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Exact propagators for Lagrangians with higher derivatives in quantum mechanics
Physica A 170 (1991) 595-611 North-Holland
EXACT PROPAGATORS FOR LAGRANGIANS WITH HIGHER DERIVATIVES IN QUANTUM MECHANICS V.V. DODONOV,
A.B. KLIMOV and V.I. MAN’KO
P. N. Lebedev
Institute,
Physical
Leninsky
prospect
53, 117924 Moscow,
USSR
Received 2.5 April 1989 Revised manuscript received 29 May 1990
1. Introduction In recent years in a series of papers [l-5] we studied the so-called quantum quadratic systems described by means of Hamiltonians of the following structure:
q=(P1,Pz,...,PN,
is the 2N-dimensional vector whose components are canonically conjugated momenta and coordinates: [ 6,) i,] = -iS,, (we assume h = l), B(t) is a 2N X 2N matrix, and C(f) is a 2Ndimensional vector. It was shown that the propagator of the Schrodinger equation with Hamiltonian (1) can be calculated exactly for quite arbitrary time-dependent matrix B(t) and vector C(f): x1,x2,...,
G(x, x’, t) = [det(-2nih,)]-1’2 X
exp
+2x’. (6, - A, . Ai1 +~)+5,.A~.A,l.fi,-218,(~).~~(T)d7)]. Here A,(t), j = 1,2,3,4
are N
x
N matrices forming the 2N
(2)
X
2N matrix
satisfying
the equation
E,. is the satisfies
unit
N x N matrix.
The
2N-dimensional
vector
d(t)
= (u, , CT:)
the equation
d=A-X-C.
A(())=().
(4)
Thus the only problem is to find the explicit solutions to eq. (3). The cases when this is possible were investigated in refs. [l-5]. In the present paper we consider a new interesting example of a quadratic system admitting the explicit propagator, namely. a system described by means of a Lagrangian containing higher derivatives. Recently Kleinert [6] who calculated
a special case of this problem was studied by the propagator corresponding to the Lagrangian
with constant coefficients in the framework of the path integral approach. Our aim is to generalize Klcinert’s result to the cases when (i) coefficients are arbitrary functions of time, (ii) the derivatives of the Nth order are present. We USC the general scheme of investigating quadratic systems [I-S] based on introducing instead of the Lagrangian the equivalent Hamiltonian in the form (i). By the way. we will demonstrate that this scheme leads to the same result as the path integral method for Lagrangians with higher derivatives as well ac in the case of usual Lagrangians with the first derivatives only. As to the physical applications of Lagrangians like (5), some references can be found in paper 161. Besides, different aspects of the quantum mechanical description ot systems possessing Lagrangians with higher derivatives were discussed in refs. [7-Y]. The most wide application of such Lagrangians can be found in the statistical physics of polymers [IO-131. This item will be considered in section 7.
2. Quadratic Hamiltonians
Lagrangians
Let us consider
with higher derivatives
a Lagrangian
and their equivalent
V.V. Dodonov
et al.
/ Lagrangians
with higher derivatives
597
where xCn)is the nth time derivative of the coordinate x = x(O), {kN} is a set of time-dependent coefficients, and f(t) is an external force. To obtain an equivalent Hamiltonian of the system with Lagrangian (6) we will use the method of Ostragradski [7, 14-161. According to this method N variables x, = X@), y1= 0, 1, . . . ) N - 1, can be considered as independent degrees of freedom. Introducing Lagrangian mutipliers p, into (6) we obtain an equivalent Lagrangian
+N-2 c=o p,(X,
- x,+1) +
k,i;-,
-fx,,.
(-UN+ -y-
n
1, . . , N - 2, are nothing else but the
One can see that variables p,, n = canonical momenta:
a
P,=ax,,'
LN
n = 0, 1, . . . , N - 2 .
The (N - 1)th generalized velocity as follows: p,+,
= 2
(7)
=
momentum
(-QN+‘kNiN_,
is related to the (N - 1)th generalized
.
Performing the Legendre transformation of (7) and excluding the variable X,_ I we obtain the following expression for the Hamiltonian: N-l
H=
c P,,-f,, -
n
LN
=o
N-l =
-
Ix”(-l).+l$
+ p+
(-l)N+’
+ z:
pnxn+,
+fx,,
.
(8)
We assume that the coefficient k, never turns to zero. We see that Hamiltonian (8) is a quadratic form of 2N canonical variables. If we rewrite it in a standard form (i), then N x N-blocks of the matrix
are as follows:
b, =
b, =
...“. i
The components Cl = 0 ,
1 =b; 0i
0
of the 2N-dimensional c, = (f(r),
0,
vector
C are
, 0)
We perform quantization of Hamiltonian (8) 17, IS]. Thus we impose the commutation relations [ I’,, ’
by the
canonical
x,,,1= - i4,,H
method
(I())
and arrive at a quantum system with N degrees of freedom, but with an unusual (although quadratic) Hamiltonian. Before investigating it in the whole volume,
let us consider
3. Lagrangians
the special
case of N = 2.
with the second derivative
The equations for the blocks with eq. (3) as follows:
of A-matrix
x, =h;b,-X,.b,.
