The solutions of the Schr ödinger with more general exponential screened coulomb (MGESC), Yukawa potential (YP) and the sum of the mixed potential (MGESCY) have been presented using the Parametric Nikiforov-Uvarov Method (pNUM). The bound state energy eigenvalues and the corresponding un-normalized eigenfunctions expressed in terms of hypergeometric functions were obtained. Some derived equations were used to calculate numerical values for MGESC, YP, and MGESCY potentials for diatomic molecules with different screening parameters ( α ) for l = 0 and l = 1 state with V 0 = 2.75 MeV and V 1 = 2.075 MeV. We observed an increase in l value; the particles behave more repulsive than attractive. The numerical values for different l-states at different screening parameters for CO molecules ( r = 1.21282) and NO molecule ( r = 1.1508) were obtained using the bound state energy eigenvalue of the Schrodinger equation for MGESC, YP and MGESCY potentials. Potential variation with intermolecular distance ( r ) for some of the particles moving under the influence of MGESC, Yukawa and the mixed potential (MGESCY) were also studied. We also observed the variation of the MGESC potential with the radial distance of separation between the interacting particles ( r ) for different screening parameters ( α ) with V 0 = 2.75 MeV at l = 0 and l = 1 and YP with V 1 = 2.075 MeV at l = 0 and l = 1 as purely diatomic particles in nature. The energies plotted against the principal quantum number n for different values of ( α ) for both CO and NO show closed resemblance even at different values of the potential depth. The energy plots of the YP and MGESC potential for both CO and NO molecules as n →∞, and the energy E →0, shows exothermal behaviour. The energy expression for the mixed potentials V0 = 5 MeV and V1 = 10 MeV, shows that both diatomic molecules possesses similar behaviour.
The more general exponential screened coulomb (MGESC) potential expressed as
V ( r ) = − V 0 r ( 1 + ( 1 + α r ) e − 2 α r ) (1)
is a potential of great interest which on expansion comprises of the sum of coulomb potential, modified screened coulomb or the Yukawa potential and a modified exponential potential given as
V ( r ) = − V 0 r − V 0 r e − 2 α r − V 0 α e − 2 α r (2)
This potential is known to describe adequately the effective potential of a many-body system of a variety of fields such as the atomic, solid state, plasma and quantum field theory [
V ( r ) = − V 0 r − V 0 r ∑ i = 0 ∞ V i ( α r ) i (3)
The coefficient V i of Equation (3) can be obtained so that the perturbation method of [
The Yukawa potential, in atomics and particle physics expressed in the form
V yukawa ( r ) = − g 2 e − k m r r ≡ − V 0 r e − α r (4)
where g is the magnitude scaling constant, m is the mass of the affected particle, r is the radial distance to the particles, k is another scaling constant which was first proposed by Hideki Yukawa in 1935on the paper titled “On the interaction of Elementary Particles” a work in which, he explained the effect of heavy nuclei interaction on pions. According to Yukawa, the interactions of particles is not always accompanied by emission of light particles when heavy particles are transmitted from neutron state to proton state, but the liberated energy due to the transmission is taken up sometimes by another heavy particles, which will be transformed from proton state into neutron state [
V ( r ) = − ( g 1 e − α r + g 2 e − γ α r ) / r (5)
where g 1 and g 2 are coupling constants, α is the screening parameter and γ is the screening strength. By subjecting g 1 = 0 , the modified screened coulomb potential as well as its numerical calculations for the bound state is obtained. Pakdel et al. [
V ( r ) = − V 0 r ∑ i = 0 ∞ V i ( α r ) i (6)
and their corresponding wavefunctions as well as providing information regarding sample dipole polarizability. Onate and Ojunubah [
V ( r ) = − b r + r c e − α r − a e − 2 α r r 2 (7)
