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D-dimensional radial Schrödinger equation for sextic potential is solved by ex- tended Nikiforov-Uvarov (NU) method analytically. Using an appropriate co- ordinate transformation, the wave equation is reduced to basic equation of the extended NU method. Then eigenstate solutions are achieved in a systematic way. It is also presented that the reduced equation which gives polynomial part of eigenfunction solution, becomes biconfluent Heun equation. Therefore the eigenfunction solutions for the potential are attained in terms of biconfluent Heun polynomials when the condition of existence of polynomial solution of biconfluent Heun equation is provided simultaneously.
Solutions of fundamental dynamical equations have received great attention because of its importance in quantum field theory, molecular physics, solid state and statistical physics.
Among the different forms of physical potentials which appear in the wave equations, those with analytical solutions are very restricted. Hence approximation solution methods or numerical methods are used to obtain Preprint submitted to Turkish Journal of Physics Nisan 2019 their solutions.
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Since exactly solvable models were still employed extensively as the starting point of numerical methods, analytical analysis of the wave equati ons have been studied widely.
In addition to studies of quatum systems in real three-dimensional space, scientists need more than (3+1) dimension in order to explain complex systems in quantum mechanics [1, 2, 3, 4, 5] Since wide-spread applications of the wave equations in higher dimensions in quantum mechanics, exact solutions of these equations have great importance. The traditional so lution method for analytic treatment of higher dimensional wave equations is wavefunction ansatz method.
However it includes more details such as numerous coordinate transformations and series expansions.
Extended NU method which eliminates the detailed procedure is obtained by changing boundary conditions of the NU method [6]. NU method is based on solving hypergeometric second order differential equations with special orthogonal functions [10]. In general NU method only succeeds for any second order differential equations which have at most three singular points.
By expanding the degrees of polynomial coefficients in the basic equation of NU method, its extended form is derived in order to ob tain analytic solution for any second order differential equations which have at most four singular points [6]. Heun equation and its important confluent forms provide the boundary conditions of the extended NU method and these type equations can be solved analytically by the method [6, 7, 8, 9]. Heun equation is a generalized second order differential equation [12]. Studies on analytical solutions of Heun-type equations which are often encountered in problems in general relativity and astrophysics, have increased extraordinarily for the last two decades(See the bibliography at the website theheunproject.org) [13]. In the present study, D-dimensional radial Schrodinger equation for sextic potential is solved analytically by the extended NU method.
It is presented that the reduced equation which gives polynomial part of eigenfunction solution, becomes biconf luent Heun equation. Moreover the condition for existence of biconfluent Heu polynomials is attained simultaneously at that stages. The paper is organized as follows; In Sec.2 explanation of the extended NU method is given briefly. exact solution of D-dimensional radial Schrodinger equation for sextic potential has 2 been obtained by the extended NU method in Sec.3. Finally, the conclusions are given in results section.
NU method is a powerful method in order to achieve exact solution of secondvorder differential equations. If any second order differential equation can bvreduced to a hypergeometric type equation given by;
ψ′′(z) + τ̃(z) σ(z)
ψ′(z) + σ̃(z) σ2(z)
ψ(z) = 0, (1)
then the equation can be solved exactly by the method. Eq.(1) is called as basic equation of the NU method. The polynomial coefficients in the basic equation have specified degrees like τ̃(z) is a polynomial of at most first-degree, σ(z) and σ̃(z) are polynomials of at most second-degree. The restriction which is related to degrees of polynomial coefficients constitute boundary conditions of the method. Expanding the boundary conditions of the NU method we proposed a new basic equation which is defined as;
ψ′′(z) + τ̃e(z) σe(z)
ψ′(z) + σ̃e(z) σ2e(z)
ψ(z) = 0, (2)
where τ̃e(z), σe(z) and σ̃e(z) are polynomials of at most second, third and fourthvdegrees respectively and e stands for ”extended” [6]. Using the transformation
ψ(z) = φe(z)y(z), (3)
and the following newly defined polynomials;
τe(z) = τ̃e(z) + 2πe(z),
h(z) = hn(z) = − n 2 τ′e(z)− n(n− 1)
σ′′e (z) + Cn,
πe(z) = σ′e(z)−τ̃e(z) 2 ± √ ( σ′e(z)−τ̃e(z) 2 )2 −σ̃e(z) + g(z)σe(z),
where
g(z) = h(z)−π′e(z),
the basic equation of the extended NU method given by Eq.(2) becomes;
σe(z)y
′′(z) + τe(z)y
′(z) + h(z)y(z) = 0.