&=A,+-A,+,
& = A, - b, - A,. b,
x,=A,.b,-X,-b,.
can be represented
in accordance
(11)
with initial conditions A,(O) = A,(O) = E?, A,(O) = A,(O) = 0. Substituting the expressions for matrices b, (9) at N = 2 into eqs. (11) and solving them we obtain the following expressions:
A,
=
V.V. Dodonov et al. I Lagrangians
Functions ci(t), i = 1,2,3,4,
with higher derivatives
599
are solutions of the equation (12)
with the initial conditions go)
&,(O) = E*(O) = 1 )
= -E&O)
= - &
) 2
E;(O)
=
-
k,(O) k2(0)
g+(O)
3
=
k,(O) -j---
-
k2(0)
’
Other values E:‘(O) are equal to zero. Thus the force-independent Green function (2) assumes the form 1 G(y,,u,,t] y,,ul,O)= [det(-2nih3)]“2 X exp
- ____
i 2 det A, {Y:@4&1-
part of the
E,“4) - 2Y,Y,i4
+ Y;[/$~(E~F~- ij3~4) + k,(ii,t-, - ii;Q] - ~u,u~E~ + u;k,(Q,
- QE,) + u;(E~E~ - Eli.,)}
X exp - & ( + yIU2&4+ y,‘, detX,=
3
1
[u2y2k,(E,E, - E36,) +
U1y,(i.,E,
-
‘31 9
i3&,)
(13)
a3E4- FEES,
where the notation yr = xo(t = 0), y2 = x,(t), u, = x,(t = 0), u2 = xl(t) is introduced. In the case of time independent coefficients we make the substitutions k,lk, = co; + w;, k,lk, = &I’,. Then the expressions for the Ei-functions are as follows: 1 E, = -
1 e2 = (W2Q - w,s,> , k2S s
=
co’,
-
1 e4 = k,6 (5 -
o;,
where cr 2 = cos o1 ,t, s, 2 = sin or 2t.
(14) c2)
3
i 2k, det A; (C,,C, -- l)(Wf + w; + 2W,W,S,S,) k,6
with
This expression coincides with the result obtained by Kleincrt 161. Thus we have shown that our approach is equivalent to the path integral method for Lagrangians with higher derivatives as well as for usual Lagrangians, and that “strange” Hamiltonians like (8) correctly describe corresponding quantum systems. As an application we will obtain the Green function for the Lagrangian proposed by Caldirola [ 171 to describe the motion of a radiating (and thus damping) electron in a cluadratic potential l> = _ ie’ rr($$
+ (&‘)
(W, (1 = const.)
(16)
In refs. [l8, 191 the quantum system corresponding to this Lagrangian was investigated by the method of canonical transformations. but neither solutions nor propagators have
The equation
were
obtained.
of the scheme
described
above
we
for F, is
1 2 P, + cy i; + 7 F, + ck with the initial
In terms
conditions
w ff
c‘(
=o
( 17)
V.V. Dodonov et al. I Lagrangians
El(O)= E*(O)= 1 ) Other
values
ii;(O) = -i,(O)
601
with higher derivatives
= - 3
)
Eq(0) = - -$ . CY
(18)
e,‘“‘(0) are eq ual to zero.
The solution
of (17) with conditions
(18) is as follows:
where -1kS P, =
2a
are the roots of the characteristic
equation
2
p4+~p3+&p2+;=o.
At
Coefficients
CY
(Y
are determined
from eqs. (18),
+nP’-
A:=
mf1 P, -Pj’ Af=~m2,~,(Pm-Pj)-’ m, II, j =
All other Green
values
function
7
C ( m#n#j
PmPn
IYI (Pj-Pm)-‘~ mfl
,$=a-3
II
> ??I#,
(Pm-Pj)Y’?
1,2,3,4.
A:
are obtained
(13) thus assume
by cyclic the form
rearrangements
of indices.
The
V.V. Dociorwv
602
et ul. )I Lqrangians
with higher
deriwtiLy.\
detX,=&,T;:j, 1.A
In the case of the full Lagrangian
(5) with f(t) # 0 one has to calculate
besides
matrices structure solutions
h,(t) also vectors 6,(t) satisfying cq. (4). Due to the very simple of the vector C(t) in the canonical form (i) of Hamiltonian (8). the of this vector equation can be expressed immediately in terms of the
functions
.5,(t) as follows:
The full Green function expression ( 13). and
equals
G = G,,G,,
where
the function
G,, is given
i -ix,
j
dt’ ~,f’-
0
iv,
i
dt’ ~~1’
I
0
i xexp
i
- ~ 2 det h,
(x2&, ~ v?E~)
j 0
i dt’ ~,j’ + (v,E,
~ ylF1)
J (I
di’ ~,,.f
by
V.V. Dodonov et al. I Lagrangians
with higher derivatives
603
4. Adiabatic approximation Let us consider a special case of the Lagrangian L, = ;(-k2X2
(5),
+ k,i2 - kox2),
(22)
when the coefficients obey the conditions fC2= 1;, = 0 )
k0
k
= w2(t) )
2
(23) p,+,
We1, co2
2
which means that the frequency s-function has the form
changes adiabatically.
The equation
d2e $y+k”,-g + w2(t) &=O. Let us putt= d4e
(24)
TT, T%=l. Then - d2c
T-4 d74 + k, z
We represent
for the
+ W’(T) E = 0.
(25)
the solution in the form
E(T) = W(7) sin( T [ dT1 S(7’))
(26)
and search for the functions W and S in the form of developments
in powers of
T-‘: S(7) = So(T) + y
+ ...,
W(7) = We(T) + y
+. . .
Substituting (24) into (23) we obtain from the coefficient at sin(T I,’ d7’ S(T’)) of the order To the equation
whose solution is (27)
V. V. Lkxionov
60-l
From
the coefficient -2&k”,
ef ul.
at cos(T
- 2s;)
! hgrungiuns
with higher derivutitm
]’ dr’ S(T))) of the order
dW,, = W,, d&k”,
T we have
~ 2.~;) ,
(28) Wi = const where
X
‘-, w
“I
k”f # 4w ‘.
Expressions (13), (26)-(27) Green function in the zeroth
enable order
us to solve the problem adiabatic
of calculating
5. The Green function of a charged particle moving in a magnetic in the presence of the second derivatives in the Lagrangian The Green functions of magnetic held were found function when the second Lagrangian. We start from
L = +k,x’-
/g’+
the
approximation.
field
a charged particle moving in a non-stationary in refs. [I-S]. Here WC will obtain the Green time derivative terms arc added to the standard a classical Lagrangian.
