Development and refinement of the Variational Method based on Polynomial Solutions of Schrödinger Equation

Fethi Maiz

doi:10.1515/eng-2020-0052

Article Open Access

Development and refinement of the Variational Method based on Polynomial Solutions of Schrödinger Equation

Fethi Maiz

Published/Copyright: May 30, 2020

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From the journal Open Engineering Volume 10 Issue 1

Abstract

The variational method is known as a powerful and preferred technique to find both analytical and numerical solutions for numerous forms of anharmonic oscillator potentials. In the present study, we considered certain conditions for the choice of the trial wave function. The current form of the trial wave function is based on the possible polynomial solutions of the Schrödinger equation. The advantage of our modified variational method is its ability to reduce the calculation steps and hence computation time. Also, we compared the results provided by our modified method with the results obtained by different methods in general but particularly Numerov method for the same problem.

Keywords: Variational Methods; Anharmonic Oscillator; Energy Eigenvalues; Polynomial Solutions; Schrödinger Equation

1 Introduction

The precise solution of the Schrodinger equation is possible only in few cases such as infinite square well and harmonic oscillator potentials. However, the complete spectra of the anharmonic oscillators are not fully solved yet. In most cases, the conventional approximate methods discussed in most standard textbooks are either unsatisfactory or computationally complicated. Several techniques of approximations have been used over the years to determine the spectral energies of various anharmonic oscillators. These approximations lead to developing many models for the study of many problems in physics which are tedious computationally. However, we observed some very simple and effective models in literature for the same purpose [1].

Here, we present some insight from available literature about variational principle together with appropriate approximations for the electron-electron interactions which are the basis for most practical approaches to solving the Schrödinger equation in condensed matter physics.

For the generalized anharmonic oscillator in D dimensions, Popescu et al. [2, 3] used approximation method along with variational method for the calculation of the ground energy state and first even-parity excited state of a single-well and found improved results. The energy levels of one dimensional quartic anharmonic oscillator were obtained by using neural network system [4], however, the analytical solutions were given by the triconfluent Heun functions [5]. Later on, Popescu et al. [6] considered a different form of the successive variational method based on a solution of a differential equation. They successfully combined the variational method which uses variational global parameter with the finite element method for the study of the generalized anharmonic oscillator in D dimensions [7]. Further, Cooper et al. [8] in another work used a newly suggested algorithm of Gozzi, Reuter and, Thacker to determine the excited states of one-dimensional systems. They determined approximated eigenvalues and eigenfunctions of the anharmonic oscillator. While Karl & Novikov [9] calculated the energies of excited states for two- and three-particle systems with arbitrary blocking potential within the framework of a simple variational approach. In anotherwork, Mei, W. N. [10] used variational method and analytical wave functions which have extremely accurate expectation values for the quartic or sextic oscillators. The Variational Method was also applied within the context of Super-symmetric Quantum Mechanics [11, 12, 13] to provide information to Morse and Hulthén potentials for several diatomic molecules and the results were in agreement with established results.

Borges et al. [14] suggested a method for constructing trial eigenfunctions for excited states to be used in the variational method. This method is a generalization of the one that uses super-potential to obtain the trial functions for the ground state. The first four eigenvalues for a quartic double-well potential were calculated at different values of the potential parameter.

By means of a collocation approach based on little Sinc functions (LSF), Amore and Fernández [15] obtained accurate eigenvalues and eigenfunctions of the stationary Schrödinger equation for systems of coupled oscillators. Gribov and Prokof’eva [16] proposed a variational method of the solutions of anharmonic problems in the theory of molecular vibrations in curvilinear coordinates taking into account the kinematic anharmonicity. VEGA and FLORES [17] used the variational method and super-symmetric quantum mechanics to calculate in an approximate way, the eigenvalues, eigenfunctions and wave functions at the origin of the Cornell potential. Payandeh and Mohammadpour used the Delta method to evaluate the energy of ground and excited stationary states in quantum mechanics. The advantage of the Delta method compared to the variational method is its simplicity and reduction of the calculation procedures [18].