and obtained bound state energy eigenvalue calculations. They deduced three different energy representations for the following potentials namely the coulomb, Yukawa and inversely quadratic Yukawa as they obtained their normalized wavefunctions and energy eigenvalues, compared to other related work via the NU and AP method in their literature and the values obtained yielded reasonable result. Ita et al. [
V ( r ) = − [ C e − α r + D e − 2 α r ( 1 − e − α r ) 2 ] − V 0 e − α r r − V ′ 0 e − 2 α r r 2 (8)
From their calculations, they deduced three different potentials such as the Manning-Rosen, Yukawa and inversely quadratic Yukawa potential and obtained bound state energy eigenvalues as well as wave functions for different principal quantum number n for the s-state. In view of the relativistic quantum mechanics, a particle moving in a potential field is described particularly with the Klein-Gordon (KG) equation. Over the years numerous works have been reported concerning the Klein-Gordon equations for various kinds of potentials by using different Methods such as supersymmetry [
V ( r , q ) = 4 V 1 e − 2 α r ( 1 − q e − 2 α r ) − V 2 1 + q e − 2 α r ( 1 − q e − 2 α r ) (9)
via the NU Method. Both energy equation as well as the un-normalized wave functions expressed in terms of the Jacobi polynomial were obtained. Ikot et al. [
V ( r ) = A + B ( q + e 2 α r ) + C ( q + e 2 α r ) 2 + F b e 2 α r ( q + e 2 α r ) + G b e 2 α r ( q + e 2 α r ) 2 (10)
where A,B,C,F and G are potential parameters, q is the deformation parameter, b = e 2 α r e , r e is the distance from equilibrium position and α is the screening parameter. In their work, they obtained both bound and scattering state with energy spectrum of some special potential such as Hulthen, Manning-Rosen, Eckart and Wood-Saxon potential. Hansabadi et al. [
of the Klein-Gordon equation with position dependent mass m ( r ) = m 0 + m 1 r
as well as the wavefunction.Since then many literatures have reported different special case of potential such as, Poschl-Teller potential [
The Nikiforov-Uvarov method is based on the solutions to a second-order linear differential equation with special orthogonal function [
Given a second order differential equation of the form
ψ ″ ( s ) + τ ¯ ( s ) σ ( s ) ψ ′ ( s ) + σ ¯ ( s ) σ 2 ( s ) ψ ( s ) = 0 (11)
In order to find the exact solutions to Equation (11), we set the wavefunction as
ψ ( s ) = ϕ ( s ) χ ( s ) (12)
where ϕ ( s ) and χ ( s ) are the hypergeometric-type functions
And on substituting Equation (12) into Equation (11), then Equation (11) reduces to hypergeometric type
σ ( s ) χ ″ ( s ) + τ ( s ) χ ′ ( s ) + λ χ ( s ) = 0 (13)
where the wave function ϕ ( s ) is defined as the logarithmic derivative,
ϕ ′ ( s ) ϕ ( s ) = π ( s ) σ ( s ) (14)
where π ( s ) is at most first order polynomials.
Likewise, the hypergeometric type function ϕ ( s ) in Equation (13) for a fixed n is given by the Rodriques relation as
χ n ( s ) = B n ρ ( S ) d n d s n [ σ n ( s ) ρ ( s ) ] (15)
where B n is the normalization constant and the weight function ρ ( s ) must satisfy the condition
d d s [ σ n ( s ) ρ ( s ) ] = τ ( s ) ρ ( s ) (16)
with
τ ( s ) = τ ¯ ( s ) + 2 π ( s ) (17)
In order to accomplish the condition imposed on the weight function ρ ( s ) , it is necessary that the classical or polynomials τ ( s ) be equal to zero to some point of an interval ( a , b ) and its derivative at this interval at σ ( s ) > 0 will be negative, that is
d τ ( s ) d s < 0 . (18)
Therefore, the function π ( s ) and the parameters λ required for the NU method are defined as follows:
π ( s ) = σ ′ − τ ˜ 2 ± ( σ ′ − τ ˜ 2 ) 2 − σ ¯ + k σ (19)
λ = k + π ′ ( s ) (20)
The s-values in Equation (19) are possible to evaluate if the expression under the square root be square of polynomials. This is possible, if and only if its discriminant is zero. With this, the new eigenvalues equation becomes
λ = λ n = − n d τ d s − n ( n − 1 ) 2 d 2 σ d s 2 , n = 0 , 1 , 2 , ⋯ (21)
On comparing Equation (20) and Equation (21), we can obtain the energy eigenvalues.