/?,.r’+ fqyJ - k,,x’-
K,$) + a(x,;
Xy).
(29)
(For another example of a two-dimensional effective Lagrangian with higher derivatives see ref. [20].) We will use the method which has been worked out in section 2. The Hamiltonian corresponding to Lagrangian (29) is as follows:
It can be represented
in the standard
form
( I) with the matrices
V.V. Dodonov
et al. I Lagrangians
605
with higher derivatives
We combine canonical coordinates and momenta in the vector 4 = ( P,~~,plX, POy>Ply7 x0, x,, y,, y,). Commutation relations take the form
with the matrix ZSdefined in eqs. (3). The solutions of the equation (3) are as follows:
kzi,
d(K,&) - __ dt
k,i,
d(&) ___dt
K16, - aE3
k2 F4
d(K&) ____dt
K, 6, - aE4
- ~~8, - aE,
The expressions for matrices A, and A, are the same as those for the matrices A, and A2 respectively, but with the substitution (E, , Ed, F~, Ed, S,, CT,, S,, h)+(Es, qj’ E7, %, 6,, 6,, a,, 8,). Functions &iand a,, i = 1,2, . . . ,8, are the solutions of the ordinary differential equations
(32) $
&ii)
+ z
+ k,E; = -28;a - 6;ri )
(k,ii)
with the initial conditions
q(O) = 1 >
ii;(O)= -
... 49 6(O)= - g-@j
1 k2(0)
3
&Z(O) = 1>
)
2
ii;(O)
...
=
k (0) k2(0)
3
K,(O)
6,(O) = - r%(o) ’
G(O)
=
a(o) k2(0)
9
6,(O)
1 4(O) = - k2(0)
?
=
8,(O)
=
1)
v. v. Dcxlonov et ul.
606
I
Lagrangians
6,(O) = ~ ~
with higher cirrivcr~ivrc
1
1
6x(o) = k?(O)
‘G(o) ’
’
k(O)
43(O)= - K
(3.3)
7
(all other initial values of these functions and their first three derivatives equal zero). The A-matrix and hence the Green function of the Hamiltonian (30) arc completely
determined
6. The Green function In this section
by eqs.
(2).
(31)-(32).
for the Lagrangian
we will obtain
with (A-‘“‘)’
the Green
function
for the complete
Lagran-
gian (6) in the case when k,Y = . . = k, = 0. As we have seen, the problem of finding the Green function of the Hamiltonian (8) can be reduced to the problem of finding the corresponding A-matrix. Coefficients k,v always can be turned to unity by a change of variable t = T(k,\,)“‘)‘” and by the r-e-definition of the coefficients k,, i # N. The equations for the A-matrix elements arc as follows:
If we introduce A;’ = 6, AY’
the notation ..,
Ai” = F’L ‘) .
= (-I)*‘+‘(F(~+‘)
The expression
A f ’ = F, then
+ k,v
for the function
Al.* = (_l)‘\‘+lF’.~”
.
,F(“’ I’),
A,‘IV ” in terms
of the E-function
is
V.V. Dodonov et al. I Lagrangians
with higher derivatives
607
Then A;’ =
E(2N-1)
(_I)“+‘(
+ “c’
(_l)“+lk,_,,E(2NZ”-l))
.
(36)
m=l
From (34), (36) it is easy to obtain the equation for the c-function, N-l
(-1)$(2~)
+
c m=l
E(2“+‘)kNpm
+koE=O +1)
(-I,-
(Na2).
The initial conditions have the form E(0) = 1
q))
=
pyo)
. . . =
7
= 0.
To determine the values of the remaining N derivatives t = 0 we use eq. (35), which reads as follows at t = 0: 0 = $“+m) + i
of the E-function at
(_l)m+1kN_mE(N+n-2m)(0) .
m=l
If N + n - 2m = 0, then from the condition c(0) = 1 we obtain a system of equations for the values of the derivatives of the c-function at t = 0, k,_,(-1)’
=
E
W+n)(o)
+
i
(_l)m+1kN_mE(N+n-2m)(o)
,
m=l
where p = 1+ [n/2]. At m > p we have ~(~‘(0) = 0, p c N - 1, whereas q is determined condition N-n-2q=O,
by the
naqa1.
N-lans0,
Since there exist no IZ- and q-values satisfying these conditions, then ~‘~‘(0) = . . . ~~~~~~~ ( 0) = 0. Similarly one can consider the sth equation for the function h;‘(l~s~N), i;;’ = /q
/ii’ = k,A;’
)
,
lip = -(k,A”,2 + A;‘), ‘sN_1
=
*1 A‘sN
Then
if
=
(-;;“l’h;u , h;N
A”,’ =
sN-n A2
sN
=
E,,
(-l)“tl(
.
. . ,
=
-(kN_l,y;N(_l)N
+
h;N-I).
hfN = E:~-') We have E;f”+n)
+
m=l
(-l~‘“+1kN__8j”+“2”))
.
The equation
for the function
(-1)~Q~‘~)
The initial
+ ‘i’ F;zsv 2,,rlk,V ,,,(_l)“” ,,i I
conditions
‘)(O) =
F,“-
The values determined
F, has the form 1) + /Q\
=().
(37)
are as follows: OcasN-I.
F~“‘(O)= 0
1.
of the remaining from the equations
N derivatives
,Y k ,v ,,(- 1)” = Ei’\‘+‘l’(o) + i F ,,I I
4 ,,
‘,,r ,
a#.~-
of the
(0) k \
I.
(38)
F,-function
at f = 0 are
’.
(30)
,,,(- 1I”‘_
where 11 = 1 + [n/2], ~2s N - I and q is determined by the condition N + II ~ 2y = s - 1. The matrices X, and A, are described by the same equations as those describing A, and A,. but with different initial conditions
;;‘“‘(o) = 0 where
,
()=z~~cN-l.
