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Harmonic approximation of the functional integral and the schroedinger equation
Volume 58B, number 3
PHYSICS LETTERS
15 September 1975
HARMONIC APPROXIMATION OF THE FUNCTIONAL INTEGRAL AND THE SCHROEDINGER EQUATION C. FAN0 and G. TURCHETTI Institute di Fish dcll’Universitd, Bobgna, Italy Received 4 July 1975 A new approximation for the density matrix e-@H(x, y) of a quantum-mechanical system is proposed. Starting from the Feynman-Kac formula we expand the potential up to second order around the linear path connecting x and y. As an example, the first eigenvalues of the anharmonic oscillator are computed. The analogue procedure in field theory is briefly discussed.
For the numerical search of the eigenvalues of the Schrodinger equation, we have recently proposed method which combines the formalism of the functional integration with the Pad6 technique. The basic tool was Trotter’s formula in the form: exp{+(T+
,
V)} = s lim A@/nr n+-
[ 1,2] a
(1)
where A(P) = exp (-0 v/2) exp (-00
exp (- PV2)
,
(2)
and T = --imA and V are the kinetic energy and potential operators of a quantum-mechanical system. See also the works of Grimm and Storer [3,4], where Trotter’s formula was first introduced in order to solve the Schrodinger equation. The kernel exp (-pH) (x,~) (where x,y are points in the configuration space and H = T + V) can be represented as a path integral
exp(- MO(x9v) = jdpxy(w) exp [ - j U4# 0
d
T],
(3)
where dPx,,(o) denotes the Wiener measure defined on all continuous paths O(T) with w(O) = x and w(J) =y. For simplicity we consider only the one-dimensional case. The trapezoidal rule with n + 1 points in [0, /I] for the integral I = _I$V(w (7)) d r g’Ives rise to Trotter’s formula. In order to improve the convergence rate of (1) we replace A(P/n) by a modified approximation of exp (-@Y/n), obtained from (3) in the following way: We denote by wL the linear path from x toy:
and we make the following change of variable: W(T) = WL(7) + w’(7).
(5)
Eq. (3) with /I’ in the place of fl becomes: exp(--P’H)(x,y)
=
exp
-
0, - x)2m 2p,
Y(wL+ w’) dr
1 .
341
Volume 58B, number 3
Expanding the potential up to second order in w’ we frequency a(t) = [ V”(cdL(t))/m] U2 in a time-dependent In order to evaluate the functional integral, we follow given by Dashen, Hasslacher and Neveu [5] (we also use existence in our case of the linear term f(t) ** . The final result is:
obtain the density matrix of an harmonic oscillator with external field f(t) = -V ‘(wL(r))* ’ . in the one-dimensional case the same procedure as that their notation). The only modification arises from the
-‘i;)2m)exp [--_/‘v(wL(T))dT]
ew-fl’H)(x,y)=exp
15 September 1975
PHYSICS LETTERS
G(x,y),
0
~(x,y)=
(T$)1’2(E
m 0-g (l/N(~))h(r) --,_
p$2dl)lii_p[-z
where N(T) is any solution of the equation ;hi = a2(t)N
dQg’NW-2
dr.l 1, (8)
with N(0) # 0, and h(t) is given by:
A simplified version of (8) which is suitable for numerical constant value ao*j. Eq. (8) becomes, choosing N(T) = cash (CLOT):
G(x,
d7j2 - _f$W2 _ _f{‘NO)-2 dp
calculation
can be obtained
sinh [fi,@‘--
ma0
u) = 2n sinh(B’Q)
by replacing a(t) with a
7)] sinh [Qs]
(10)
%
which is the familiar expression of the Green function of a forced harmonic oscillator. See ref. [6]. Our method works as long as the potential V(o) can be represented in the “relevant region” with sufficient acV(O~(T)). By the term “relevant region” we mean the incuracy by the osculating parabola at the point (We, m since most paths o = wL+ o’ satisfy the condition lw’l < 3JK terval [wL(7) - 3Jm, ~~(7) + 3m] In fig. 1 it is shqwn how our approximation improves Trotter’s formula; the latter is obtained by taking in eq. (7) G(x,y) = 1 and J$ V(aL(~))dT -3 [V(x) + I+)]$. Using the approximate expression (7) for exp (-P’H) (x,~) we evaluate the moments cc, = (9, (exp(-P’H))‘J/),
r = 1,2,3
.
