On the Variational Treatment of a Class of Double-Well Oscillators

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

$$H=-\frac{d^2}{d{x}^2}+V\left(x\right),V\left(x\right)=\sum _{j=0}^{2K-1}{A}_{2j}{x}^{2j},K=2,3,$$

(1)

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

$${\psi }_0\left(x\right)=e^{-F\left(x\right)},$$

(2)

we can obtain a reference potential $V_0\left(x\right)$ as follows

$$\frac{{\psi }_0^{\prime \prime }}{{\psi }_0}=F^{\prime 2}-F^{\prime \prime }=V_0-E_0,F\left(x\right)=\sum _{j=0}^K{F}_j{x}^{2j},F_K>0.$$

(3)

For some particular values of the coefficients $A_j$ we can choose $F_j$ so that $V_0\left(x\right)=V\left(x\right)$. In this case both ${\psi }_0\left(x\right)$ and $E_0$ are exact and the SSQMGS is expected to yield the most accurate results.

For example, when $K=2$ the requirement for $V_0\left(x\right)=V\left(x\right)$ is that the coefficients $A_j$ satisfy

$$4a_2{a}_6-a_4^2+12a_6^{3/2}=0,$$

(4)

while for $K=3$ we have two restrictive conditions

$$\begin{array}{ccc}\hfill 4a_{10}{a}_6{a}_8-40a_{10}^{5/2}-8a_{10}^2{a}_4-a_8^3& =& 0,\hfill \\ \hfill 16a_{10}^2{a}_6^2-64a_{10}^3{a}_2-96a_8{a}_{10}^{5/2}-8a_{10}{a}_6{a}_8^2+a_8^4& =& 0.\hfill \end{array}$$

(5)

The role of $A_0$ is irrelevant because it is just a shift of $E_0$.

Since the potential $V\left(x\right)$ 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

$${\phi }^{\left[N\right]}=\sum _{j=0}^{N-1}{c}_j{f}_j,$$

(6)

where $B=\left\{f_0,f_1,\dots \right\}$ is a complete set of basis functions. The variational principle leads to a secular equation of the form

$$\left(\mathbf{H}-E\mathbf{S}\right)\mathbf{c},$$

(7)

where the $N\times N$ matrices $\mathbf{H}$ and $\mathbf{S}$ have elements $H_{ij}=\left.〈f_i\right|H\left|f_j\right.〉$ and $S_{ij}=〈f_i\left|f_j\right.〉$, respectively, and $\mathbf{c}$ is a column vector of the expansion coefficients $c_j$[4,5]. The approximate eigenvalues $E_i^{\left[N\right]}$, $i=0,1,\dots ,N-1$, are roots of the secular determinant $\left|\mathbf{H}-E\mathbf{S}\right|$ that yields the characteristic polynomial $p\left(E\right)$. It can be proved that $E_n^{[N-1]}>E_n^{\left[N\right]}>E_n$, where $E_n$ is an exact eigenvalue of H[6,7]. For each $E_n^{\left[N\right]}$ we obtain ${\phi }_n^{\left[N\right]}$ and it can be proved that $〈{\phi }_i^{\left[N\right]}\left|{\phi }_j^{\left[N\right]}\right.〉=0$ if $E_i^{\left[N\right]}\ne E_j^{\left[N\right]}$. Obviously, in the case of present one-dimensional toy models there are no degenerate states and $〈{\phi }_i^{\left[N\right]}\left|{\phi }_j^{\left[N\right]}\right.〉=0$ if $i\ne j$. In particular, when the basis set B is orthonormal then $\mathbf{S}=\mathbf{I}$ (the $N\times N$ identity matrix).

It follows from $\left.〈{\phi }_i^{\left[N\right]}\right|H\left|{\phi }_j^{\left[N\right]}\right.〉=〈{\phi }_i^{\left[N\right]}\left|{\phi }_j^{\left[N\right]}\right.〉E_i^{\left[N\right]}$[7] that

$$E_i^{\left[N\right]}=\frac{\left.〈{\phi }_i^{\left[N\right]}\right|H\left|{\phi }_i^{\left[N\right]}\right.〉}{〈{\phi }_i^{\left[N\right]}\left|{\phi }_i^{\left[N\right]}\right.〉}>E_i,$$

(8)

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

$$H_{HO}=-\frac{d^2}{d{x}^2}+{\omega }^2{x}^2.$$

(9)

Although the asymptotic behaviour of the eigenfunctions of $H_{HO}$ 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 $H_{ij}$ 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 $\omega$; here we arbitrarily resort to the condition

$$\frac{d}{d\omega }\sum _{j=0}^M{H}_{jj}\left(\omega \right)=0,M\le N.$$

(10)

3. Results and discussion

We first consider the example with $K=2$, $A_0=1$, $A_2=A_4=-2$ and $A_6=1$ that allows an exact ground state ${\psi }_0$ with $E_0=0$. 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 $c_0=0$ that leads to the exact ground-state eigenfunction ${\psi }_0$ (of even parity) but it is not clear why they kept the nonlinear parameter $c_n$ in the exponential factors of the other trial functions ${\psi }_n$. 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 $[-1,1]$ when in the present case the variable interval is $(-\infty ,\infty )$.

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 $\omega$ from equation (10) with $M=10$ and arbitrarily chose an integer value close to the real root. Although the rate of convergence depends on $\omega$, 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 $E_n$ of this model but ${\lambda }_n=E_n/2$ 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 $\mathbf{H}$ with elements $H_{ij}$ 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 $K=2$, but since $A_0=0$, $A_2=-26$, $A_4=6$ and $A_6=1$ 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 $\omega$ as in the previous example. In this case, we appreciate that the SSQMGS eigenvalues $E_4$ and $E_6$ 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 $K=3$, $A_0=0$, $A_2=3/2$, $A_4=-5/2$, $A_6=1/4$, $A_8=-1/2$ and $A_{10}=1/4$. In this case there is an exact ground state ${\psi }_0$ with $E_0=0$. 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 ${\psi }_0$. Batael and Drigo Filho chose a particular class of potentials for which one can obtain the exact ${\psi }_0$ 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.