A;’ = g,.
The values equations
of the remaining
The final expressions P,
A,=
(30)
. . .
for the A-matrix
(A’
I I
(N
I)
F,
I F,,’
PC
N derivatives
_
elements
at I = 0 are determined
turn
by the
out to be as follows:
V.V. Dodonov et al. I Lagrangians
609
with higher derivatives
where the functions I, are determined by eqs. (37) and by the initial conditions (38), (39). The expressions for A, and A, are the same as those for X, and A, respectively, with a substitution E,+ EJ, where KY(t)satisfies eq. (37) with the initial conditions (40), (41).
7. The Green function for Lagrangians statistics
with higher derivatives
and
of polymers
In the problems connected with the investigation of the statistical properties of polymers (and membranes) [lo-131 a necessity to calculate the functions G(R, R’, u, u’, L), which have the meaning of distribution functions, arise. These functions depend on the coordinates R and R’ of the initial and final points of a flexible polymer chain of length L and on the parameters u and U’ which determine the flexion of the chain at the initial and final points,
R')= dsW)
u’=u(r=
u =
u(r = R) =
,
?-=R’
%$IrzR ,
where r is the coordinate of the point of chain, s is the natural parameter along the chain. All the mean characteristics of the line are determined by the Green function G(u, u’, R, R’, L) in the following manner:
I
(f)=
dR dR’ du du’ G(u, u’, R, R’, L) f(R, R’, u, u’) (42)
dR dR’ du du’ G(u, u’, R, R’, L)
I
The Green function is determined
usually by the path integral [II]
G=h’jD”“exp(-idsY’(s)), 0
R
(43) 2
=
2
i”(s)
+
g
@(s)
-
W(r(s))
)
where e, 5 are the force constants, p = 1 lkT, W(r(s)) describes the influence of the external forces on the polymer chain, and dots denote the derivatives with respect to the parameter S. The term with the second order derivative appears
in the
Lagrangian
proportional
(43)
the torsion
of the chain
The general Y = ,*,
the
potential
curvature.
then terms
of the polymer
of the
flexed
in the Lagrangian chain
chain
is
into account (43).
is [ 111
g,,[r”‘J(s>J2 - W(r(s)) no external
represented
force
acts on the chain
which coincides
the Green
function
is
In this case [II,
131
in the form i,
~(u,
energy
i.e. to Y’(S). If one takes
like -yr’ can appear
form of the Lagrangian
In the case when usually
because
to the squared
I.
with the Green
u’. L) = @&j’
function
’ cxp(-
&
of an oscillator.
[(u’ + IL”) ch a - 2~47)
.
I ,2
3
)
a=Lz i
h=(F)
,
If, however. the chain is subjected to some external force for example, if it is placed in some held with the potential W(Y) = iwir2, then the Green function (42) does not coincide with that of an oscillator. In case of the Lagrangian
with 5, e. w depending
on the parameter
(3) in which the functions
with the initial
E, satisfy
S, the Green
the following
function
is given by cq.
equation:
conditions
e,(O) = 1 , iuu F;(O) = 5(o)
&((I) = -i ,
,
F>(O) = i E-(O)
3i F?(O) = ~ P5(0)
1
1 Ed(O) = ~ PtYO) ’
V.V. Dodonov et al. I Lagrangians
with higher derivatives
611
Other values c?)(O) are equal to zero. In the case of constant 5, e, o0 one can use eq. (15) with the substitution wy+w;=-pp
3
2 c+;=-~,
t=-is.
9. Conclusion Thus we have shown that the exact Green functions (propagators) can be calculated for arbitrary quadratic Lagrangians with higher derivatives. They are expressed through solutions of certain ordinary differential equations of the 2Nth order, if the Lagrangian contains derivatives of the Nth order. In the case of time-dependent coefficients of the Lagrangian one can obtain a rather simple and symmetric explicit expression for the Green function.
References
[ll PI
V.V. Dodonov, I.A. Malkin and V.I. Man’ko, Physica 59 (1972) 241. V.V. Dodonov, I.A. Malkin and V.I. Man’ko, Physica A 82 (1976) 113. V.V. Dodonov and V.I. Man’ko, Physica A 115 (1982) 215. 131 E.A. Akhundova, [41 V.V. Dodonov and V.I. Man’ko, Physica A 137 (1986) 306. 151 V.V. Dodonov and V.I. Man’ko, Proc. Lebedev Physics Institute, vol. 176 (supplemental volume) (Nova Science, Commack, 1988) p. 197. Fl H. Kleinert, J. Math. Phys. 27 (1986) 3003. [71 J. Kruger, Physica 65 (1973) 1. PI C. Battle, J. Gomis, J. Pons and N. Roman, J. Phys. A 21 (1988) 2693. J. Phys. A 22 (1989) 1673. [91 V.V. Nesterenco, [lOI R. Harris and H. Hearst, J. Chem. Phys. 44 (1966) 2595. [Ill K. Freed, Adv. Chem. Phys. 22 (1972) 1. [=I W. Janke and H. Kleinert, Phys. Lett. A 117 (1986) 353. and J. Thomchick, J. Phys. A 10 (1977) 1115. [I31 G.J. Papadopoulos A Treatise on the Analytical Dynamics of Particles and Rigid Bodies [I41 E.T. Whittaker, (Cambridge Univ. Press, Cambridge, 1959) p. 266. of Systems with Constraints (Nauka, [151 D.M. Gitman and I.V. Tyutin, Canonical Quantization Moscow, 1986) p. 190-202 (in Russian). [161 F. Riahi, Am. J. Phys. 40 (1972) 386. iI71 P. Caldirola, Rend. 1st. Lomb. Accad. Sci. Lett. A 93 (1959) 439. [la G. Valentini, Nuovo Cimento 19 (1961) 1280. u91 M. Battezzati, Canad. J. Phys. 58 (1980) 1691. PO1 B.-G. Englert, Ann. Phys. 129 (1980) 1.