J/ being any “reasonable” wave function. From the moments /.+ we compute the [(N - 1)/N] Pad6 approximant, whose poles are the approximate eigenvalues of exp (- /3%r) [ 11. In table 1 we compare the eigenvalues of the anharmonic oscillator obtained from Trotter’s formula and from our modified approximation (7), (10). The method being non-perturbative, the accuracy is independent of the coupling strength. *I This interpretation holds for V”(wr (t))> 0. In the opposite case the formulas can be analytically continued to imaginary 51. ** Formula (2.7) of ref. [S] must be modified as follows in our case: y(s) = X(T)- 15 {&(&V(r)) x(p) dN+ @h(b) dw where h is given by (9). *’ The value of Re which minimizes the error introduced by the replacement n(t) + ~20is given by inn;=
12(&)3
[(Tm)
(y-x)
-JV(()dl]
.
With this choice for Q, it is possible to prove that the R.H.S. of (7) differs from exp (-P’H) (x,y) by the small term @‘3/240m2) X d4 V/dx4 Ix+ + 0(p’4) where F is a point in [x,y]. Details of the calculations will be given elsewhere. See also ref. [2].
342.
15 September 1975
PHYSICS LETTERS
Volume 58B, number 3
Fig. 1. For a given potential V(w) the osculating parabola at the point [wL(r), V(wL(r))] and the “Trotter’s approximation” are shown. A1 is the difference between V(w) and its quadratic approximation, and A2 is the difference between I’(w) and the “Trotter’s approximation”.
In field theory the analogue of this approximation can be explicitly worked out. Let Z(J)/Z(O) be the generating functional of the Green-function for an Euclidean field cpself-interacting with a “potential” V(q) and with an external source J. We have formally:
(12) where x E Rd. The classical field is: Q,+g~‘(Vc)=J;
K=-A+m2.
(13)
Usually [S, 71 one expands V(q) around the classical solution qc, putting cp= ~c + 9’. The analogue of the linear path (4) is the field solution qL with g = 0; hence KtpL = J. We make the change of variable cp= qL + up’and we expand V(V, + cp’) up to second order. After performing the functional integration we obtain the result: W(J)=
~p(lpL)=?i~~K~L-g~Y(~L)dx+ig2~‘(~L)~-1II+~~”(~L)K-1l-1~‘(~L)
,
(14)
where e.g. qLKqL means / dx (pL(x) (- A t m2) opt, etc. and WamP. generates the connected amputated Schwinger functions. It can be easily seen by expanding [ 1+ g V”((p,)K-’ ] -I in powers of g that (14) generates a subset of the tree-diagrams. This subset differs from the complete set only to order g4: This is clear from the diagrammatic point of view since we neglect terms with V”‘(pL), V”“(~,), .... Of course the situation in field theory is much less satisfactory since we cannot easily iterate the “kernel” as in the Schrbdinger equation. We wish to thank Dr. A..Neveu for correspondence
and Dr. F. Guerra for stimulating
discussions.
Table 1 The first three eigenvalues of the anharmonic oscillator V(x) = 3 (x2 + x4), m = 1, computed with different approximations compared. We hate chosen P’ = 0.2, N= 6 and 48 mesh points. Eigenvalues
Eo ____
Exact ‘Trotter’s approximation” Approximation (es. (10))
El
are
E2
-
0.696 1 0.6919 0.6948
2.324 2.306 2.324
4.327 4.285 4.329
343
Volume 58B, number 3
PHYSICS LETTERS
15 September 1975
References (11 [2] [3] [4] [5] [6] (71
344
A. Bove, G. Fano, G. Turchetti and A.G. Teolis, Journ. of Math. Phys. 16 (1975) 268. A. Bove, G. Fano, G. Turchetti and A.G. Teolis, Nuovo Cimento, to be published. R.G. Storer, Phys. Rev. 176 (1968) 326. R.C. Grimm and R.G. Storer, J. Comp. Phys. 7 (1971) 134. R.F. Dashen, B. Hasslacher and A. Neveu, Phys. Rev. DlO (1974) 4114. R.P. Feynman and A.R. Hibbs, Quantum mechanics and path integral (McGraw-Hill, N.Y., 1954). J. Iliopoylos, C. Hzykson, Functional methods and perturbation theory, to appear in Review of Mod. Phys.