The polynomials τe(z) with πe(z) and h(z) with g(z) which are defined in or der to rearrange the polynomial coefficients of the basic equation, are at most second-degree and first degree respectively. To determine all possible values of πe(z), the polynomial g(z) must be known explicitly. Due to the condition on the degree of πe(z) the expression under the square root sign must be square of a polynomial which is at most second-degree.
Thus g(z) must be specified appropriately [6]. Each polynomial πe(z) produces different eigenstate solutions. In order to obtain the eigenvalue solution, the relationship between h(z) and hn(z) must be set up by means of Eq.(5). For the eigenfunction solution, the function φe in the beginning transformation is given by
φ′(z) φ(z) = π(z) σ(z)
and y(z) = yn(z) which is equal to polynomial solution of Eq.(8).
Solution of D-dimensional radial Schrodinger equation for sextic potential
Schrödinger equation which conserves invariance in spatial rotation for spherically symmetric central fields, is given as for h̄ = µ = 1 in D-dimensional84
space; [ − 12 ∇2D + V (r) ]
ψ(r) = Eψ(r). (10)
The solution of the D- dimensional Schrödinger equation can be expressed as;86
ψ(r) = r−(D−1)/2U(r)Y llD−1...l1(x̂), (11)
where Y llD−1...l1(x̂) is the generalized spherical harmonics [1]. Substituting this87solution into Eq.(10) the radial ̈dinger equation can be obtained a [
d2 dr2 − l(l +D − 2) + (D − 1)(D − 3)/4 r2 ]
U(r) = −2(E − V (r))U(r). (12)
By taking κ = (l − 1 +D/2) Eq.(12) can be rearranged as [1]; [ d2 dr2 −κ 2 − 1/4 r2 ]
U(r) = −2(E − V (r))U(r). (13)
Since the anharmonic oscillator potentials are of great attention for scientists due to their important role in the evolution of many branches in quantum physics, solution of the D-dimensional radial ̈dinger equation for the spotential;
V (r) = ar6 + br4 + cr2, a > 0 (14)
is an interesting physical problem. For this potential Eq.(13) becomes;
[ d2 dr2 −κ 2 − 1/4 r2 + 2E − 2ar6 − 2br4 − 2cr2 ]
U(r) = 0. (15)
By using the transformation (a/2)1/4r2 = z, a comprehensive form of Eq.(15) is achieved as follows;
[ d2 dr2 + 1 2z d dz + −4z4 − 2b( 2a ) 3/4z3 − 2c( 2a ) 1/2z2 + 2E( 2a ) 1/4z − (κ2 − 1/4) 4z2] U(z) = 0.
This equation is compared with the basic equation of the extended NU method and the polynomial coefficients in Eq.(2) are attained in terms of physical parameters of the problem;
τ̃e(z) = 1,
σe(z) = 2z,
σ̃e(z) = −4z4 − 2b(v2vav)3/4z3 − 2c( 2vav)1/2z2 + 2E( 2 a )1/4z − (κ2 − 1/4).(17)
Substituting these polynomials into Eq.(6) the polynomial πe(z) can be deter mined for appropriately specified polynomials g(z) which makes the expression under the square root sign in Eq.(6) equal to square of a second order polyno mial;
πe1(z) = 1 2 − 2z2 − b 2 ( 2 a )3/4z −κ, (18)
πe2(z) = 12 + 2z2 + b 2 ( 2 a )3/4z + κ (19)
for g1(z) = [ b2 8 ( 2 a ) 3/2 + 2κ− c( 2a ) 1/2 ] z + b2 ( 2 a ) 3/4κ+ E( 2a ) 1/4, and
πe3(z) = 1 2 + 2z2 + b 2 ( 2 a )3/4z −κ, (20)
πe4(z) = 1 2 − 2z2 − b 2 ( 2 a )3/4z + κ (21)
for g2(z) = [ b2 8 ( 2 a )3/2 − 2κ− c( 2a ) 1/2 ] z − b2 ( 2 a ) 3/4κ+ E( 2a ) 1/4.