P. M. Gaiki and P. M. Gade [19] demonstrated how a freeware, SAGE, can be employed for the variational solution of simple and complex Hamiltonians in one dimension to estimate the ground state energy. FM Fernández and J Garcia [20] considered Rayleigh-Ritz variational computations with non-orthogonal basic sets with the correct asymptotic behavior. This approach is illustrated by the construction of appropriate basis sets for one-dimensional models such as the two double-well oscillators recently examined by other authors. The convergence rate of the variational method is considerably greater than that of orthogonal.

S. Khuri and A.Wazwaz [21] applied an amended variational scheme for the solution of a second-order nonlinear boundary value problem. However, the variational iteration method was used for solving linear and nonlinear ODEs and scientific models with variable coefficients [22, 23] and the asymptotic iteration method was applied to certain quasinormal modes and non Hermitian systems [24].

In this paper, we start by formulating the problem in Sec. 2. Then, we show in Sec. 3 and 4, that under certain conditions, the harmonic plus linear term and the sextic anharmonic potentials energy is easily solvable by the variational method. In Sec. 5, we explore the non-polynomial exactly solvable quartic potential. The development of this variational method is explained in Sec. 6. Sec. 7 is devoted to some applications and discussions about anharmonic potentials. Finally, the conclusion of the work is presented in Sec. 8.

2 Problem formulation

2.1 Schrödinger equation

Let us consider the one-dimensional time-independent Schrödinger equation:

(1)−ℏ22md2ψnxdx2+Vxψnx=Enψnx

Where E_n is the system’s energy and ψ_n is the wave function (n^th eigenstates). V (x) = α₁x + α₂x² + α₃x³ + α₄x⁴ + · · · + α_N x^N, α2=12k>0.k = mω², m is the particle mass and ω the angular frequency. V (x) is anharmonic potential energy, the coefficients α_p_<N are real and α_N is positive. This potential energy form was chosen because using the Lagrange interpolation, we can approach any potential energy with high accuracy in each continuous potential energy to polynomial potential energy [25]. Dividing Eq. (1) by ħω, putting λ=ℏ2ωmand En=Enℏω(to express the energy parameters in the unit of ħω₀) and moving to the variable, y=2ωmℏ1/2x,we obtain the dimensionless equation:

(2)d2ψnydy2+[En−(α1ℏωλ12y+α2ℏωλy2+α3ℏωλ32y3+α4ℏωλ2y4+⋯+αNℏωλN/2yN)]ψny=0,

and

(3)d2ψnydy2+En−v(y)ψny=0

wherevy=b1y+b2y2+b3y3+b4y4+…+biyi+⋯+bNyN,bi=∝iℏωλi/2

and i = 1, 2, . . . N

The Hamiltonian system is consequently:

(4)H=−d2dy2+v(y)

Under certain conditions applied on the parameters of the potentials as described in earlier work of Maiz et al. [26], some potentials are exactly solvable. However, if there is no exact solution, then we applied an approximation approach like perturbation theory, variational and WKB methods and numerical method such as Numerov method, Airy function approach, and the asymptotic iteration method. The variational method (VM) will be modified to carry out calculations in this work.

To explore the conditions of the existence of polynomial solutions, we use the trial normalized wavefunctionsin the form of:

(5)ψny=Afnyexp−hy

The expectation value of the energy is E = 〈ΨHΨ〉 . The substitution of the wave function Ψ_n leads to the following relation:

(6)E=〈Afnyexp−hy−d2dy2+v(y)Afnyexp−hy〉

Applying the Hamiltonian gives:

(7)E=〈Afn(y)e(−h(y))|(−d2f(y)dy2+2dh(y)dydf(y)dy+fn(y)(d2h(y)dy2−dh(y)dy2+v(y)fn(y))|Ae(−h(y))〉

Eq. 7 states the conditions of the polynomial solution existence; it depends on the potential energy expression.

