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Relation between two variational methods to calculate the energy levels
Volume 144, number
I
PHYSICS
LETTERS A
12 February
1990
RELATION BETWEEN TWO VARIATIONAL METHODS TO CALCULATE THE ENERGY LEVELS ’
Luis VAZQUEZ
Departamento de Fisica Tedrica, Facultad de Ciencias Fisicas, Universidad Complutense, 28040 Madrid, Spain Received I 1 September 1989; revised manuscript Communicated by A.R. Bishop
received
I6 November
1989; accepted
for publication
6 December
I989
For one-dimensional quantum systems, we establish the relation between two energy spectrum variational methods, associated to the SchrGdinger and Heisenberg equations of motion. We show that by working in the Heisenberg picture we are able to obtain energy levels straightforwardly and not only energy differences.
The finite difference schemes to solve the Heisenberg equations can be used to obtain estimations of the energy spectrum [ l-31. Here we extend the method to obtain estimations of the energy levels directly and not only energy differences. It turns out that such a generalization is the same method that was introduced by Dias de Deus et al. [4-61 in the Schrodinger picture. The method in the Heisenberg picture is based on a variational approach to the matrix element
as well as the accuracy of the approximated eigenstates. To illustrate the method let us consider the one-dimensional quantum system:
H=$p2+ V(q).
(2)
A. Computation of the energy levels. In this case we consider the matrix element between genstate which using ( 1 ) yields
the same ei-
(E,l~(t)lE,)=(E,l~(O)lE,).
(3)
Now let us consider the operator s(t) =q( t)p( t) and with the help of the unitary
=<&1@(0)1~,)
C [-i(~,-~,)~lkl~!,
where I,!?,) and 1E,) are eigenvectors of the Hamiltonian H and 8(t) is any operator associated to the quantum system. Any discretization of the Heisenberg equations defines an approximation to eq. ( 1). On the other hand we have to consider a suitable approximation to the eigenstates 1E,). From this approximation and by consistency arguments we obtain estimations of the energy levels and the energy differences independently. The accuracy of the method is related to the accuracy of the Heisenberg finite difference scheme
0375-9601/90/$
(4)
k
Pn+
cia e Investigation
[ 71
%I+1 -4n =Pn+l > T (1)
’ Partially supported
scheme
by the Comision Interministerial of Spain under grant PB86-0005.
03.50 0 Elsevier Science Publishers
de Cien-
I -Pn =-v’(4n) r
>
(5)
we obtain
+r<&
Ip(0)2-dO)v’
(0) IEn> +@r’)
.
(6)
The scheme is accurate to order 7, so by consistency and in agreement with (3) we impose the condition
(E,lp(0)2-q(0)V’(q(O))IE,)=0.
(7)
A general way to satisfy ( 7 ) is by introducing a variational parameter y which is compatible with the commutator [q(O), p(O)]=O,
q(O)+w
B.V. (North-Holland
)
>
(8) 15
Volume
144. number
I
PHYSICS
Y(O)-Pl;*.
(9)
In this way we get
(E,,I~‘IE,,)=;I~(E,,I.~I”(IJ.~)IE,,).
LETTERS
v ( I’.\-1 .
(10)
anharmonic
State n
Exact results
Estimated
0
0.6680 2.3936 4.6968 1.3361 10.244 13.379
0.6814 2.4237 4.6850 7.291 1 10.167 13.267
1 2 3 4 5
(11)
That allows us to estimate the energy levels of H in terms of a variational parameter. The accuracy will be related to the accuracy of the variational states 1E,). Thus we get a justification of the variational method of refs. [4-61 in the context of the Heisenberg picture. On the other hand it represents a generalization of the method of refs. [ l-31 which only allows us to compute the energy differences. B. Computation of the energy>differences /l-3]. By using the same unitary difference scheme as before and considering the operators q( t ) and p( t ). we obtain
I2 February
Table I Estimation of the energy levels of the quartic lator lq4 in units 2”‘.
which is the expression of the virial theorem and it turns out to be the minimization condition in 1’of the matrix element (E,,] HI E,) with I Et, > + v/,,= fi
A
WI eJz rJJ3 w
oscil-
energies
Table 2 Estimations of the adjacent energy differences w,=E,+, the quartic anharmonic oscillator lq4 in units 2”‘.