EXACT PROPAGATORS FOR LAGRANGIANS WITH HIGHER DERIVATIVES IN QUANTUM MECHANICS V.V. DODONOV,
A.B. KLIMOV and V.I. MAN’KO
P. N. Lebedev
Institute,
Physical
Leninsky
prospect
53, 117924 Moscow,
USSR
Received 2.5 April 1989 Revised manuscript received 29 May 1990
1. Introduction In recent years in a series of papers [l-5] we studied the so-called quantum quadratic systems described by means of Hamiltonians of the following structure:
q=(P1,Pz,...,PN,
is the 2N-dimensional vector whose components are canonically conjugated momenta and coordinates: [ 6,) i,] = -iS,, (we assume h = l), B(t) is a 2N X 2N matrix, and C(f) is a 2Ndimensional vector. It was shown that the propagator of the Schrodinger equation with Hamiltonian (1) can be calculated exactly for quite arbitrary time-dependent matrix B(t) and vector C(f): x1,x2,...,
G(x, x’, t) = [det(-2nih,)]-1’2 X
exp
+2x’. (6, - A, . Ai1 +~)+5,.A~.A,l.fi,-218,(~).~~(T)d7)]. Here A,(t), j = 1,2,3,4
are N
x
N matrices forming the 2N
(2)
X
2N matrix
satisfying
the equation
E,. is the satisfies
unit
N x N matrix.
The
2N-dimensional
vector
d(t)
= (u, , CT:)
the equation
d=A-X-C.
A(())=().
(4)
Thus the only problem is to find the explicit solutions to eq. (3). The cases when this is possible were investigated in refs. [l-5]. In the present paper we consider a new interesting example of a quadratic system admitting the explicit propagator, namely. a system described by means of a Lagrangian containing higher derivatives. Recently Kleinert [6] who calculated
a special case of this problem was studied by the propagator corresponding to the Lagrangian
with constant coefficients in the framework of the path integral approach. Our aim is to generalize Klcinert’s result to the cases when (i) coefficients are arbitrary functions of time, (ii) the derivatives of the Nth order are present. We USC the general scheme of investigating quadratic systems [I-S] based on introducing instead of the Lagrangian the equivalent Hamiltonian in the form (i). By the way. we will demonstrate that this scheme leads to the same result as the path integral method for Lagrangians with higher derivatives as well ac in the case of usual Lagrangians with the first derivatives only. As to the physical applications of Lagrangians like (5), some references can be found in paper 161. Besides, different aspects of the quantum mechanical description ot systems possessing Lagrangians with higher derivatives were discussed in refs. [7-Y]. The most wide application of such Lagrangians can be found in the statistical physics of polymers [IO-131. This item will be considered in section 7.
2. Quadratic Hamiltonians
Lagrangians
Let us consider
with higher derivatives
a Lagrangian
and their equivalent
V.V. Dodonov
et al.
/ Lagrangians
with higher derivatives
597
where xCn)is the nth time derivative of the coordinate x = x(O), {kN} is a set of time-dependent coefficients, and f(t) is an external force. To obtain an equivalent Hamiltonian of the system with Lagrangian (6) we will use the method of Ostragradski [7, 14-161. According to this method N variables x, = X@), y1= 0, 1, . . . ) N - 1, can be considered as independent degrees of freedom. Introducing Lagrangian mutipliers p, into (6) we obtain an equivalent Lagrangian
+N-2 c=o p,(X,
- x,+1) +
k,i;-,
-fx,,.
(-UN+ -y-
n
1, . . , N - 2, are nothing else but the
One can see that variables p,, n = canonical momenta:
a
P,=ax,,'
LN
n = 0, 1, . . . , N - 2 .
The (N - 1)th generalized velocity as follows: p,+,
= 2
(7)
=
momentum
(-QN+‘kNiN_,
is related to the (N - 1)th generalized
.
Performing the Legendre transformation of (7) and excluding the variable X,_ I we obtain the following expression for the Hamiltonian: N-l
H=
c P,,-f,, -
n
LN
=o
N-l =
-
Ix”(-l).+l$
+ p+
(-l)N+’
+ z:
pnxn+,
+fx,,
.
(8)
We assume that the coefficient k, never turns to zero. We see that Hamiltonian (8) is a quadratic form of 2N canonical variables. If we rewrite it in a standard form (i), then N x N-blocks of the matrix
are as follows:
b, =
b, =
...“. i
The components Cl = 0 ,
1 =b; 0i
0
of the 2N-dimensional c, = (f(r),
0,
vector
C are
, 0)
We perform quantization of Hamiltonian (8) 17, IS]. Thus we impose the commutation relations [ I’,, ’
by the
canonical
x,,,1= - i4,,H
method
(I())
and arrive at a quantum system with N degrees of freedom, but with an unusual (although quadratic) Hamiltonian. Before investigating it in the whole volume,
let us consider
3. Lagrangians
the special
case of N = 2.
with the second derivative
The equations for the blocks with eq. (3) as follows:
of A-matrix
x, =h;b,-X,.b,.