2.2 The variational method review

The variational method provides a simple way to place an upper bound on the ground state energy of any quantum system and it is particularly useful when trying to demonstrate that existence of bound states. In some cases, it can also be used to estimate higher energy levels. We start with a quantum system with Hamiltonian H, which has a discrete spectrum: H|n⟩=En|n⟩with n = 0, 1, 2, . . ., the energy eigen values are ordered such that E_n ≤ E_n₊₁. The simplest application of the variational method places an upper bound is on the value of the ground state energy E₀. If we consider an arbitrary normalized state |Ψ⟩,(⟨ΨΨ⟩=1),the expectation value of the energy obeys the inequality: E=Ψ|H|Ψ≥E0.

Let’s consider a family of normalized states, |Ψ(α)〉,depending on some number of parameters α. The expectation value of energy, E(α) obeys E(α) ≥ E₀ for all α.

The minimum value of E(α) which is E(α_min) with dE(α)dα|α=αmin=0,

offers the upper bound E₀ ≤ E(α_min). This is the soul of the variational method. But the variational method does not give the difference between the founded and exact values of energies ΔE = E(α_min)− E₀. This difference disappears when the chosen trial wave function coincides with the exact problem solution. This means that the choice of the trial wave function is decisive and, in this case, the variational method gives remarkably accurate results.

3 The harmonic oscillator potential

Considering the well-known case of harmonic oscillator potential energy plus linear term v (y) = b₁y + b₂y², the coefficient b₁ is real and b₂ is positive.

The node-less eigenfunction (ground state) is given by ψ₀ (y) = Af₀ (y) exp (−h (y)), with f₀ (y) = 1 and h (y) = ∑p=12Napyp.Introducing this solution in Eq. 7 leads to the equation:

(8)E=−4a22+b2ψ0y2ψ0+−4a1a2+b1ψ0|y|ψ0+2a2−a12

Note that for

a2=b22,anda1=b12b2,E=2a2−a12

and the eigenenergy value is constant and equal to the exact value of energy

Ee=b2−b124b2

however, the ground state function is ψ₀ (y) = A exp −b12b2y−b22y2.

These results were found to be the same as previously published for the same potential energy and level [27].

4 The closely solvable sextic anharmonic oscillator potential energy

Considering the case of sextic anharmonic oscillator potential v (y) = b₁y + b₂y² + b₃y³ + b₄y⁴ + b₅y⁵ + b₆y⁶, the coefficients b_i_<6 is real and b₆ is positive.

The node-less eigenfunction (ground state) is given by ψ₀ (y) = Af₀ (y) exp (−h (y)), with f₀ (y) = 1 and h (y) = ∑p=12Napyp. Introducing this solution in Eq. 7 offers the energy expectation value:

(9)E=b6−16a42ψ0y6ψ0+b5−24a3a4ψ0y5ψ0+b4−16a2a4−9a32ψ0y4ψ0+b3−12a2a3−8a1a4ψ0y3ψ0+b2+12a4−4a22−6a1a3ψ0y2ψ0+b1+6a3−4a1a2ψ0|y|ψ0+2a2−a12

Fora4=b61/24,a3=b56b612,a2=b44b612−b5216b632,anda1=b32b612−b4b54b632+b5316b652

and the two conditions:

b2=4a22+6a1a3−12a4andb1=4a1a2−6a3,

The expectation value of energy remains E_e = 2a2−a12,it is constant and constitutes an exact solution. The polynomial coefficients a_i_=1,2,3,4 are the only function of the potential parameters b_i_=3,4,5,6 and not the function of the two firsts parameters b_i₌₂,3. Furthermore, for any quadruplet of real (a₁, a₂, a₃, a₄ > 0), the sextic anharmonic potential energy:

v6ey=(4a1a2−6a3)y+(4a22+6a1a3−12a4)y2+(12a2a3+8a1a4)y3+(9a32+16a2a4)y4+24a3a4y5+16a42y6admits an exact ground state energy E0=2a2−a12.For non-closely solvable sextic anharmonic oscillator potential energy, we use quartic anharmonic potential, this will be explained in the forthcoming section.

