1. Introduction
There has been some interest in the calculation of the eigenvalues and eigenfunctions of rather simple one-dimensional Hamiltonians with double-well potentials because they are supposed to be useful for the calculation of the probability density for the one-dimensional Fokker-Planck equation[
1,
2]. In a paper appeared recently, Batael and Drigo Filho[
3] proposed a variational method that is supposed to yield upper bounds to all the eigenvalues of the Hamiltonian. They constructed the variational wavefunctions by means of supersymmetric quantum mechanics (SSQM) and the Gram-Schmidt (GM) orthogonalization method but did not provide a plausible proof for those bounds. From now, on we will refer to this approach as SSQMGS.
The well known Rayleigh-Ritz variational method (RRVM), discussed in most textbooks on quantum chemistry[
4,
5], is known to provide upper bounds to all the eigenvalues of a given Hamiltonian operator[
6] (see also a recent simpler proof of the RRVM upper bounds[
7]). In our opinion, it is interesting to compare the SSQMGS and the RRVM because they apparently exhibit somewhat similar features.
In
Section 2 we discuss the one-dimensional quantum-mechanical models and the approximate method used for obtaining their eigenvalues. In
Section 3 we compare and discuss the eigenvalues provided by SSQMGS and RRVM.
2. Models and methods
Batael and Drigo Filho[
3] obtained some eigenvalues of a particular class of simple quantum-mechanical models of the form
that are amenable for the application of SSQM. We can provide an alternative sound reason for this choice without resorting to SSQM.
From a square-integrable exponential function
we can obtain a reference potential
as follows
For some particular values of the coefficients we can choose so that . In this case both and are exact and the SSQMGS is expected to yield the most accurate results.
For example, when
the requirement for
is that the coefficients
satisfy
while for
we have two restrictive conditions
The role of is irrelevant because it is just a shift of .
Since the potential is parity invariant, then the eigenfunctions of H have definite parity (they are either even or odd). This fact enables us to apply the approximate methods to each symmetry thus reducing considerably the computation time.
The RRVM is based on trial functions of the form
where
is a complete set of basis functions. The variational principle leads to a secular equation of the form
where the
matrices
and
have elements
and
, respectively, and
is a column vector of the expansion coefficients
[
4,
5]. The approximate eigenvalues
,
, are roots of the secular determinant
that yields the characteristic polynomial
. It can be proved that
, where
is an exact eigenvalue of
H[
6,
7]. For each
we obtain
and it can be proved that
if
. Obviously, in the case of present one-dimensional toy models there are no degenerate states and
if
. In particular, when the basis set
B is orthonormal then
(the
identity matrix).
It follows from
[
7] that
that resembles the SSQMGS expression (Equation (
8)) reported by Batael and Drigo Filho[
3] without proof. More precisely, one can prove rigorously that the bounds proposed by these authors apply to the ground state and first-excited state that have the smallest energy for each symmetry (provided, of course, that the trial functions have the appropriate symmetry). However, as far as we know, there is no proof for the remaining states (as in the case of the RRVM[
6,
7]). The outcome of upper bounds and orthogonal approximate wavefunctions make the RRVM and the SSQMGS look similar, with the difference that in the latter case the upper bounds have not been rigorously proved, except in the two cases just mentioned.
When the potential is parity invariant, we can apply the RRVM to each kind of symmetry thus reducing the dimension of the matrices involved in the calculation.
The simplest basis set is given by the eigenfunctions of the Harmonic oscillator
Although the asymptotic behaviour of the eigenfunctions of is quite different from that of the problems discussed here, such eigenfunctions exhibit two advantages. First, we already know that this orthonormal basis set is complete and, second, the matrix elements can be calculated exactly without difficulty.
In principle, we can resort to a set of basis functions with suitable asymptotic behaviour[
8] but it is not necessary for present discussion.
There are many ways of obtaining a suitable value of
; here we arbitrarily resort to the condition
3. Results and discussion
We first consider the example with
,
,
and
that allows an exact ground state
with
. The ansatz used by Batael and Drigo Filho[
3] is a curious linear combination of functions with no definite parity which is not convenient for a parity-invariant Hamiltonian operator. They chose the nonlinear parameter
that leads to the exact ground-state eigenfunction
(of even parity) but it is not clear why they kept the nonlinear parameter
in the exponential factors of the other trial functions
. Although Batael and Drigo Filho mentioned the advantage of the separate treatment of even and odd states, their ansätze do not reflect this fact. Another curious feature of their approach is the choice of Legendre Polynomials that are known to be orthogonal in the interval
when in the present case the variable interval is
.
