ORIGINAL ARTICLE: Lina Ellis, Ikumi Ellis, Christoph Koutschan, Sergei K. Suslov, On Potentials Integrated by the Nikiforov-Uvarov Method, arXiv:2303.02560v3 [quant-ph]. https://doi.org/10.48550/arXiv.2303.02560.
Article Abstract: We discuss basic potentials of the nonrelativistic and relativistic quantum mechanics that can be integrated in the Nikiforov and Uvarov paradigm with the aid of a computer algebra system. This approach may help the readers to study modern analytical methods of quantum physics.
1. Introduction
1. Introduction
We discuss basic potentials of the nonrelativistic and relativistic quantum mechanics that can be integrated in the Nikiforov and Uvarov paradigm with the aid of the computer algebra system Mathematica.
In Section 2 below, the general formulas are derived. They are stored in global variables that will be used in all the subsequent sections. For this purpose, allow Mathematica to evaluate all initialization cells. After that you will be able to run each integrable case independently from the others.
Use the following command, if you need to save/export graphs created by Mathematica into the current notebook directory:
In Section 2 below, the general formulas are derived. They are stored in global variables that will be used in all the subsequent sections. For this purpose, allow Mathematica to evaluate all initialization cells. After that you will be able to run each integrable case independently from the others.
Use the following command, if you need to save/export graphs created by Mathematica into the current notebook directory:
SetDirectory[NotebookDirectory[]]
2. Summary of the Nikiforov-Uvarov approach
2. Summary of the Nikiforov-Uvarov approach
This is the generalized equation of hypergeometric type (2.1).
eqnGHyp=u''[x]+([x]/σ[x])*u'[x]+([x]/(σ[x])^2)*u[x]
τ
σ
Out[]=
u[x][x]
σ
2
σ[x]
τ
′
u
σ[x]
′′
u
This is substitution (2.2) into the generalized equation of hypergeometric type,yielding the transformation to (2.9).
eqnGHypTrans1=Collect[(eqnGHyp/.uFunction[x,φ[x]*y[x]])/φ[x],(y|y'|y'')[x],Expand]
Out[]=
′
y
τ
σ[x]
2[x]
′
φ
φ[x]
′′
y
σ
2
σ[x]
τ
′
φ
σ[x]φ[x]
′′
φ
φ[x]
Differential equation (2.10) for φ.
eqnPhi[x_]=(φ'[x](ϖ[x]/σ[x])*φ[x])
Out[]=
′
φ
ϖ[x]φ[x]
σ[x]
This is substitution (2.10) into (2.9), yielding (2.12).
eqnGHypTrans2=Collect[eqnGHypTrans1//.Derivative[n_][φ][x]D[eqnPhi[x][[2]],{x,n-1}],(y|y'|y'')[x],Together]
Out[]=
(2ϖ[x]+[x])[x]
τ
′
y
σ[x]
y[x](+[x]+ϖ[x][x]+σ[x][x]-ϖ[x][x])
2
ϖ[x]
σ
τ
′
ϖ
′
σ
2
σ[x]
′′
y
The quantization rule (2.7).
Energy=λ+n*τ'[x]+(n*(n-1)/2)*σ''[x]
Out[]=
λ+n[x]+(-1+n)n[x]
′
τ
1
2
′′
σ
Differential equation (2.8) for ρ.
eqnRho[x_]=(D[(σ[x]*ρ[x]),x]τ[x]*ρ[x])
Out[]=
σ[x][x]+ρ[x][x]ρ[x]τ[x]
′
ρ
′
σ
Definition (2.13).
σ
Out[]=
2
ϖ[x]
σ
′
ϖ
τ
′
σ
Equation (2.13) with the substitution (2.16) yields (2.15).
Simplify[[x]λ*σ[x]/.ϖ'[x]λ-k]
σ
Out[]=
2
ϖ[x]
σ
τ
′
σ
Solution of Eqn. (2.15) is the quadratic formula (2.6).
Solve[%,ϖ[x]]
Out[]=
ϖ[x]-[x]-[x],ϖ[x]-[x]+[x]
1
2
τ
-4(-kσ[x]+[x])+
+
σ
2
([x]-[x])
τ
′
σ
′
σ
1
2
τ
-4(-kσ[x]+[x])+
+
σ
2
([x]-[x])
τ
′
σ
′
σ
Write the quadratic polynomial p[x] in general form.