5 The quartic anharmonic potential

Studying the case of quartic anharmonic oscillator potential v (y) = b₁y + b₂y² + b₃y³ + b₄y⁴, the coefficients b_i_<4 is real and b₄ is positive. It is well known that this potential has no exactly polynomial solutions [26]. To estimate the energy expectation value, we apply the normalized wave function: ψ_n (y) = Af_n (y) exp (−h (y)).

The node-less eigenfunction is given by ψ0(y) = Af₀ (y) exp (−h (y)), with f₀ (y) = 1 and hy=∑p=12Napyp.Eq. (7) leads to the following expression:

(10)E0=−16a42ψ0y6ψ0+−24a3a4ψ0y5ψ0+b4−16a2a4−9a32ψ0y4ψ0+b3−12a2a3−8a1a4ψ0y3ψ0+b2+12a4−4a22−6a1a3ψ0y2ψ0+b1+6a3−4a1a2ψ0|y|ψ0+2a2−a12

For

(11)a4=13a22+13a2a12−112a1b1−112b2,

(12)a3=23a1a2−16b1

and the two conditions:

(13)b4=283a12a22−103a1a2b1+14b12+163a23−43a2b2

(14)b3=323a1a22−2a2b1+83a13a2−23a12b1−23a1b2,

The expectation value of energy is

E0=−16a42ψ0y6ψ0+−24a3a4ψ0y5ψ0+2a2−a12.

Equations (11-12) show the expressions of the coefficients a₃ and a₄ as a function of the coefficients a₁, a₂ and the two fists potential energy parameters b₁ and b₂. It should be noted here that the coefficients a₁ and a₂ are the acceptable solutions of equations (13-14), they are a function of the potential’s parameters. The minimum of E₀ in the vicinity of the coefficients a_i_(i=1,2,3,4) values give a good estimation of the ground state energy value. This minimum is noted E_vm and equal to

−16a4∗2φ0y6φ0+−24a3∗a4∗φ0y5φ0+2a2∗−a1∗2

where the associated wave function is φ (y) = A exp (−h (y)), with hy=∑p=12Nap∗yp.For each integer p, the value of the coefficient ap∗is in the vicinity of the a_p one.

6 The variational method development

As we noted in section 2.a, using the Lagrange interpolation, a given potential energy v (y) may be approached with high accuracy to a polynomial potential energy v_n (y) of degree n [25]:

vy→vny=∑i=1nbiyi

For non-exactly solvable energy potential, we considered just the fourth firsts terms for a given potential energy, v₄ (y), which is the corresponding quartic anharmonic potential energy. As mentioned in the preceding paragraph, for the ground state energy the expectation value of energy is:

E0=−16a42ψ0y6ψ0+−24a3a4ψ0y5ψ0+2a2−a12.

The coefficients a_i_(i=1,2,3,4) were derived in paragraph 4. Using the variational method for the given potential energy v (y), the expectation energy level value is E =

φ|H|φ.The trial normalized wave function is φ (y) = A exp (−h (y)), with hy=∑p=12Nap∗yp.For the given potential v (y), E may be obtained by differentiating this previous expectation value of energy with respect to the four coefficients ai=1,2,3,4∗.Canceling different equations leads to a system of four equations on various ai=1,2,3,4∗:

∂E∂a1∗=0∂E∂a2∗=0∂E∂a3∗=0∂E∂a4∗=0

To calculate the ground state energy value E for the given potential energy v (y), we solve this previous system for the coefficients ai=1,2,3,4∗in the vicinity of the previous coefficients a_i_=1,2,3,4 for the corresponding quartic anharmonic potential energy. By limiting the calculations in the vicinity of the previous coefficients a_i₌₁,2,3,4, we were successful in reducing computation time. Finally, we deduce the estimated ground state energy value: E_vm = Emin(ai=1,2,3,4∗)and compare this result with available ones for the same potential energy [27].