Table 1 and
Table 2 show the convergence of the lowest RRVM eigenvalues towards results that are supposed to be accurate up to the last digit. We estimated
from equation (
10) with
and arbitrarily chose an integer value close to the real root. Although the rate of convergence depends on
, the choice of an optimal value of this adjustable parameter is not that relevant. As expected, the RRVM eigenvalues converge from above[
4,
5,
6,
7]. Note that Batael and Drigo Filho[
3] did not report the eigenvalues
of this model but
as in reference[
1]. Although the RRVM requires about 25 basis functions for a ten-digit accuracy, the calculation is extremely simple because it only requires the diagonalization of matrices
with elements
that can be obtained analytically. On the other hand, the SSQMGS is considerably cumbersome because it requires the numerical calculation of all the integrals and minimization of the approximate energy that is a function of nonlinear parameters. Besides, the accuracy of these results cannot be improved any further.
In the second example we also have
, but since
,
,
and
then there is no exact ground state.
Table 3 and
Table 4 show the the convergence of the lowest RRVM eigenvalues towards results that are also supposed to be accurate up to the last digit. We estimated
as in the previous example. In this case, we appreciate that the SSQMGS eigenvalues
and
do not provide upper bounds which suggests that the Equation (
8) of Batael and Drigo Filho[
3] does not hold. This fact is not surprising because, as stated above, such bounds were not proved rigorously.
The last example is given by
,
,
,
,
,
and
. In this case there is an exact ground state
with
. The RRVM eigenvalues are shown in
Table 5 and
Table 6 together with those of Batael and Drigo Filho.
It is worth comparing the performances of the RRVM and the SSQMGS. The former approach provides eigenvalues of unlimited accuracy (depending only on hardware and software facilities) that converge towards the exact energies from above. On the other hand, the accuracy of the SSQMGS eigenvalues is determined by the accuracy of the initial ansatz
. Batael and Drigo Filho chose a particular class of potentials for which one can obtain the exact
or at least a sufficiently accurate trial function with the appropriate asymptotic behaviour. Such models are of the form illustrated in
Section 2. Batael and Drigo Filho reported more digits than the actual accuracy of their results. Present RRVM eigenvalues are even more accurate than those used by Batael and Drigo Filho as benchmark. While it has already been proved that the RRVM provides upper bounds to the energies of all the states[
6,
7] such proof is lacking in the case of the SSQMGS and we have already pointed out two cases in which the latter approach fails to provide such bounds. As is well known, one counterexample is sufficient to prove an statement false.
References
- Borges G R P, Filho E D, and Riccota R M 2010 Physica A 389 3892.
- Shizgal B D 2016 J. Stat. Phys. 164 1379.
- Batael H O and Filho E D 2023 J. Phys. A 56 285201.
- Pilar F L 1968 Elementary Quantum Chemistry (McGraw-Hill, New York).
- Szabo A and Ostlund N S 1996 Modern Quantum Chemistry (Dover Publications, Inc., Mineola, New York).
- MacDonald J K L 1933 Phys Rev. 43 830.
- Fernández, F.M. Dimensionless equations in non-relativistic quantum mechanics. arXiv 2020, arXiv:2005.05377. [Google Scholar]
- Fernández F M and Garcia J 2014 Cent. Eur. J. Phys. 12 554.
Table 1.
RRVM even-state eigenvalues for , , with
Table 1.
RRVM even-state eigenvalues for , , with
N |
|
|
|
|
5 |
0.02 |
4.677918651 |
14.53469054 |
28.3757404 |
10 |
6.6 × 10−6
|
4.62986462 |
14.35154075 |
27.52416887 |
15 |
1.5 × 10−8
|
4.629826578 |
14.3509522 |
27.51712162 |
20 |
8.6 × 10−11
|
4.629826494 |
14.35095078 |
27.51709995 |
25 |
9.7 × 10−13
|
4.629826493 |
14.35095078 |
27.5170999 |
30 |
2.5 × 10−15
|
4.629826493 |
14.35095078 |
27.5170999 |
|
|
2.314913246 |
7.17547539 |
13.75854995 |
Ref.[3] |
0 |
2.31799 |
7.18145 |
13.7670 |
Table 2.
RRVM odd-state eigenvalues for , , with
Table 2.
