V)e7I’dV]dT}.
According to eq. (2) the propagator becomes: K(q”, t”;q’, t’)
x
[f~
2exp(~
(r)e~Tdrq” _f~(r)e7T dTq’]
f expt~-(e_7t~— e_7t)k2 +ik[q”
—q’
_J~2J” e77[J~(v)e~ dV]
dl)
—~(J”e_1TJ~(v)e~1~’dv)]}dk;
(14)
the above integral is of Frensel type [10] and eq. (14) can finally be written as: K(q”, t”;q’, t’)= [2~ih(e~
exp
e_1t’)]
_f
x exp{~[f’~(r)e7Tdrqh1
~(r)e7~drq~]
L
2h(e7t”
~
The case where the function d(t) is constant, d(t) = ~
_
fe~
e_7~’)[q” _q9_~(fe_YTf~(~e~vdv)]2)
[J~(V)e7”dV]dr).
(15)
has been studied by Moreira [11
4. The forced harmonic oscillator. H
—(h2/2m)a2/8q2
+
~m w(t)2q2 —f(t)q
(16)
.
According to Husimi [9] the Schrodinger equation with a hamiltonian given by eq. (16) has a solution of gaussian type, namely ‘T’k (q, t) = exp {(i/2h) [a(t)q2 + 2~(t)q + ~‘(t)] — (ik2/2mh)A (t) + (ik/l1) [qB(t) where the function a (t) satisfies the following Ricatti equation: m~da/dt
=
—a2/m2
exp [_~fa(r)dr]ff(r)exp[~-f
~(t)
=
~!~~~fa(r)dr _-Lfdr exp
[_±j
a(r’) dr’]dr,
(19)
[—--~-f a(r’)dr] {ff(r’)exP [-~-f a(r”)dr”] drt}~
t a(r’)dr’]dr,
B(t)= exp [_~f
By means of eq. (2) the propagator can be expressed as:
(20)
(21,22)
a(r)dr],
r(t)=~fdrexP[_~fa(r’)dr]{ff(r’)exP[~fa(r”)dr”]dr’}
8
(17) (18)
=
exp
[‘(t)] } ,
— w2(t) ,
~(t)
A(t)f
+
.
(23)
Volume 74A, number 1,2
PHYSICS LETTERS
29 October 1979
K(q”, t”;q’, t’)= exp~(i/2h)[(a(t”)q”2 + 2~(t”)q”+~(t”)) (a(t’)q’2 —
f
X~
If we set k given by
exp{—(ik2/2mh)[A(t”) —A(t’)] -~
+ (ik/h)
+ 2~(t’)q’+~(t’))]}
[q”B(t”) q’B(t’) + F(t”) —
—
F(t’)] } dk
(24)
.
hk in the integral of the r.hs. of eq. (24), this is again of Frensel type and the propagator is now
K(q”, t”, q’, t’)exp f(i/2h)[(a(t”)q”2 ÷2~(t”)q+ ~(t’)) ex {2irih[A(t”)—A(t’)]}1’2
—
(a(t’)q’2
+ 2~(t’)q’+
im [q”B(t”)—q’B(t’)+1’(t”)—r’(t’)]2 2h~ A(t”)A(t’)
~
(25)
In the same way, we can study the case of the damped and forced harmonic oscillator. The hamiltonian has now the form: H_(h2/2m)e_7ta2/aq2 +e7t [~mw2(t)q2 —f(t)q]
(26)
.
With the help of the following contact transformation [7] Wk(Qt)
q _~e_7ti’2Q, ~Il(q,t)exp[~7t+(im7/4fl)e7tq2]
(27,28)
the wave function satisfies the Schrodinger equation =
—(fl2/2m) a2”4’/aQ’2
+
[~m cl(t)2Q’2
—
F(t)Q’] XI’,
(29)
where =
w(t)2 _~72,
F(t)= e7t/2f(t).