The parametric form is simply using parameters to obtain explicitly energy eigenvalues and it is still based on the solutions of a generalized second order linear differential equation with special orthogonal functions. The hypergeometric NU method has shown high utility in calculating the exact energy levels of all bound states for some solvable quantum systems.
Given a second order differential equation of the form
ψ ″ ( s ) + τ ¯ ( s ) σ ( s ) ψ ′ ( s ) + σ ¯ ( s ) σ 2 ( s ) ψ ( s ) = 0 (22)
where σ ( s ) and σ ¯ ( s ) are polynomials at most second degree and τ ˜ ( s ) is first degree polynomials. The parametric generalization of the N-U method is given by the generalized hypergeometric-type equation
Ψ ″ ( s ) + c 1 − c 2 s s ( 1 − c 3 s ) Ψ ′ ( s ) + 1 s 2 ( 1 − c 3 s ) 2 [ − ϵ 1 s 2 + ϵ 2 s − ϵ 3 ] Ψ ( s ) = 0 (23)
Thus Equation (22) can be solved by comparing it with Equation (23) and the following polynomials are obtained
τ ¯ ( s ) = ( c 1 − c 2 s ) , σ ( s ) = s ( 1 − c 3 s ) , σ ¯ ( s ) = − ϵ 1 s 2 + ϵ 2 s − ϵ 3 (24)
The parameters obtainable from Equation (23) serve as important tools to finding the energy eigenvalue and eigenfunctions.
Now substituting Equation (24) into Equation (19):
σ ¯ ( s ) = c 4 + c 5 s ± [ ( c 6 − c 3 k ± ) s 2 + ( c 7 + k ± ) s + c 8 ] 1 2 (25)
where
c 4 = 1 2 ( 1 − c 1 ) , c 5 = 1 2 ( c 2 − 2 c 3 ) , c 6 = c 5 2 + ϵ 1 , c 7 = 2 c 4 c 5 − ϵ 2 , c 8 = c 4 2 + ϵ 3 , c 9 = c 3 c 7 + c 3 2 c 8 + c 6 (26)
The resulting value of k in Equation (25) is obtained from the condition that the function under the square root is square of a polynomials and it yields,
k ± = − ( c 7 + 2 c 3 c 8 ) ± 2 c 8 c 9 (27)
The new π ( s ) for k − becomes
π ( s ) = c 4 + c 5 s − [ ( c 9 + c 3 c 8 ) s − c 8 ] (28)
for the k − value,
k − = − ( c 7 + 2 c 3 c 8 ) − 2 c 8 c 9 (29)
Using Equation (17), we obtain
τ ( s ) = c 1 + 2 c 4 − ( c 2 − 2 c 5 ) s − 2 [ ( c 9 + c 3 c 8 ) s − c 8 ] . (30)
The physical condition for the bound state solution is τ ′ < 0 and thus
τ ′ ( s ) = − 2 c 3 − 2 ( c 9 + c 3 c 8 ) < 0 (31)
with the aid of Equations (20) and (21), we obtain the energy equation as
( c 2 − c 3 ) n + c 3 n 2 − ( 2 n + 1 ) c 5 + ( 2 n + 1 ) ( c 9 + c 3 c 8 ) + c 7 + 2 c 3 c 8 + 2 c 8 c 9 = 0 (32)
The weight function ρ(s) is obtained from Equation (16) as
ρ ( s ) = s c 10 − 1 ( 1 − c 3 s ) c 11 c 3 − c 10 − 1 (33)
and together with Equation (15), we have
χ n ( s ) = P n ( c 10 − 1 , c 11 c 3 − c 10 − 1 ) ( 1 − 2 c 3 s ) (34)
where
c 10 = c 1 + 2 c 4 + 2 c 8 , c 11 = c 2 − 2 c 5 + 2 ( c 9 + c 3 c 8 ) (35)
and P n ( α , β ) are the Jacobi polynomials. The second part of the wave function is obtained from Equation (14) as
ϕ ( s ) = s c 12 ( 1 − c 3 s ) − c 12 − c 13 c 3 (36)
where
c 12 = c 4 + c 8 , c 13 = c 5 − ( c 9 + c 3 c 8 ) (37)
Thus, the total wave function becomes
ψ ( s ) = N n s c 12 ( 1 − c 3 s ) − c 12 − c 13 c 3 P n ( c 10 − 1 , c 11 c 3 − c 10 − 1 ) ( 1 − 2 c 3 s ) (38)
where N n is the normalization constant.