WI)
1990
-E,
of
Exact results
Energy differences from the estimated levels
Energy differences obtained directly
1.7256 2.3032 2.6399 2.9073 3.1350
I .7423 2.2448 2.606 I 2.8759 3.1000
1.8171 2.2894 2.6207 2.8845 3.1072
(E,,,Iq(r)lE,)=(E,,Iq(O)lE,) +~(E,,, IP(O) IE,, > +@(72)
.
(E,~Ip(-c)lE,)=(E,,Ip(O)lE,) +7(E,IF(q(O))lE,)+B(r’).
(12)
By imposing the consistency of ( 1) and ( 12) up to the first order in 7 we obtain
“=i
1
1 (EmIf-(q(O))l&) ‘j’=i (E,,, Ip(0) IL?,)
I am very much indebted to Professor A. Martin at CERN for the information about the Dias de Deus method as well as for helpful discussions.
References
(13)
where o= E,,- E,. In order to satisfy this we introduce the same variational parameters as in (7). Thus we obtain (14) .4s an illustration, in tables 1 and 2 we show the values of the energy levels and adjacent energy differences. obtained with the above methods in the case of the quartic anharmonic oscillator. The states of the harmonic oscillator are used as variational states
16
I E,). As we can see the obtained values are a remarkably good approximation. Similar estimations can be also obtained in the case of spherical three-dimensional potentials [ 8 1.
[I ] C.M. Bender, K.A. Milton, D.H. Sharp, L.M. Simmons Jr. and R. Strong, Phys. Rev. D 32 (1985) 1476. [2] L. Vazquez, Phys. Rev. D 35 (1987) 3274. [3] M.J. Rodriguez and L. Vazquez. in: Lecture notes in mathematics, Vol. 1394. Nonlinear semigroups, partial differential equations and attractors, eds. T.L. Gill and W.W. Zachary (Springer, Berlin, 1989) pp. 131-135. [4] J. Dias de Deus, A.B. Henriques and J.M.R. Pulido, Z. Phys. c7 (1981) 157. [ 51 J. Dias de Deus, Phys. Rev. D 26 ( 1982) 2782. [ 61 J. Dias de Deus and A.B. Henriques, Port. Phys. 16 ( 1985) 105. [7] L. Vazques, Z. Naturforsch. 41a (1986) 788. [ 81 L. Vazques, Spectral estimations for spherical potentials, in preparation.
I
PHYSICS
LETTERS A
12 February
1990
RELATION BETWEEN TWO VARIATIONAL METHODS TO CALCULATE THE ENERGY LEVELS ’
Luis VAZQUEZ
Departamento de Fisica Tedrica, Facultad de Ciencias Fisicas, Universidad Complutense, 28040 Madrid, Spain Received I 1 September 1989; revised manuscript Communicated by A.R. Bishop
received
I6 November
1989; accepted
for publication
6 December
I989
For one-dimensional quantum systems, we establish the relation between two energy spectrum variational methods, associated to the SchrGdinger and Heisenberg equations of motion. We show that by working in the Heisenberg picture we are able to obtain energy levels straightforwardly and not only energy differences.
The finite difference schemes to solve the Heisenberg equations can be used to obtain estimations of the energy spectrum [ l-31. Here we extend the method to obtain estimations of the energy levels directly and not only energy differences. It turns out that such a generalization is the same method that was introduced by Dias de Deus et al. [4-61 in the Schrodinger picture. The method in the Heisenberg picture is based on a variational approach to the matrix element
as well as the accuracy of the approximated eigenstates. To illustrate the method let us consider the one-dimensional quantum system:
H=$p2+ V(q).
(2)
A. Computation of the energy levels. In this case we consider the matrix element between genstate which using ( 1 ) yields
the same ei-
(E,l~(t)lE,)=(E,l~(O)lE,).
(3)
Now let us consider the operator s(t) =q( t)p( t) and with the help of the unitary
=<&1@(0)1~,)
C [-i(~,-~,)~lkl~!,
where I,!?,) and 1E,) are eigenvectors of the Hamiltonian H and 8(t) is any operator associated to the quantum system. Any discretization of the Heisenberg equations defines an approximation to eq. ( 1). On the other hand we have to consider a suitable approximation to the eigenstates 1E,). From this approximation and by consistency arguments we obtain estimations of the energy levels and the energy differences independently. The accuracy of the method is related to the accuracy of the Heisenberg finite difference scheme
0375-9601/90/$
(4)
k
Pn+
cia e Investigation
[ 71
%I+1 -4n =Pn+l > T (1)
’ Partially supported
scheme
by the Comision Interministerial of Spain under grant PB86-0005.