7 Results and discussions

We applied our developed variational method (DVM) to study three classes of potential designed by the following eight potentials energies:

v1y=y+y2,v2y=y2+y4+y6,v3y=y2−y3+y4−y5+y6,v4y=y+y2+y3+y4+y5+y6,v5y=−y+2y2−3y3+4y4−5y5+6y6,v6y=y+y2+y3+y4,v7y=y2+y3+y4,andv8y=y3+y4.

The well-known harmonic plus linear term potential constitutes the first class; it is exactly solvable potential as described in paragraph 3. It was chosen to make the first test to our developed variational method. The second class is presented by the four sextic anharmonic potentials, while the last quartic formed the last class of potential energy. Profiles of these potentials energy are shown in Figure 1.

Figure 1

Studied potentials energy profiles

7.1 Harmonic plus linear term

The harmonic plus linear potential energy is expressed by v₁ (y) = y + y², it is exactly solvable as shown in paragraph 3. The ground state energy value is Ee=b2−b124b2,which numerically equal to 0.75. The corresponding exactly solvable potential energy is clearly the same: v_e (y) = v₁ (y) = y + y². The associated normalized wavefunction is written as φ (y) = A exp (−h (y)), with hy=∑p=12Nap∗yp,and N = 1. The coefficients a1∗and a2∗values are obtained which are nearly equal as 0.5000000002 and 0.4999999999 respectively. Furthermore, the obtained ground state energy value is E_vm = 0.749999 while Numerov method gives E_nm = 0.750009. These values are in agreement with the exact solution derived in paragraph 3. The relative energy difference between them ΔE/E is in the range of 10⁻⁴. Physical parameters for the potential energy v₁ (y) = y + y² are collected in Table 1.

Table 1

Physical parameters for the potential energy v₁ (y) = y + y²

potential energy		polynomial coefficients		ground state energy values			10⁵ΔE/E
v₁ (y)	y + y²	a1∗=0.500000	a2∗=0.499999	E_vm	E_nm	E_e	1.3

v _e(y)	y + y²	a₁ = 0.5	a₂ = 0.5	0.749999	0.750009	0.750

7.2 Sextic potential energy

Some sextic potential energy is exactly solvable. This depends on relations between the potential energy parameters [26]. Sextic potential were found reliable as a potential model for quark confinement in quantum chromodynamics [28]. Furthermore, this model is very important when trying to understand many theories including molecular spectroscopy, quantum-tunneling time, and field theories [29]. In paragraph 4, the study of the sextic anharmonic potential energy demonstrate that under some conditions on its parameters, it may be exactly solvable. In general, we applied numerical and approximated methods to solve the Schrödinger equation. To verify our developed variational method for the non-exactly solvable anharmonic sextic potential energy, we propose to study four examples of potential energy. We start with the symmetric sextic potential as v₂ (y) = x² + x⁴ + x⁶. The corresponding exactly solvable potential energy is:

vey=x2+x4+12826105077x6with(a1,a2,a3,a4)=0,1943927166,0,11955136873

and E_e = 1.431127. For the potential energy v₂ (y), the associated normalized wavefunction is φ (y) = A exp (−h (y)), with hy=∑p=12Nap∗yp,and N = 2. The coefficients a1∗and a3∗values are null because of the potential’s symmetry. The coefficients a2∗anda4∗values are obtained as 1131314116and208112474,respectively.

Furthermore, the obtained ground state energy value is E_vm = 1.615237 while, Numerov method offers E_nm = 1.614889. These values agree very closely and the relative energy difference is a less than 10⁻³. Physical parameters for the potential energy v₂ (y) = y² + y⁴ + y⁶ are collected in Table 2.