Given the radial Schrodinger equation as [
d 2 R n l ( r ) d r 2 + 2 r d R n l ( r ) d r + 2 μ ћ 2 [ E − V ( r ) − λ ћ 2 2 μ r 2 ] R n l ( r ) = 0 , (39)
λ = l ( l + 1 ) and V ( r ) is the potential energy function given as
V ( r ) = − V 0 r ( 1 + ( 1 + α r ) e − 2 α r ) , (40)
where V 0 is the potential depth of the MGESC potential and a is an adjustable positive parameter and takes any value between zero and infinity
V e f f ( r ) = V ( r ) + l ( l + 1 ) ћ 2 2 μ r 2 = − V 0 r ( 1 + ( 1 + α r ) e − 2 α r ) + l ( l + 1 ) ћ 2 2 μ r 2 (41)
Substituting the effective potential of Equation (41) into radial Schrodinger equation of Equation (39), we obtain
d 2 R n l ( r ) d r 2 + 2 r d R n l ( r ) d r + 2 μ ћ 2 [ E + V 0 r ( 1 + ( 1 + α r ) e − 2 α r ) − l ( l + 1 ) ћ 2 2 μ r 2 ] R n l ( r ) = 0 (42)
d 2 R n l ( r ) d r 2 + 2 r d R n l ( r ) d r + 2 μ ћ 2 [ E + V 0 r + V 0 r e − 2 α r + V 0 α e − 2 α r − l ( l + 1 ) ћ 2 2 μ r 2 ] R n l ( r ) = 0 (43)
d 2 R n l ( r ) d r 2 + 2 r d R n l ( r ) d r + 1 r 2 [ 2 μ ћ 2 ( E + V 0 α e − 2 α r ) r 2 + 2 μ ћ 2 ( V 0 + V 0 e − 2 α r ) r − γ ] R n l ( r ) = 0 (44)
Introducing the following dimensional parameters,
− β 2 = 2 μ ћ 2 ( E + V 0 α e − 2 α r ) ε = 2 μ ћ 2 ( V 0 + V 0 e − 2 α r ) γ = l ( l + 1 ) (45)
d 2 R n l ( r ) d r 2 + 2 r d R n l ( r ) d r + 1 r 2 [ − β 2 r 2 + ε r − γ ] R n l ( r ) = 0 (46)
Comparing Equation (46) with Equation (32) yields the following dimensional parameters
τ ¯ ( r ) = ( c 1 − c 2 r ) = 2 , σ ( r ) = r ( 1 − c 3 r ) = r , σ ¯ ( r ) = − β 2 r 2 + ε r − γ (47)
c 1 = 2 , c 2 = c 3 = 0 , c 4 = 1 2 ( 1 − c 1 ) = − 1 2 , c 5 = 1 2 ( c 2 − 2 c 3 ) = 0 , c 6 = c 5 2 + ϵ 1 = β 2 , c 7 = 2 c 4 c 5 − ϵ 2 = ε , c 8 = c 4 2 + ϵ 3 = 1 4 + γ , c 9 = c 3 c 7 + c 3 2 c 8 + c 6 = β 2 (48)
Substituting the polynomial of Equation (47) into Equation (28), the following is obtained
π ( r ) = 1 2 ± 1 2 1 4 + β 2 r 2 − ε r + γ + k r = − 1 2 ± 1 2 4 β 2 r 2 + 4 ( k − ε ) r + 4 γ + 1 (49)
The discriminant of the expression under the square root in Equation (49) has to be zero for it to have equal roots. Therefore, we obtain
( 4 ( k − ε ) ) 2 − 4 ( 4 β 2 ) ( 4 γ + 1 ) = 0 (50)
On solving Equation (50), the following is obtained for
k ± = − ( c 7 + 2 c 3 c 8 ) ± 2 c 8 c 9 = − ε ± β ( 4 γ + 1 ) (51)
where
k − = − ε − β ( 4 γ + 1 ) (51a)
k + = − ε + β ( 4 γ + 1 ) (51b)