03.50 0 Elsevier Science Publishers
de Cien-
I -Pn =-v’(4n) r
>
(5)
we obtain
+r<&
Ip(0)2-dO)v’
(0) IEn> +@r’)
.
(6)
The scheme is accurate to order 7, so by consistency and in agreement with (3) we impose the condition
(E,lp(0)2-q(0)V’(q(O))IE,)=0.
(7)
A general way to satisfy ( 7 ) is by introducing a variational parameter y which is compatible with the commutator [q(O), p(O)]=O,
q(O)+w
B.V. (North-Holland
)
>
(8) 15
Volume
144. number
I
PHYSICS
Y(O)-Pl;*.
(9)
In this way we get
(E,,I~‘IE,,)=;I~(E,,I.~I”(IJ.~)IE,,).
LETTERS
v ( I’.\-1 .
(10)
anharmonic
State n
Exact results
Estimated
0
0.6680 2.3936 4.6968 1.3361 10.244 13.379
0.6814 2.4237 4.6850 7.291 1 10.167 13.267
1 2 3 4 5
(11)
That allows us to estimate the energy levels of H in terms of a variational parameter. The accuracy will be related to the accuracy of the variational states 1E,). Thus we get a justification of the variational method of refs. [4-61 in the context of the Heisenberg picture. On the other hand it represents a generalization of the method of refs. [ l-31 which only allows us to compute the energy differences. B. Computation of the energy>differences /l-3]. By using the same unitary difference scheme as before and considering the operators q( t ) and p( t ). we obtain
I2 February
Table I Estimation of the energy levels of the quartic lator lq4 in units 2”‘.
which is the expression of the virial theorem and it turns out to be the minimization condition in 1’of the matrix element (E,,] HI E,) with I Et, > + v/,,= fi
A
WI eJz rJJ3 w
oscil-
energies
Table 2 Estimations of the adjacent energy differences w,=E,+, the quartic anharmonic oscillator lq4 in units 2”‘.
WI)
1990
-E,
of
Exact results
Energy differences from the estimated levels
Energy differences obtained directly
1.7256 2.3032 2.6399 2.9073 3.1350
I .7423 2.2448 2.606 I 2.8759 3.1000
1.8171 2.2894 2.6207 2.8845 3.1072
(E,,,Iq(r)lE,)=(E,,Iq(O)lE,) +~(E,,, IP(O) IE,, > +@(72)
.
(E,~Ip(-c)lE,)=(E,,Ip(O)lE,) +7(E,IF(q(O))lE,)+B(r’).
(12)
By imposing the consistency of ( 1) and ( 12) up to the first order in 7 we obtain
“=i
1
1 (EmIf-(q(O))l&) ‘j’=i (E,,, Ip(0) IL?,)
I am very much indebted to Professor A. Martin at CERN for the information about the Dias de Deus method as well as for helpful discussions.
References
(13)
where o= E,,- E,. In order to satisfy this we introduce the same variational parameters as in (7). Thus we obtain (14) .4s an illustration, in tables 1 and 2 we show the values of the energy levels and adjacent energy differences. obtained with the above methods in the case of the quartic anharmonic oscillator. The states of the harmonic oscillator are used as variational states
16
I E,). As we can see the obtained values are a remarkably good approximation. Similar estimations can be also obtained in the case of spherical three-dimensional potentials [ 8 1.
[I ] C.M. Bender, K.A. Milton, D.H. Sharp, L.M. Simmons Jr. and R. Strong, Phys. Rev. D 32 (1985) 1476. [2] L. Vazquez, Phys. Rev. D 35 (1987) 3274. [3] M.J. Rodriguez and L. Vazquez. in: Lecture notes in mathematics, Vol. 1394. Nonlinear semigroups, partial differential equations and attractors, eds. T.L. Gill and W.W. Zachary (Springer, Berlin, 1989) pp. 131-135. [4] J. Dias de Deus, A.B. Henriques and J.M.R. Pulido, Z. Phys. c7 (1981) 157. [ 51 J. Dias de Deus, Phys. Rev. D 26 ( 1982) 2782. [ 61 J. Dias de Deus and A.B. Henriques, Port. Phys. 16 ( 1985) 105. [7] L. Vazques, Z. Naturforsch. 41a (1986) 788. [ 81 L. Vazques, Spectral estimations for spherical potentials, in preparation.