&=A,+-A,+,
& = A, - b, - A,. b,
x,=A,.b,-X,-b,.
can be represented
in accordance
(11)
with initial conditions A,(O) = A,(O) = E?, A,(O) = A,(O) = 0. Substituting the expressions for matrices b, (9) at N = 2 into eqs. (11) and solving them we obtain the following expressions:
A,
=
V.V. Dodonov et al. I Lagrangians
Functions ci(t), i = 1,2,3,4,
with higher derivatives
599
are solutions of the equation (12)
with the initial conditions go)
&,(O) = E*(O) = 1 )
= -E&O)
= - &
) 2
E;(O)
=
-
k,(O) k2(0)
g+(O)
3
=
k,(O) -j---
-
k2(0)
’
Other values E:‘(O) are equal to zero. Thus the force-independent Green function (2) assumes the form 1 G(y,,u,,t] y,,ul,O)= [det(-2nih3)]“2 X exp
- ____
i 2 det A, {Y:@4&1-
part of the
E,“4) - 2Y,Y,i4
+ Y;[/$~(E~F~- ij3~4) + k,(ii,t-, - ii;Q] - ~u,u~E~ + u;k,(Q,
- QE,) + u;(E~E~ - Eli.,)}
X exp - & ( + yIU2&4+ y,‘, detX,=
3
1
[u2y2k,(E,E, - E36,) +
U1y,(i.,E,
-
‘31 9
i3&,)
(13)
a3E4- FEES,
where the notation yr = xo(t = 0), y2 = x,(t), u, = x,(t = 0), u2 = xl(t) is introduced. In the case of time independent coefficients we make the substitutions k,lk, = co; + w;, k,lk, = &I’,. Then the expressions for the Ei-functions are as follows: 1 E, = -
1 e2 = (W2Q - w,s,> , k2S s
=
co’,
-
1 e4 = k,6 (5 -
o;,
where cr 2 = cos o1 ,t, s, 2 = sin or 2t.
(14) c2)
3
i 2k, det A; (C,,C, -- l)(Wf + w; + 2W,W,S,S,) k,6
with
This expression coincides with the result obtained by Kleincrt 161. Thus we have shown that our approach is equivalent to the path integral method for Lagrangians with higher derivatives as well as for usual Lagrangians, and that “strange” Hamiltonians like (8) correctly describe corresponding quantum systems. As an application we will obtain the Green function for the Lagrangian proposed by Caldirola [ 171 to describe the motion of a radiating (and thus damping) electron in a cluadratic potential l> = _ ie’ rr($$
+ (&‘)
(W, (1 = const.)
(16)
In refs. [l8, 191 the quantum system corresponding to this Lagrangian was investigated by the method of canonical transformations. but neither solutions nor propagators have
The equation
were
obtained.
of the scheme
described
above
we
for F, is
1 2 P, + cy i; + 7 F, + ck with the initial
In terms
conditions
w ff
c‘(
=o
( 17)
V.V. Dodonov et al. I Lagrangians
El(O)= E*(O)= 1 ) Other
values
ii;(O) = -i,(O)
601
with higher derivatives
= - 3
)
Eq(0) = - -$ . CY
(18)
e,‘“‘(0) are eq ual to zero.
The solution
of (17) with conditions
(18) is as follows:
where -1kS P, =
2a
are the roots of the characteristic
equation
2
p4+~p3+&p2+;=o.
At
Coefficients
CY
(Y
are determined
from eqs. (18),
+nP’-
A:=
mf1 P, -Pj’ Af=~m2,~,(Pm-Pj)-’ m, II, j =
All other Green
values
function
7
C ( m#n#j
PmPn
IYI (Pj-Pm)-‘~ mfl
,$=a-3
II
> ??I#,
(Pm-Pj)Y’?
1,2,3,4.
A:
are obtained
(13) thus assume
by cyclic the form
rearrangements
of indices.
The
V.V. Dociorwv
602
et ul. )I Lqrangians
with higher
deriwtiLy.\
detX,=&,T;:j, 1.A
In the case of the full Lagrangian
(5) with f(t) # 0 one has to calculate
besides
matrices structure solutions
h,(t) also vectors 6,(t) satisfying cq. (4). Due to the very simple of the vector C(t) in the canonical form (i) of Hamiltonian (8). the of this vector equation can be expressed immediately in terms of the
functions
.5,(t) as follows:
The full Green function expression ( 13). and
equals
G = G,,G,,
where
the function
G,, is given
i -ix,
j
dt’ ~,f’-
0
iv,
i
dt’ ~~1’
I
0
i xexp
i
- ~ 2 det h,
(x2&, ~ v?E~)
j 0
i dt’ ~,j’ + (v,E,
~ ylF1)
J (I
di’ ~,,.f
by
V.V. Dodonov et al. I Lagrangians
with higher derivatives
603
4. Adiabatic approximation Let us consider a special case of the Lagrangian L, = ;(-k2X2
(5),
+ k,i2 - kox2),
(22)
when the coefficients obey the conditions fC2= 1;, = 0 )
k0
k
= w2(t) )
2
(23) p,+,
We1, co2
2
which means that the frequency s-function has the form
changes adiabatically.
The equation
d2e $y+k”,-g + w2(t) &=O. Let us putt= d4e
(24)
TT, T%=l. Then - d2c
T-4 d74 + k, z
We represent
for the
+ W’(T) E = 0.
(25)
the solution in the form
E(T) = W(7) sin( T [ dT1 S(7’))
(26)
and search for the functions W and S in the form of developments
in powers of
T-‘: S(7) = So(T) + y
+ ...,
W(7) = We(T) + y
+. . .
Substituting (24) into (23) we obtain from the coefficient at sin(T I,’ d7’ S(T’)) of the order To the equation
whose solution is (27)
V. V. Lkxionov
60-l
From
the coefficient -2&k”,
ef ul.
at cos(T
- 2s;)
! hgrungiuns
with higher derivutitm
]’ dr’ S(T))) of the order
dW,, = W,, d&k”,
T we have
~ 2.~;) ,
(28) Wi = const where
X
‘-, w
“I
k”f # 4w ‘.
Expressions (13), (26)-(27) Green function in the zeroth
enable order
us to solve the problem adiabatic
of calculating
5. The Green function of a charged particle moving in a magnetic in the presence of the second derivatives in the Lagrangian The Green functions of magnetic held were found function when the second Lagrangian. We start from
L = +k,x’-
/g’+
the
approximation.
field
a charged particle moving in a non-stationary in refs. [I-S]. Here WC will obtain the Green time derivative terms arc added to the standard a classical Lagrangian.