(30)
So we can easily find the propagator: ~(q”, t”, q’, t’) = exp [h(t” + t’) ÷ (im7/4h) (e7t q”2
—
e7t q’~2)jK(Q”, t”; Q’, t’) ,
(31)
where K(Q”, t”; Q’, t’) = K (elt ‘2q”, t”, e7t’t2qt, t’) is the propagator (25). The evaluation of the propagator for the hamiltonian (26) has been carried out by Khandekar and Lawande [2, 3] by means of the known method of path integrals [11] , which has been used for the same purpose by other authors [12,13]. 5. Damped harmonic oscillator in a uniform magnetic field. Another interesting case is that of the damped harmonic oscillator in a uniform magnetic field, which we will study now. Accoding to Jannussis et al. [14] , the Hamilton operator is given by the relation H(t) = (l/2m) [p + (e/c)H(t)X q] 2 e_7t
+
~e7tmw2(q~
+ q~+
q~),
(32)
where H(t) = He7t and His the constant intensity of the magnetic field. Using the contact transformation q eltI’2Q, the solution of the time-dependent SchrOdinger equation has the following form: ‘P(Q, t) exp[~’yt (im’y/411)Q2] F 1(Q1 , Q2, t)F3(Q3, t), =
—
where 2
F1(Q1, Q2, t) = (m~l)~~’
exp ~
[(mfZ/h)r2] e~e~t~’n1,
(33)
9
Volume 74A, number 1,2
PHYSICS LETTERS
29 October 1979
m~23 mm !)_~2 exp(_~-~_ Q~)~m[(m~l
1/2
F3(Q3, t) = (~~) (2
3/1’1)Q3] e_(1~tEm
and L~(x)are the generalized Laguerre polynomials. The above solutions have been expressed in polar coordinates, that is Q1 = rcos ~, Q2 = rsin s~,Q3 = Q3. The energy eigenvalues are as follows: En =h~l(2n+l+l)±flwLl,
~2 ~
_~72,
eH/2mc,
WL
(34)
2—~y2. Em=h~l3(m+~), ~2~=w The calculation of the propagator proceeds now easily through relation (2). After a certain amount of algebra we obtain m~.2 K(q”, r”;q’, t’)
+~ +
~3(t
2n~t”
— 2cos
WL(t
3 [2i~hsin ~3(t”
— t’) — t’)
1/2 i’)]
—
3~y(t”+t’)/4
2i~hsin~(t”
—
t’) exp (_~7(e7t~12 — e7t”q?P2)
2+ e7t’q~2)— 2et+t)I2q~~~2q2]
[cos ~3(t”
— 1) (e7t”q~’
{cos &~(t” t’) [e~t’(qj’2 + q~2)+ e7t’ (q~2+ q~)]
_tP)e7(t”+t~2(qjIqj + q~q~)—
2sin
WL(t”
—
(35)
t’) e7(t’ ft’)/2 (qjq~’— q~qj’)}~
It is clear from the result of the present work that the generalized propagator defined by eq. (2) considerably simplifies the evaluation of the propagator for quadratic forms with friction; moreover, it can be used for time-dependent hamiltonians of a more general form. References [1] [2] [3] [4]
[5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
10
IC. Moreira, Lett. Nuovo Cimento 23 (1978) 294. D.C. Khandekar and S.V. Lawande, Phys. Lett. 67A (1975) 175. D.C. Khandekar and S.V. Lawande, J. Math. Phys. 16 (1975) 384. B. Kursunoglou, Modern quantum theory (Freedman, London, 1962). V.V. Dodonov and VI. Man’co, Nuovo Cimento 44B (1978) 265. F. Bopp, Z. Angew. Phys. 14 (1962) 699. E.H. Kerner, Can. J. Phys. 36 (1958) 371. L.H. Buch and H.H. Denman, Am. J. Phys. 42 (1974) 304. K. Husimi, Prog. Theor. Phys. 9 (1953) 381. W. Magnus, F. Oberbettinger and R. Soni, Formulae and theorems for the special functions of Mathematical Physics (Springer, 1966) § 9.2. R.P. Feynman and A. Hibbs, Quantum mechanics and path integrals (New York, 1965). G.J. Papadopoulos, J. Phys. A7 (1974) 209; Al (1968) 593. A.V. Jones and G. Papadopoulos, J. Phys. A4 (1971) 86. A. Jannussis, G. Brodimas and A. Streclas, Lett. Nuovo Cimento 25 (1979) no. 9.