Table 2

Physical parameters for the potential energy v₂ (y) = y² + y⁴ + y⁶

potential energy	polynomial coefficients				ground state energy values			103ΔEE
v₂ (y)	a1∗=0	a2∗=0.715	a3∗=0	a4∗=0.087	E_vm	E_nm	E_e	2

v_e(y)	a₁ = 0	a2=1943927166	a₃ = 0	a2=11955136873	1.615237	1.614889	1.431127

Secondly, we studied the anharmonic sextic potential v₃ (y) = y² − y³ + y⁴ − y⁵ + y⁶. The corresponding potential energy exactly solvable is:

vey=y2−0.999999y3+0.999999y4−1130054669y5+1023793042y6,

where the exact ground state energy value is E_e = 1.308642 and the associated wavefunction: ψny= Af_n (y) exp (−h (y)), with hy=∑p=12Napyp,and (a₁, a₂, a₃, a₄) = (−948441429,11951756,−355934268,232027977).

For the potential energy v₃ (y), the associated normalized wavefunction is: φ (y) = A exp (−h (y)), with h (y) = ∑p=12Nap∗yp.The quadruplet (a1∗,a2∗,a3∗,a4∗)is equal to −948441429,11951756,−1458493615,1723986619.

Furthermore, the obtained ground state energy value is E_vm = 1.472721 while, Numerov method offers E_nm = 1.471141. These values agree very closely and the relative energy difference is approximately equal to 10-3. Table 3. shows these parameters.

Table 3

Physical parameters for the potential energy v₃ (y) = x² − x³ + x⁴ − x⁵ + x⁶

potential energy	polynomial coefficients				ground state energy values			103ΔEE
v₃ (y)	a1∗=−948441429	a2∗=11951756	a3∗=−1458493615	a4∗=1723986619	E_vm	E_nm	E_e	1

v_e(y)	a1=−948441429	a2=11951756	a3=−355934268	a4=232027977	1.472721	1.471141	1.308642

Thirdly, we explore the anharmonic sextic potential defined by: v₄ (y) = y + y² + y³ + y⁴ + y⁵ + y⁶. The corresponding potential energy exactly solvable is:

vey=y+y2+0.999y3+y4+1381185343y5+209116849y6,

where the exact ground state energy value is E_e = 1.049869. Additionally, the obtained ground state energy value is equal to E_vm = 1.230 although, the numerical Numerov method gives: E_nm = 1.220. The relative energy difference is in the range of 10-3 (see Table 4).

Table 4

Physical parameters for the potential energy v₄ (y) = y + y² + y³ + y⁴ + y⁵ + y⁶

potential energy	polynomial coefficients				ground state energy values			103ΔEE
v₄ (y)	a1∗=55558102365	a2∗=1985029529	a3∗=1985029529	a4∗=803846095	E_vm	E_nm	E_e	2

v _e(y)	a1=55558102365	a2=1985029529	a3=7758101329	a4=13321151254	1.223769	1.220971	1.049869

Finally, we investigate the anharmonic sextic potential: v₅ (y) = −y + 2y² − 3y³ + 4y⁴ − 5y⁵ + 6y⁶. The corresponding potential energy exactly solvable as:

vey=−y+2y2−2.999999y3+3.999999y4−5306156984y5+3544242859y6,

where the exact ground state energy value is E_e = 1.812756. Additionally, the obtained ground state energy value is E_vm = 2.045453 although, the numerical Numerov method gives E_nm = 2.045895. The relative energy difference between these results is less than 10-3. The results are collected in Table 5.