Substituting k± into Equation (49), gives the following four possible solutions obtained for π(r) as
π ( r ) = − 1 2 ± { β r − 1 2 ( 4 γ + 1 ) for k − = − ε − β ( 4 γ + 1 ) β r + 1 2 ( 4 γ + 1 ) for k + = − ε + β ( 4 γ + 1 ) (52)
From the four possible forms of π(r) in Equation (34), we select the one for which the function τ(s) in Equation (19) has a negative derivative. τ(s) satisfies these requirements with:
τ ( s ) = 1 − 2 β r + 4 γ + 1 (53)
and
τ ′ ( s ) = − 2 β < 0 (54)
Hence the new π ( r ) for which k − becomes
π ( r ) = − 1 2 − β r + 1 2 ( 4 γ + 1 ) (55)
π ′ ( r ) = − β (56)
From Equation (20),
λ = k + π ′ ( s ) = − ε − β ( 4 γ + 1 ) − β (57)
and also from Equation (21),
λ = λ n = − n d τ d s − n ( n − 1 ) 2 d 2 σ d s 2 = 2 n β ( n = 0 , 1 , 2 , ⋯ ) (58)
solving Equation (57) and Equation (58) explicitly, we obtain
β 2 = ( ε 2 n + 1 + ( 4 γ + 1 ) ) 2 (59)
Substituting the values of β 2 , ε and γ of Equation (45) into Equation (59) yield
E n , l = − V 0 α e − 2 α r − μ 2 ћ 2 ( ( V 0 + V 0 e − 2 α r ) n + l + 1 ) 2 (60)
To obtain the radial wave function o Equation (3), where c 3 → 0 , the following expressions are obtained
χ n ( r ) = L n c 10 − 1 ( c 11 r ) (61a)
ϕ ( r ) = r c 12 e c 13 r (61b)
where
lim c 3 → 0 P n ( c 10 − 1 , c 11 c 3 − c 10 − 1 ) ( 1 − 2 c 3 r ) = L n c 10 ( c 11 r ) (61c)
lim c 3 → 0 ( 1 − c 3 r ) − c 12 − c 13 c 3 = e c 13 s (61d)
Then, the radial wave function can be expressed
R n l ( r ) = N n r c 12 e c 13 r L n c 10 ( c 11 r ) (62)
where
c 10 = 1 + 4 l ( l + 1 ) + 1 , c 12 = − 1 2 + 1 2 4 l ( l + 1 ) + 1 , c 13 = − β , c 11 = 2 β (63)
Then the wave functions for the MGESC potential is expressed below as
R n l ( r ) = N n r − 1 2 + 1 2 4 l ( l + 1 ) + 1 e − β r L n 1 + 4 l ( l + 1 ) + 1 ( 2 β r ) (64)
if r = ( 2 β ) − 1 v and α = 1 2 4 l ( l + 1 ) + 1 (65)
Substituting Equation (65) into Equation (64), we obtain
R n l ( r ) = N n , l ( 2 β ) 1 2 − α v − 1 2 + α e − v 2 L n 1 + 2 α (v)
where N n , l is the normalization constant.
Given the radial Schrodinger equation as
d 2 R n l ( r ) d r 2 + 2 μ ћ 2 [ E − V e f f ( r ) ] R n l ( r ) , (67)
where V e f f ( r ) is the effective potential energy function given as
V ( r ) = − V 1 e − α r r + l ( l + 1 ) ћ 2 2 μ r 2 , (68)
V 1 s the potential depth of the YP and α is an adjustable positive parameter.