/?,.r’+ fqyJ - k,,x’-
K,$) + a(x,;
Xy).
(29)
(For another example of a two-dimensional effective Lagrangian with higher derivatives see ref. [20].) We will use the method which has been worked out in section 2. The Hamiltonian corresponding to Lagrangian (29) is as follows:
It can be represented
in the standard
form
( I) with the matrices
V.V. Dodonov
et al. I Lagrangians
605
with higher derivatives
We combine canonical coordinates and momenta in the vector 4 = ( P,~~,plX, POy>Ply7 x0, x,, y,, y,). Commutation relations take the form
with the matrix ZSdefined in eqs. (3). The solutions of the equation (3) are as follows:
kzi,
d(K,&) - __ dt
k,i,
d(&) ___dt
K16, - aE3
k2 F4
d(K&) ____dt
K, 6, - aE4
- ~~8, - aE,
The expressions for matrices A, and A, are the same as those for the matrices A, and A2 respectively, but with the substitution (E, , Ed, F~, Ed, S,, CT,, S,, h)+(Es, qj’ E7, %, 6,, 6,, a,, 8,). Functions &iand a,, i = 1,2, . . . ,8, are the solutions of the ordinary differential equations
(32) $
&ii)
+ z
+ k,E; = -28;a - 6;ri )
(k,ii)
with the initial conditions
q(O) = 1 >
ii;(O)= -
... 49 6(O)= - g-@j
1 k2(0)
3
&Z(O) = 1>
)
2
ii;(O)
...
=
k (0) k2(0)
3
K,(O)
6,(O) = - r%(o) ’
G(O)
=
a(o) k2(0)
9
6,(O)
1 4(O) = - k2(0)
?
=
8,(O)
=
1)
v. v. Dcxlonov et ul.
606
I
Lagrangians
6,(O) = ~ ~
with higher cirrivcr~ivrc
1
1
6x(o) = k?(O)
‘G(o) ’
’
k(O)
43(O)= - K
(3.3)
7
(all other initial values of these functions and their first three derivatives equal zero). The A-matrix and hence the Green function of the Hamiltonian (30) arc completely
determined
6. The Green function In this section
by eqs.
(2).
(31)-(32).
for the Lagrangian
we will obtain
with (A-‘“‘)’
the Green
function
for the complete
Lagran-
gian (6) in the case when k,Y = . . = k, = 0. As we have seen, the problem of finding the Green function of the Hamiltonian (8) can be reduced to the problem of finding the corresponding A-matrix. Coefficients k,v always can be turned to unity by a change of variable t = T(k,\,)“‘)‘” and by the r-e-definition of the coefficients k,, i # N. The equations for the A-matrix elements arc as follows:
If we introduce A;’ = 6, AY’
the notation ..,
Ai” = F’L ‘) .
= (-I)*‘+‘(F(~+‘)
The expression
A f ’ = F, then
+ k,v
for the function
Al.* = (_l)‘\‘+lF’.~”
.
,F(“’ I’),
A,‘IV ” in terms
of the E-function
is
V.V. Dodonov et al. I Lagrangians
with higher derivatives
607
Then A;’ =
E(2N-1)
(_I)“+‘(
+ “c’
(_l)“+lk,_,,E(2NZ”-l))
.
(36)
m=l
From (34), (36) it is easy to obtain the equation for the c-function, N-l
(-1)$(2~)
+
c m=l
E(2“+‘)kNpm
+koE=O +1)
(-I,-
(Na2).
The initial conditions have the form E(0) = 1
q))
=
pyo)
. . . =
7
= 0.
To determine the values of the remaining N derivatives t = 0 we use eq. (35), which reads as follows at t = 0: 0 = $“+m) + i
of the E-function at
(_l)m+1kN_mE(N+n-2m)(0) .
m=l
If N + n - 2m = 0, then from the condition c(0) = 1 we obtain a system of equations for the values of the derivatives of the c-function at t = 0, k,_,(-1)’
=
E
W+n)(o)
+
i
(_l)m+1kN_mE(N+n-2m)(o)
,
m=l
where p = 1+ [n/2]. At m > p we have ~(~‘(0) = 0, p c N - 1, whereas q is determined condition N-n-2q=O,
by the
naqa1.
N-lans0,
Since there exist no IZ- and q-values satisfying these conditions, then ~‘~‘(0) = . . . ~~~~~~~ ( 0) = 0. Similarly one can consider the sth equation for the function h;‘(l~s~N), i;;’ = /q
/ii’ = k,A;’
)
,
lip = -(k,A”,2 + A;‘), ‘sN_1
=
*1 A‘sN
Then
if
=
(-;;“l’h;u , h;N
A”,’ =
sN-n A2
sN
=
E,,
(-l)“tl(
.
. . ,
=
-(kN_l,y;N(_l)N
+
h;N-I).
hfN = E:~-') We have E;f”+n)
+
m=l
(-l~‘“+1kN__8j”+“2”))
.
The equation
for the function
(-1)~Q~‘~)
The initial
+ ‘i’ F;zsv 2,,rlk,V ,,,(_l)“” ,,i I
conditions
‘)(O) =
F,“-
The values determined
F, has the form 1) + /Q\
=().
(37)
are as follows: OcasN-I.
F~“‘(O)= 0
1.
of the remaining from the equations
N derivatives
,Y k ,v ,,(- 1)” = Ei’\‘+‘l’(o) + i F ,,I I
4 ,,
‘,,r ,
a#.~-
of the
(0) k \
I.
(38)
F,-function
at f = 0 are
’.
(30)
,,,(- 1I”‘_
where 11 = 1 + [n/2], ~2s N - I and q is determined by the condition N + II ~ 2y = s - 1. The matrices X, and A, are described by the same equations as those describing A, and A,. but with different initial conditions
;;‘“‘(o) = 0 where
,
()=z~~cN-l.
A;’ = g,.