Normal[Series[p[x],{x,0,2}]]
Out[]=
p[0]+x[0]+[0]
′
p
1
2
2
x
′′
p
Determine the remainder from completing the square (2.20).
PolynomialRemainder[%,x+p'[0]/p''[0],x]//Together
Out[]=
-[0]+2p[0][0]
2
′
p
′′
p
2[0]
′′
p
Its numerator yields (2.22).
rem=-Numerator[%]/.pFunction[x,q[x]+kσ[x]]
Out[]=
2
([0]+k[0])
′
q
′
σ
′′
q
′′
σ
General quadratic equation (2.23).
quadEqn=a*k^2+2b*k+c
Out[]=
c+2bk+a
2
k
Compute the coefficients a, b, c in (2.24) - (2.26).
TableForm[abcSol=First[Solve[CoefficientList[quadEqn-rem,k]0,{a,b,c}]]]
Out[]//TableForm=
a 2 ′ σ ′′ σ |
b ′ q ′ σ ′′ q ′′ σ |
c 2 ′ q ′′ q |
Discriminant d, as given in (2.29).
(σ[0]q''[0]-σ''[0]q[0])^2-2(σ'[0]q''[0]-σ''[0]q'[0])(σ[0]q'[0]-σ'[0]q[0])
Out[]=
2
(σ[0][0]-q[0][0])
′′
q
′′
σ
′
q
′
σ
′
σ
′′
q
′
q
′′
σ
Save it for later use.
abcd=Append[{a,b,c}/.abcSol,%];
Check correctness of the discriminant formula.
Expand[abcd[[4]]b^2-ac/.abcSol]
Out[]=
True
Formula (2.27), solution of equation (2.23) when a=0.
Solve[quadEqn0/.a0,k]
Out[]=
k-
c
2b
We defined to be this solution.
k
0
k0=k/.First[%]/.abcSol
Out[]=
-[0]-2q[0][0]
2
′
q
′′
q
2([0][0]-σ[0][0]-q[0][0])
′
q
′
σ
′′
q
′′
σ
Formula (2.28), solution of equation (2.23) when .
a≠0
Solve[quadEqn0,k]
Out[]=
k,k
-b--ac
2
b
a
-b+-ac
2
b
a
We denote these two solutions by and .
k
1
k
2
{k1,k2}=k/.Reverse[%]/.abcSol
Out[]=
[0]-2σ[0][0]-[0][0]+σ[0][0]+q[0][0]+-([0]-2q[0][0])([0]-2σ[0][0]),[0]-2σ[0][0]-[0][0]+σ[0][0]+q[0][0]--([0]-2q[0][0])([0]-2σ[0][0])
1
2
′
σ
′′
σ
′
q
′
σ
′′
q
′′
σ
2
([0][0]-σ[0][0]-q[0][0])
′
q
′
σ
′′
q
′′
σ
2
′
q
′′
q
2
′
σ
′′
σ
1
2
′
σ
′′
σ
′
q
′
σ
′′
q
′′
σ
2
([0][0]-σ[0][0]-q[0][0])
′
q
′
σ
′′
q
′′
σ
2
′
q
′′
q
2
′
σ
′′
σ
Define q[x] as in (2.19).
Define p[x] as in (2.18).
Define the two solutions for ϖ of (2.15) in terms of rootp = Sqrt[p[x]], as in (2.6).
Define τ[x] as in (2.11).
Define λ as in (2.16).
Protect global variables to prevent the user to inadvertently change them.
3. Harmonic Oscillator
3. Harmonic Oscillator
Stationary Schrödinger equation for the harmonic oscillator
Stationary Schrödinger equation for the harmonic oscillator
Those are data for harmonic oscillator:
Those are equations (2.24)-(2.26) and (2.29) for harmonic oscillator
These are equations (2.27) and (2.30) for harmonic oscillator when a=0:
These are equations (2.6) and (2.5) etc. for harmonic oscillator:
This is equation (2.16) for harmonic oscillator
Solving equations (2.10) and (2.8) for harmonic oscillator
This is solution of equation (2.7) for the energy levels of harmonic oscillator
Time-dependent Schrödinger equation for the harmonic oscillator
Time-dependent Schrödinger equation for the harmonic oscillator
This is the rescaled PDE for the Harmonic Oscillator.