Inserting Equation (68) into Equation (67), we obtain
d 2 R n l ( r ) d r 2 + 2 μ ћ 2 [ E + V 1 e − α r r − l ( l + 1 ) ћ 2 2 μ r 2 ] R n l ( r ) , (69)
Equation (69) cannot be solved exactly for l ≠ 0 hence to overcome this barrier, we introduce an approximation of the pekeris type for the centrifugal [
1 r 2 = α 2 ( 1 − e − α r ) 2 ; 1 r = α ( 1 − e − α r )
making the transformation s = e − α r Equation (68) becomes
V ( s ) = − V 1 α s ( 1 − s ) + l ( l + 1 ) ћ 2 α 2 2 μ ( 1 − s ) 2 (70)
Again, applying the transformation s = e − α r to get the form that NU method is applicable, Equation (67) gives a generalized hypergeometric-type equation as
d 2 R ( s ) d s 2 + ( 1 − s ) ( 1 − s ) s d R ( s ) d s + 1 ( 1 − s ) 2 s 2 [ − ( β 2 + B ) s 2 + ( 2 β 2 + B ) s − ( β 2 + λ ) ] R ( s ) = 0 (71)
where
− β 2 = 2 μ E α 2 ћ 2 , B = 2 μ V 1 α ћ 2 (72)
Comparing Equation (71) with Equations (32), (26), (35) and (37) yields the following parameters
c 1 = c 2 = c 3 = 1 , c 4 = 0 , c 5 = − 1 2 , c 6 = 1 4 + β 2 + B , c 7 = − 2 β 2 − B , c 8 = β 2 + λ , c 9 = 1 4 + λ , c 10 = 1 + 2 β 2 + λ , c 11 = 2 + 2 ( 1 4 + λ + β 2 + λ ) c 12 = β 2 + λ , c 13 = − 1 2 − ( 1 4 + λ + β 2 + λ ) , ϵ 1 = β 2 + B , ϵ 2 = 2 β 2 + B , ϵ 3 = β 2 + λ (73)
Now using Equations (32), (72) and (73) we obtain the energy eigen spectrum of the YP as
β 2 = [ 2 λ − B + ( n 2 + n + 1 2 ) + ( 2 n + 1 ) 1 4 + λ ( 2 n + 1 ) + 2 1 4 + λ ] 2 − λ , (74)
Equation (74) can be solved explicitly and the energy eigen spectrum of YP becomes
E = − 2 μ ћ 2 [ V 1 2 ( n + l + 1 ) − α ћ 2 ( n + l + 1 ) 2 μ ] 2 (75)
We now calculate the radial wave function of the YP as follows:
Using Equation (73), the weight function ρ ( s ) of Equation (33) is given as
ρ ( s ) = s 2 U ( 1 − s ) 2 V , (76)
where U = β 2 + λ and V = 1 4 + λ
Also we obtain the wave function χ ( s ) as
χ ( s ) = P n ( 2 U , 2 V ) ( 1 − 2 s ) , (77)
where P n ( 2 U , 2 V ) are Jacobi polynomials
Lastly,
φ ( s ) = s c 12 ( 1 − c 3 s ) − c 12 − c 13 c 3 , (78)
And using Equation (14) we get
φ ( s ) = s U ( 1 − s ) 1 2 + V , (79)
we then obtain the radial wave function from the equation
R n ( s ) = N n s U ( 1 − s ) 1 2 + V P n ( 2 U , 2 V ) ( 1 − 2 s ) , (80)
where n is a positive integer and N n is the normalization constant.
Combining the potential of Equation (40) and the Yukawa potential of Equation (4), we obtain
V ( r ) = − V 0 r ( 1 + ( 1 + α r ) e − 2 α r ) − V 1 e − α r r (81)
Considering the MGESCY potential expression of Equation (81), on substitution into Equation (39) given as
d 2 R n l ( r ) d r 2 + 2 r d R n l ( r ) d r + 2 μ ћ 2 [ E + V 0 r ( 1 + ( 1 + α r ) e − 2 α r ) + V 1 r e − α r − l ( l + 1 ) ћ 2 2 μ r 2 ] R n l ( r ) = 0 (82)
we obtained both bound state solution as well as un-normalized wave function of the Schrodinger equation after solving Equation (82) explicitly by applying the NU method as
E n , l = − V 0 α e − 2 α r − μ 2 ћ 2 ( ( V 0 + V 0 e − 2 α r + V 1 e − α r ) n + l + 1 ) 2 (83)
And the radial wave function expressed as
R n l ( r ) = N n r − 1 2 + 1 2 4 l ( l + 1 ) + 1 e − β r L n 1 + 4 l ( l + 1 ) + 1 ( 2 β r ) (84)
if r = ( 2 β ) − 1 v and α = 1 2 4 l ( l + 1 ) + 1 (85)
substituting Equation (85) into Equation (84), we obtain
R n l ( r ) = N n , l ( 2 β ) 1 2 − α v − 1 2 + α e − v 2 L n 1 + 2 α ( v ) (86)
where N n , l is the normalization constant.