PHYSICS LETTERS
15 September 1975
HARMONIC APPROXIMATION OF THE FUNCTIONAL INTEGRAL AND THE SCHROEDINGER EQUATION C. FAN0 and G. TURCHETTI Institute di Fish dcll’Universitd, Bobgna, Italy Received 4 July 1975 A new approximation for the density matrix e-@H(x, y) of a quantum-mechanical system is proposed. Starting from the Feynman-Kac formula we expand the potential up to second order around the linear path connecting x and y. As an example, the first eigenvalues of the anharmonic oscillator are computed. The analogue procedure in field theory is briefly discussed.
For the numerical search of the eigenvalues of the Schrodinger equation, we have recently proposed method which combines the formalism of the functional integration with the Pad6 technique. The basic tool was Trotter’s formula in the form: exp{+(T+
,
V)} = s lim A@/nr n+-
[ 1,2] a
(1)
where A(P) = exp (-0 v/2) exp (-00
exp (- PV2)
,
(2)
and T = --imA and V are the kinetic energy and potential operators of a quantum-mechanical system. See also the works of Grimm and Storer [3,4], where Trotter’s formula was first introduced in order to solve the Schrodinger equation. The kernel exp (-pH) (x,~) (where x,y are points in the configuration space and H = T + V) can be represented as a path integral
exp(- MO(x9v) = jdpxy(w) exp [ - j U4# 0
d
T],
(3)
where dPx,,(o) denotes the Wiener measure defined on all continuous paths O(T) with w(O) = x and w(J) =y. For simplicity we consider only the one-dimensional case. The trapezoidal rule with n + 1 points in [0, /I] for the integral I = _I$V(w (7)) d r g’Ives rise to Trotter’s formula. In order to improve the convergence rate of (1) we replace A(P/n) by a modified approximation of exp (-@Y/n), obtained from (3) in the following way: We denote by wL the linear path from x toy:
and we make the following change of variable: W(T) = WL(7) + w’(7).
(5)
Eq. (3) with /I’ in the place of fl becomes: exp(--P’H)(x,y)
=
exp
-
0, - x)2m 2p,
Y(wL+ w’) dr
1 .
341
Volume 58B, number 3
Expanding the potential up to second order in w’ we frequency a(t) = [ V”(cdL(t))/m] U2 in a time-dependent In order to evaluate the functional integral, we follow given by Dashen, Hasslacher and Neveu [5] (we also use existence in our case of the linear term f(t) ** . The final result is:
obtain the density matrix of an harmonic oscillator with external field f(t) = -V ‘(wL(r))* ’ . in the one-dimensional case the same procedure as that their notation). The only modification arises from the
-‘i;)2m)exp [--_/‘v(wL(T))dT]
ew-fl’H)(x,y)=exp
15 September 1975
PHYSICS LETTERS
G(x,y),
0
~(x,y)=
(T$)1’2(E
m 0-g (l/N(~))h(r) --,_
p$2dl)lii_p[-z
where N(T) is any solution of the equation ;hi = a2(t)N
dQg’NW-2
dr.l 1, (8)
with N(0) # 0, and h(t) is given by:
A simplified version of (8) which is suitable for numerical constant value ao*j. Eq. (8) becomes, choosing N(T) = cash (CLOT):
G(x,
d7j2 - _f$W2 _ _f{‘NO)-2 dp
calculation
can be obtained
sinh [fi,@‘--
ma0
u) = 2n sinh(B’Q)
by replacing a(t) with a
7)] sinh [Qs]
(10)
%
which is the familiar expression of the Green function of a forced harmonic oscillator. See ref. [6]. Our method works as long as the potential V(o) can be represented in the “relevant region” with sufficient acV(O~(T)). By the term “relevant region” we mean the incuracy by the osculating parabola at the point (We, m since most paths o = wL+ o’ satisfy the condition lw’l < 3JK terval [wL(7) - 3Jm, ~~(7) + 3m] In fig. 1 it is shqwn how our approximation improves Trotter’s formula; the latter is obtained by taking in eq. (7) G(x,y) = 1 and J$ V(aL(~))dT -3 [V(x) + I+)]$. Using the approximate expression (7) for exp (-P’H) (x,~) we evaluate the moments cc, = (9, (exp(-P’H))‘J/),
r = 1,2,3
.