RRVM odd-state eigenvalues for , , with
N |
|
|
|
|
5 |
0.8655650394 |
9.111949632 |
20.98289274 |
36.23196314 |
10 |
0.8459004855 |
9.007614525 |
20.55620684 |
35.17488491 |
15 |
0.8458893236 |
9.007557826 |
20.55577168 |
35.16841201 |
20 |
0.845889291 |
9.007557632 |
20.55577029 |
35.16839427 |
25 |
0.8458892907 |
9.00755763 |
20.55577028 |
35.16839416 |
30 |
0.8458892907 |
9.00755763 |
20.55577028 |
35.16839416 |
|
0.4229446453 |
4.503778815 |
10.27788514 |
17.58419708 |
Ref.[3] |
0.42388 |
4.50813 |
10.2852 |
17.5941 |
Table 3.
RRVM even-state eigenvalues for , , , with
Table 3.
RRVM even-state eigenvalues for , , , with
N |
|
|
|
|
5 |
-14.39416156 |
-2.418081882 |
6.897731829 |
23.83165889 |
10 |
-14.47163202 |
-2.523730405 |
6.599680377 |
21.61724028 |
15 |
-14.47165595 |
-2.523911539 |
6.59851881 |
21.60602543 |
20 |
-14.47165597 |
-2.523911704 |
6.598517525 |
21.60600654 |
25 |
-14.47165597 |
-2.523911705 |
6.598517524 |
21.60600652 |
30 |
-14.47165597 |
-2.523911705 |
6.598517524 |
21.60600652 |
Ref.[3]. |
-14.4483 |
-2.42763 |
6.596869 |
21.56765 |
Table 4.
RRVM odd-state eigenvalues for , , , with
Table 4.
RRVM odd-state eigenvalues for , , , with
N |
|
|
|
|
5 |
-14.3640557 |
-0.4515691057 |
13.89792265 |
32.55127969 |
10 |
-14.42792517 |
-0.6900912613 |
13.35318022 |
30.73875482 |
15 |
-14.42794579 |
-0.690175821 |
13.3524621 |
30.7269972 |
20 |
-14.42794583 |
-0.6901759943 |
13.3524612 |
30.72698225 |
25 |
-14.42794583 |
-0.6901759952 |
13.35246119 |
30.72698222 |
30 |
-14.42794583 |
-0.6901759952 |
13.35246119 |
30.72698222 |
Ref.[3]. |
-14.4135 |
-0.65821 |
13.36402 |
|
Table 5.
RRVM even-state eigenvalues for , , , , , with
Table 5.
RRVM even-state eigenvalues for , , , , , with
N |
|
|
|
|
5 |
0.09 |
4.573017185 |
16.36066839 |
34.15352004 |
10 |
0.002 |
4.32310851 |
15.61645666 |
31.68651075 |
15 |
6.1 × 10−5
|
4.315907553 |
15.58461237 |
31.54805825 |
20 |
1.8 × 10−7
|
4.315700166 |
15.58363087 |
31.54320834 |
25 |
2.1 × 10−7
|
4.31569472 |
15.58360629 |
31.54308125 |
30 |
7.6 × 10−10
|
4.315694041 |
15.58360331 |
31.54306785 |
35 |
1.0 × 10−9
|
4.315694019 |
15.58360321 |
31.54306732 |
40 |
1.1 × 10−10
|
4.315694016 |
15.58360319 |
31.54306723 |
45 |
8.5 × 10−12
|
4.315694015 |
15.58360319 |
31.54306722 |
50 |
7.6 × 10−13
|
4.315694015 |
15.58360319 |
31.54306722 |
Ref.[3]. |
0 |
4.31612 |
15.5851 |
31.5460 |
Table 6.
RRVM odd-state eigenvalues for , , , , , with
Table 6.
RRVM odd-state eigenvalues for , , , , , with
N |
|
|
|
|
5 |
1.256573678 |
9.855150686 |
24.44520554 |
44.68121271 |
10 |
1.048870482 |
9.357073321 |
23.02789536 |
41.29594435 |
15 |
1.046988529 |
9.351398959 |
23.00064258 |
41.16116412 |
20 |
1.046927491 |
9.351217587 |
22.9997951 |
41.15687172 |
25 |
1.046922323 |
9.351202299 |
22.99972988 |
41.15661593 |
30 |
1.046922115 |
9.351201593 |
22.99972602 |
41.15659472 |
35 |
1.046922092 |
9.351201522 |
22.99972569 |
41.15659332 |
40 |
1.046922091 |
9.351201519 |
22.99972568 |
41.15659324 |
45 |
1.046922091 |
9.351201519 |
22.99972568 |
41.15659323 |
50 |
1.046922091 |
9.351201519 |
22.99972568 |
41.15659323 |
Ref.[3]. |
1.04703 |
|
|
|
|
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).