Table 5

Physical parameters for the potential energy v₅ (y) = −y + 2y² − 3y³ + 4y⁴ − 5y⁵ + 6y⁶

potential energy	polynomial coefficients				ground state energy values			103ΔEE
v₅ (y)	a1∗=−444010019	a2∗=2682723733	a3∗=−1849860217	a4∗=107987250000	E_vm	E_nm	E_e	1

v_e(y)	a1=−80994164489	a2=3175230899	a3=−26735156656	a4=24910109571	2.045453	2.043194	1.812756

7.3 Quartic potential energy

Anharmonic octic potential energy has no exactly polynomials solutions. For this class of potentials, we start by studying the given potential energy: v₆ (y) = y + y²+y³+y⁴. The corresponding potential energy is the exactly solvable potential as:

vey=y+y2+0.999y3+y4+1381185343y5+209116849y6,

where the exact ground state energy value is E_e = 1.049869897. Moreover, the developed variational and the Numerov methods evaluate, the ground state energy as E_vm = 1.034086 and E_nm = 1.034035 respectively. The relative energy difference between these results is less than 10⁻⁴. We have shown our findings in Table 6.

Table 6

Physical parameters for the potential energy v₆ (y) = y + y² + y³ + y⁴

potential energy	polynomial coefficients				ground state energy values			103ΔEE
v₆ (y)	a1∗=67843125000	a2∗=3360549991	a3∗=13459140633	a4∗=8807100000	E_vm	E_nm	E_e	1

v_e(y)	a1=55558102365	a2=1985029529	a3=7758101329	a4=13321151254	1.035545	1.034035	1.049869897

The non-exactly polynomials solutions potential energy: v₇ (y) = y² + y³ + y⁴ is also investigated. v_e (y) = y² + 0.999999y³+0.999999y⁴+ 1130054669y5+1023793042y6presents the corresponding potential energy. For this potential, the exact ground state energy is evaluated to be, E_e = 1.308642. In addition, the developed variational and the Numerov methods estimate the ground state energy is equal to E_vm = 1.311036 and E_nm = 1.31026 respectively. Table 7 shows our estimated results.

Table 7

Physical parameters for the potential energy v₇ (y) = y² + y³ + y⁴

potential energy	polynomial coefficients				ground state energy values			104ΔEE
v₇ (y)	a1∗=20603100000	a2∗=1407820687	a3∗=292131250	a4∗=708294891	E_vm	E_nm	E_e	5

v_e(y)	a1=948441429	a2=11951756	a3=355934268	a4=232027977	1.311036	1.31026	1.308642

Finally, we explore the non-exactly polynomials solutions potential energy given by: v₈ (y) = y³ + y⁴. Here the corresponding potential energy is found to be:

vey=10−9y2+0.999999y3+0.999999y4+1211340787y5+830645161y6.

The exact ground state energy value is closed to E_e = 0.912247.Moreover, the use of approximated and numerical methods offers the two following values for the ground

state energy: E_vm = 0.905322 and E_nm = 0.905322. Physical parameters values are collected in Table 8.

Table 8

Physical parameters for the potential energy v₈ (y) = y³ + y⁴

potential energy	polynomial coefficients				ground state energy values			104ΔEE
v₈ (y)	a1∗=2460381013	a2∗=1974838491	a3∗=793476381	a4∗=621064357	E_vm	E_nm	E_e	2

v _e(y)	a1=27698206	a2=1338126081	a3=487142204	a4=50284469003	0.907941	0.905322	0.912247

8 Conclusion

The polynomial solutions of the Schrödinger equation for some anharmonic potential energy helped us to our modified and improved variational method. This study shows that under appropriate conditions on the potential’s parameters, the choice of the suitable trial wave function becomes easy. Focusing on the anharmonic potential problem, we have also presented a comparison between the solutions provided by this developed variational method and the results for the same problem by different methods such as the Numerov method. The obtained results for the ground state energy values are found to be accurate. Finally, our results agreed well with those available in literature [27, 30, 31]. We believe that this study will encourage the researchers and scientists for further investigation of general polynomial potentials.

Acknowledgement

The authors would like to express their gratitude to King Khalid University, Saudi Arabia for providing administrative and technical support.

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Received: 2020-01-06

Accepted: 2020-04-29

Published Online: 2020-05-30

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/eng-2020-0052

Keywords for this article

Variational Methods; Anharmonic Oscillator; Energy Eigenvalues; Polynomial Solutions; Schrödinger Equation

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