J/ being any “reasonable” wave function. From the moments /.+ we compute the [(N - 1)/N] Pad6 approximant, whose poles are the approximate eigenvalues of exp (- /3%r) [ 11. In table 1 we compare the eigenvalues of the anharmonic oscillator obtained from Trotter’s formula and from our modified approximation (7), (10). The method being non-perturbative, the accuracy is independent of the coupling strength. *I This interpretation holds for V”(wr (t))> 0. In the opposite case the formulas can be analytically continued to imaginary 51. ** Formula (2.7) of ref. [S] must be modified as follows in our case: y(s) = X(T)- 15 {&(&V(r)) x(p) dN+ @h(b) dw where h is given by (9). *’ The value of Re which minimizes the error introduced by the replacement n(t) + ~20is given by inn;=
12(&)3
[(Tm)
(y-x)
-JV(()dl]
.
With this choice for Q, it is possible to prove that the R.H.S. of (7) differs from exp (-P’H) (x,y) by the small term @‘3/240m2) X d4 V/dx4 Ix+ + 0(p’4) where F is a point in [x,y]. Details of the calculations will be given elsewhere. See also ref. [2].
342.
15 September 1975
PHYSICS LETTERS
Volume 58B, number 3
Fig. 1. For a given potential V(w) the osculating parabola at the point [wL(r), V(wL(r))] and the “Trotter’s approximation” are shown. A1 is the difference between V(w) and its quadratic approximation, and A2 is the difference between I’(w) and the “Trotter’s approximation”.
In field theory the analogue of this approximation can be explicitly worked out. Let Z(J)/Z(O) be the generating functional of the Green-function for an Euclidean field cpself-interacting with a “potential” V(q) and with an external source J. We have formally:
(12) where x E Rd. The classical field is: Q,+g~‘(Vc)=J;
K=-A+m2.
(13)
Usually [S, 71 one expands V(q) around the classical solution qc, putting cp= ~c + 9’. The analogue of the linear path (4) is the field solution qL with g = 0; hence KtpL = J. We make the change of variable cp= qL + up’and we expand V(V, + cp’) up to second order. After performing the functional integration we obtain the result: W(J)=
~p(lpL)=?i~~K~L-g~Y(~L)dx+ig2~‘(~L)~-1II+~~”(~L)K-1l-1~‘(~L)
,
(14)
where e.g. qLKqL means / dx (pL(x) (- A t m2) opt, etc. and WamP. generates the connected amputated Schwinger functions. It can be easily seen by expanding [ 1+ g V”((p,)K-’ ] -I in powers of g that (14) generates a subset of the tree-diagrams. This subset differs from the complete set only to order g4: This is clear from the diagrammatic point of view since we neglect terms with V”‘(pL), V”“(~,), .... Of course the situation in field theory is much less satisfactory since we cannot easily iterate the “kernel” as in the Schrbdinger equation. We wish to thank Dr. A..Neveu for correspondence
and Dr. F. Guerra for stimulating
discussions.
Table 1 The first three eigenvalues of the anharmonic oscillator V(x) = 3 (x2 + x4), m = 1, computed with different approximations compared. We hate chosen P’ = 0.2, N= 6 and 48 mesh points. Eigenvalues
Eo ____
Exact ‘Trotter’s approximation” Approximation (es. (10))
El
are
E2
-
0.696 1 0.6919 0.6948
2.324 2.306 2.324
4.327 4.285 4.329
343
Volume 58B, number 3
PHYSICS LETTERS
15 September 1975
References (11 [2] [3] [4] [5] [6] (71
344
A. Bove, G. Fano, G. Turchetti and A.G. Teolis, Journ. of Math. Phys. 16 (1975) 268. A. Bove, G. Fano, G. Turchetti and A.G. Teolis, Nuovo Cimento, to be published. R.G. Storer, Phys. Rev. 176 (1968) 326. R.C. Grimm and R.G. Storer, J. Comp. Phys. 7 (1971) 134. R.F. Dashen, B. Hasslacher and A. Neveu, Phys. Rev. DlO (1974) 4114. R.P. Feynman and A.R. Hibbs, Quantum mechanics and path integral (McGraw-Hill, N.Y., 1954). J. Iliopoylos, C. Hzykson, Functional methods and perturbation theory, to appear in Review of Mod. Phys.