The aim of this report is to obtain both bound state and their corresponding eigenfunctions of the Schrodinger for the Mixed Potential (MGESCY) potential. To fulfil this aim, we now use some of the previously derived equations to calculate numerical values for the MGESC potential, Yukawa potential and also the sum of both potential known as the MGESCY potential for diatomic molecules with different screening parameters a for l = 0 and l = 1 state using python program.
Considering the bound state energy eigenvalue equation expressed in Equation (60), we obtained numerical values for different l-states at different screening parameters for CO molecule ( r = 1.21282 ) and NO molecule ( r = 1.1508 ) as shown in Tables 1-6.
To obtain the bound state energy eigen values of the Yukawa potential for diatomic molecules, we considered Equation (83) by subjecting V0 = 0 for two states
For CO and NO diatomic molecules moving under the influence of the mixed potential, we obtained the Energy eigenvalues given in Equation (83) and computed numerical values for different screening parameter.
The r values for N2 (1.0940), CO (1.21282) and NO (1.1508) were adapted from M. Karplus and R. N. Porter, Atoms and Molecules [
n | α = 0.01 | α = 0.03 | ||||
---|---|---|---|---|---|---|
CO | NO | CO | NO | |||
V 0 = 2.75 MeV | V 0 = 2.75 MeV | V 0 = 5 MeV | V 0 = 10 MeV | V 0 = 5 MeV | V 0 = 10 MeV | |
1 2 3 4 5 6 7 8 | −3.7180178 −1.6673640 −0.9496352 −0.6174293 −0.4369717 −0.3281615 −0.2575395 −0.2091213 | −3.7225783 −1.6694094 −0.9508002 −0.6181869 −0.4375080 −0.3285644 −0.2578557 −0.2093782 | −11.777558 −5.3119550 −3.0489938 −2.0015661 −1.4325929 −1.0895201 −0.6685268 −0.7141926 | −46.8312884 −20.9668751 −11.9170305 −7.7273196 −5.45142726 −4.07913595 −3.1884661 −2.57782580 | −11.8199261 −5.3310739 −3.05997575 −2.00878169 −1.43776271 −1.09345627 −0.86998815 −0.71677913 | −46.9997199 −21.0443114 −11.9599184 −7.7551422 −5.4710662 −4.0934052 −3.1996031 −2.5713199 |
n | α = 0.03 | |||
---|---|---|---|---|
CO | NO | |||
V 0 = 5 MeV | V 0 = 10 MeV | V 0 = 5 MeV | V 0 = 10 MeV | |
1 2 3 4 5 6 7 8 | −5.4720184 −3.0993611 −2.0011597 −1.4046059 −1.0449027 −0.8114416 −0.6513814 −0.5369127 | −21.7904705 −12.2998410 −7.9070355 −5.5202019 −4.0820076 −3.1481629 −2.5079220 −2.0499615 | −5.4787305 −3.1031632 −2.0036148 −1.4063294 −1.0461850 −0.8124375 −0.6521810 −0.3755048 | −21.8171974 −12.3149287 −7.9167348 −5.5275928 −4.0701559 −3.1520255 −2.5109943 −2.0524772 |
n | α = 0.01 | |||
---|---|---|---|---|
CO | NO | |||
V 1 = 2.075 MeV | V 1 = 5 MeV | V 1 = 2.075 MeV | V 1 = 5 MeV | |
1 2 3 4 5 6 7 8 | −0.5253053 −0.2334690 −0.1313263 −0.0840488 −0.0536725 −0.0422066 −0.0323158 −0.0259410 | −3.0501106 −1.3556047 −0.7625276 −0.4881771 −0.3389011 −0.2489886 −0.1906319 −0.1506227 | −0.5259573 −0.2337588 −0.1314893 −0.0841531 −0.0584397 −0.0429353 −0.0328723 −0.0259732 | −3.0538963 −1.3572873 −0.7634741 −0.4886234 −0.3393218 −0.2492976 −0.1908685 −0.1508026 |
n | α = 0.01 | |||
---|---|---|---|---|
CO | NO | |||
V 1 = 2.075 MeV | V 1 = 5 MeV | V 1 = 2.075 MeV | V 1 = 5 MeV | |
1 2 3 4 5 6 7 8 | −0.2334690 −0.1313263 −0.0840488 −0.0583673 −0.0428821 −0.0328316 −0.0259410 −0.0211221 | −1.3556047 −0.7625276 −0.4880177 −0.3389011 −0.2489806 −0.1906319 −0.1506227 −0.1220044 | −0.2337588 −0.1314893 −0.0841531 −0.0584397 −0.0429352 −0.0328723 −0.0259732 −0.0210329 | −1.3572872 −0.7634740 −0.4886234 −0.3393218 −0.2492976 −0.1908685 −0.1500969 −0.1221555 |