For the polynomial πe1(z) the analytical solution is given explicitly: The polynomials h(z) and hn(z) are determined by using Eq.(7) and Eq.(5) respec tively;
h(z) = [b2 8 ( 2 a )3/2 + 2κ− c( 2 a )1/2 − 4 ] z + b 2 ( 2 a )3/4(κ− 1) + E( 2 a )1/4, (22)
hn(z) = 4nz + nb 2 ( 2 a )3/4 + Cn1. (23)
By taking h(z) = hn(z) one can reach to the following two equations;
b2 2(2a)3/2 + κ− c√ 2a − 2 = 2n, (24)
b 2 ( 2 a )3/4(κ− n− 1) + E( 2 a )1/4 = Cn1. (25)
Eq.(24) implies a restriction on the parameters of the potential and κ. Eq.(25) gives energy eigenvalue solution when the integration constant Cn1 is equal to zero;
E = − b√ 2a (n− κ+ 1). (26)
For the eigenfunction solution, the function φe is determined from Eq.(9) recal
ling; (a/2)1/4r2 = z;
φe(r) = (a/2) 1 8 (−κ+ 1 2 )r−κ+ 1 2 exp [ − b 2 √ 2a r2 − √ 2a 4 r4 ] . (27)
When h(z) = hn(z) Eq.(8) which gives polynomial part of eigenfunction solution is reduced to the following equation;
zy′′(z)+ [ 1−κ− b 2 ( 2 a )3/4z−2z2 ] y′(z)+ [ 2nz+ b 4 ( 2 a )3/4(κ−1)+E 2 ( 2 a )1/4 ] y(z) = 0. (28)
Eq.(28) is biconfluent form of the Heun equation;
xy′′ + (1 + α− βx− 2x2)y′ + { (γ − a− 2)x− 1 2 [δ + (1 + α)β] } y = 0, (29) where α = −κ, β = b2 ( 2 a ) 3/4, γ = b 2 2(2a)3/2 − c√ 2a
and δ = −E( 2a ) 1/4.
Polynomial solution represented with N(α, β, γ, δ, z) of degree n of the biconfluent Heun equation can be achieved when γ−a−2 = 2n is provided. This relation is directly satisfied in solution processes of the extended NU method. Thus eigenfunction solution of the problem can be obtained completely;
U(r) = (a/2) 1 8 (−κ+ 1 2 )r−κ+ 1 2 exp [ − b 2 √ 2a r2 − √ 2a 4 r4 ] N ( − κ, b 2 ( 2 a )3/4, b2 2(2a)3/2 − c√ 2a ,−E( 2 a )1/4, ( a 2 )1/4r2 ) . (30)
By taking the integration constants equal to zero, other eigenstate solutions for127 the polynomials πe2(z), πe3(z) and πe4(z) can be derived by above mentioned128 procedure;
For Eq. (19);
E = − b√ 2a (n+ κ+ 1), (31) U2(r) = (a/2) 1 8 (κ+ 1 2 )rκ+ 1 2 exp [ b 4 ( 2 a ) 1 2 r2 + √ a 2 √ 2 r4 ] p2(r). (32)
For Eq. (20);
E = b√ 2a (κ− n− 1), (33) U3(r) = (a/2) 1 8 (−κ+ 1 2 )r−κ+ 1 2 exp [ b 4 ( 2 a ) 1 2 r2 + √ a 2 √ 2 r4 ]
p2(r).p3(r). (34)
For Eq. (21);
E = b√ 2a (n+ κ+ 1), (35) U4(r) = (a/2) 1 8 (κ+ 1 2 )r−κ+ 1 2 exp [ − b 2 √ 2a r2 − √ 2a 4 r4 ] N ( κ, b 2 ( 2 a )3/4, b2 2(2a)3/2 − c√ 2a ,−E( 2 a )1/4, ( a 2 )1/4r2 ) . (36)
Exact eigenstate solutions of D-dimensional radial Schrödinger equation for sextic potential have been achieved by extended NU method. It is demonstr ted that the Schrödinger equation is reduced to biconfluent Heun equation. The condition of existence of polynomial solutions of biconfluent Heun equation is satisfied simultaneously for the solutions obtained in terms of biconfluent Heunpolynomials. Elimination of detailed procedures in power series technique or wavefunction ansatz method is priority of the method. Since eigenstate solutions are achieved via extended NU method by the first time for D-dimensional wave equation in an easy way, the extended NU method can be handled as and efficient method for exact solutions of other higher dimensional wave equations for different type potentials.
Analytical Solutions of D-Dimensional Radial Schrödinger Equation for Sextic Potential using Extended Nikiforov-Uvarov Method. (2024, Feb 22). Retrieved from https://studymoose.com/document/analytical-solutions-of-d-dimensional-radial-schrodinger-equation-for-sextic-potential-using-extended-nikiforov-uvarov-method
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