The values equations
of the remaining
The final expressions P,
A,=
(30)
. . .
for the A-matrix
(A’
I I
(N
I)
F,
I F,,’
PC
N derivatives
_
elements
at I = 0 are determined
turn
by the
out to be as follows:
V.V. Dodonov et al. I Lagrangians
609
with higher derivatives
where the functions I, are determined by eqs. (37) and by the initial conditions (38), (39). The expressions for A, and A, are the same as those for X, and A, respectively, with a substitution E,+ EJ, where KY(t)satisfies eq. (37) with the initial conditions (40), (41).
7. The Green function for Lagrangians statistics
with higher derivatives
and
of polymers
In the problems connected with the investigation of the statistical properties of polymers (and membranes) [lo-131 a necessity to calculate the functions G(R, R’, u, u’, L), which have the meaning of distribution functions, arise. These functions depend on the coordinates R and R’ of the initial and final points of a flexible polymer chain of length L and on the parameters u and U’ which determine the flexion of the chain at the initial and final points,
R')= dsW)
u’=u(r=
u =
u(r = R) =
,
?-=R’
%$IrzR ,
where r is the coordinate of the point of chain, s is the natural parameter along the chain. All the mean characteristics of the line are determined by the Green function G(u, u’, R, R’, L) in the following manner:
I
(f)=
dR dR’ du du’ G(u, u’, R, R’, L) f(R, R’, u, u’) (42)
dR dR’ du du’ G(u, u’, R, R’, L)
I
The Green function is determined
usually by the path integral [II]
G=h’jD”“exp(-idsY’(s)), 0
R
(43) 2
=
2
i”(s)
+
g
@(s)
-
W(r(s))
)
where e, 5 are the force constants, p = 1 lkT, W(r(s)) describes the influence of the external forces on the polymer chain, and dots denote the derivatives with respect to the parameter S. The term with the second order derivative appears
in the
Lagrangian
proportional
(43)
the torsion
of the chain
The general Y = ,*,
the
potential
curvature.
then terms
of the polymer
of the
flexed
in the Lagrangian chain
chain
is
into account (43).
is [ 111
g,,[r”‘J(s>J2 - W(r(s)) no external
represented
force
acts on the chain
which coincides
the Green
function
is
In this case [II,
131
in the form i,
~(u,
energy
i.e. to Y’(S). If one takes
like -yr’ can appear
form of the Lagrangian
In the case when usually
because
to the squared
I.
with the Green
u’. L) = @&j’
function
’ cxp(-
&
of an oscillator.
[(u’ + IL”) ch a - 2~47)
.
I ,2
3
)
a=Lz i
h=(F)
,
If, however. the chain is subjected to some external force for example, if it is placed in some held with the potential W(Y) = iwir2, then the Green function (42) does not coincide with that of an oscillator. In case of the Lagrangian
with 5, e. w depending
on the parameter
(3) in which the functions
with the initial
E, satisfy
S, the Green
the following
function
is given by cq.
equation:
conditions
e,(O) = 1 , iuu F;(O) = 5(o)
&((I) = -i ,
,
F>(O) = i E-(O)
3i F?(O) = ~ P5(0)
1
1 Ed(O) = ~ PtYO) ’
V.V. Dodonov et al. I Lagrangians
with higher derivatives
611
Other values c?)(O) are equal to zero. In the case of constant 5, e, o0 one can use eq. (15) with the substitution wy+w;=-pp
3
2 c+;=-~,
t=-is.
9. Conclusion Thus we have shown that the exact Green functions (propagators) can be calculated for arbitrary quadratic Lagrangians with higher derivatives. They are expressed through solutions of certain ordinary differential equations of the 2Nth order, if the Lagrangian contains derivatives of the Nth order. In the case of time-dependent coefficients of the Lagrangian one can obtain a rather simple and symmetric explicit expression for the Green function.
References
[ll PI
V.V. Dodonov, I.A. Malkin and V.I. Man’ko, Physica 59 (1972) 241. V.V. Dodonov, I.A. Malkin and V.I. Man’ko, Physica A 82 (1976) 113. V.V. Dodonov and V.I. Man’ko, Physica A 115 (1982) 215. 131 E.A. Akhundova, [41 V.V. Dodonov and V.I. Man’ko, Physica A 137 (1986) 306. 151 V.V. Dodonov and V.I. Man’ko, Proc. Lebedev Physics Institute, vol. 176 (supplemental volume) (Nova Science, Commack, 1988) p. 197. Fl H. Kleinert, J. Math. Phys. 27 (1986) 3003. [71 J. Kruger, Physica 65 (1973) 1. PI C. Battle, J. Gomis, J. Pons and N. Roman, J. Phys. A 21 (1988) 2693. J. Phys. A 22 (1989) 1673. [91 V.V. Nesterenco, [lOI R. Harris and H. Hearst, J. Chem. Phys. 44 (1966) 2595. [Ill K. Freed, Adv. Chem. Phys. 22 (1972) 1. [=I W. Janke and H. Kleinert, Phys. Lett. A 117 (1986) 353. and J. Thomchick, J. Phys. A 10 (1977) 1115. [I31 G.J. Papadopoulos A Treatise on the Analytical Dynamics of Particles and Rigid Bodies [I41 E.T. Whittaker, (Cambridge Univ. Press, Cambridge, 1959) p. 266. of Systems with Constraints (Nauka, [151 D.M. Gitman and I.V. Tyutin, Canonical Quantization Moscow, 1986) p. 190-202 (in Russian). [161 F. Riahi, Am. J. Phys. 40 (1972) 386. iI71 P. Caldirola, Rend. 1st. Lomb. Accad. Sci. Lett. A 93 (1959) 439. [la G. Valentini, Nuovo Cimento 19 (1961) 1280. u91 M. Battezzati, Canad. J. Phys. 58 (1980) 1691. PO1 B.-G. Englert, Ann. Phys. 129 (1980) 1.