n | α = 0.01 | α = 0.03 | ||
---|---|---|---|---|
CO | NO | CO | NO | |
1 2 3 4 5 6 7 8 | −48.8541621 −21.7400730 −12.2501418 −7.5765943 −5.4716195 −4.0329128 −3.0991367 −2.4594302 | −48.9144403 −21.7668970 −12.2652568 −7.6735482 −5.4737100 −4.0379117 −3.1029609 −2.4619773 | −46.6610242 −20.1571763 −11.7696030 −7.5292080 −5.3085336 −3.9371500 −3.0470693 −2.4368832 | −46.8319028 −20.9195255 −11.8129699 −7.6100979 −5.3279823 −3.9515762 −3.0582366 −2.4457656 |
n,l | α = 0.01 | α = 0.03 | ||||
---|---|---|---|---|---|---|
N2 | CO | NO | N2 | CO | NO | |
1.0 2.0 2.1 3.0 3.1 3.2 | −7.04486 −3.14596 −1.78139 −1.78139 −1.14977 −0.80667 | −7.02827 −3.13859 −1.77200 −1.77200 −1.14707 −0.80477 | −7.03692 −3.14245 −1.77938 −1.77938 −1.14848 −0.80576 | −6.79940 −3.06487 −1.75779 −1.75779 −1.15280 −0.82416 | −6.75229 −3.04363 −1.74560 −1.74560 −1.14480 −0.81844 | −6.77682 −3.05469 −1.75195 −1.75195 −1.14896 −0.82142 |
We start by giving an overview of the differences in the effective potential plots against internuclear distance of some particles moving under the influence of the MGESC potential in Figures 1-3 and the Yukawa potential in
The variation of the MGESC potential with the radial distance of separation between the interacting particles (r) for different screening parameters (α) with V 0 = 2.75 MeV at l = 0 and l = 1 in Figures 1-3 and the Yukawa potential with V 1 = 2.075 MeV at l = 0 and l = 1in
Comparing the energy plots of the Yukawa and the MGESC potential for both CO and NO molecules, one can see that as n→∞, the energy obtained E→0, which describes exothermal behaviour (supporting information). For the mixed potentials where V 0 = 5 MeV and V 1 = 10 MeV , the energy expression in
The analytical solutions of both Schrodinger for the more general exponential screened coulomb plus Yukawa (MGESCY) potential have been presented via the NU method. The Nikiforov-Uvarov (NU) method employed in the solutions enables us to explore an effective way of obtaining the energy eigenvalues and their corresponding eigenfunctions of the Schrödinger equations for any l-state. Finally, we calculate the energies and also obtained graphs of the MGESC potential, Yukawa potential and the mixed potential for diatomic molecules by means of Equations (60) and (83), for the l-states at different values of the screened parameters (Supporting information). The explicit values of the energy at different values of the screened parameter are shown in Tables 1-6.
The authors greatly appreciate the effort of Professor Benedict IseromIta and Nelson Nzeata-Ibe and other colleagues in Mathematics and Computer Science Department, University of Calabar, Nigeria for their relevant scientific input.
Ita, B.I., Louis, H., Akakuru, O.U., Magu, T.O., Joseph, I., Tchoua, P., Amos, P.I., Effiong, I. and Nzeata, N.A. (2018) Bound State Solutions of the Schrödinger Equation for the More General Exponential Screened Coulomb Potential Plus Yukawa (MGESCY) Potential using Nikiforov-Uvarov Method. Journal of Quantum Information Science, 8, 24-45. https://doi.org/10.4236/jqis.2018.81003