Standard harmonic oscillator solutions
Graphs of the first Hermite functions
Graphs of the first Hermite functions
Addendum: Rescaling of the stationary Schrödinger equation for the harmonic oscillator
Addendum: Rescaling of the stationary Schrödinger equation for the harmonic oscillator
This is the ODE (3.1) for the Harmonic Oscillator.
This is Substitution (3.3) for the Harmonic Oscillator.
This is the result: rescaled equation (3.4)
This is another try for the chain rule.
Here ξ[x] = x*Sqrt[(m*ω)/ℏ].
Once again, this is the result: rescaled equation (3.4)
6. Nonrelativistic Coulomb Problem
6. Nonrelativistic Coulomb Problem
Those are data for nonrelativistic Coulomb problem
Those are equations (2.24)-(2.26) and (2.29) for nonrelativistic Coulomb problem
These are equations (2.28) and (2.30) for nonrelativistic Coulomb problem
These are equations (2.6) and (2.5) etc. for nonrelativistic Coulomb problem
This is equation (2.16) for nonrelativistic Coulomb problem
Solving equations (2.10) and (2.8) for nonrelativistic Coulomb problem
This is solution of equation (2.7) for the energy levels of nonrelativistic Coulomb problem
Radial wave functions
Radial wave functions
Normalization (6.15) check
Radial density in hydrogen atom
Radial density in hydrogen atom
normalization check
normalization check
Addendum: Rescaling of the nonrelativistic Schrödinger equation for Coulomb potential
Addendum: Rescaling of the nonrelativistic Schrödinger equation for Coulomb potential
This is the left hand side of radial equation (6.2).
Using F[r] = u[x[r]] with x[r] = r*(m*e^2/ℏ^2).
This is the left-hand side of the radial equation (6.4)
check
17. Modified Hulthén Potential
17. Modified Hulthén Potential
Those are data for the modified Hulthén potential
Those are equations (2.24)-(2.26) and (2.29) for the modified Hulthen potential
These are equations (2.28) and (2.30) for the modified Hulthen potential
Use substitution for κ from (17.6).
These are equations (2.6) and (2.5) etc. for the modified Hulthen potential
This is equation (2.16) for the modified Hulthen potential
Solving equations (2.10) and (2.8) for the modified Hulthen potential
This is the solution of equation (2.7) for the energy levels of the modified Hulthen potential
Let us transform the numerator in (17.13)-(17.14) in order to obtain the previous result (17.15):
The answer in (17.15) is :
They are equal to each other
Comparison of Morse and modified Hulthén potentials
Comparison of Morse and modified Hulthén potentials
You do not have to do that, once again, if you already did set up the subdirectory for your Mathematica files.
Finding the minimum of the Morse potential.
Values of the Morse potential and its derivatives at the minimum.
Modified Hulthén potential (17.1).
Finding the minimum of the modified Hulthén potential.
Values at the minimum of modified Hulthén potential.
Solving equations (17.23).
New parameters of the Morse potential.
The Morse potential for the new parameters.
Special case of the Morse potential for the graph below (Figure 2).
A slightly different form.
Check
Check (17.24): we got here
Equation (17.24) in the paper:
Check
Special case for the graph below (Figure 2)
Molecule HCl below (eV energy units)
Monotonicity of φ[b], when b≳1.
Data in Table 15.
Compare expansions (16.4) and (18.3)
With data from Table 15.
Conclusion: the difference in approximation is large for light biatomic molecules and practically negligible for the heavy ones.
CITE THIS NOTEBOOK
CITE THIS NOTEBOOK
On Potentials Integrated by the Nikiforov-Uvarov Method
by Lina Ellis, Ikumi Ellis, Christoph Koutschan, Sergei Suslov
Wolfram Community, STAFF PICKS, April 12 2023
https://community.wolfram.com/groups/-/m/t/2897057
by Lina Ellis, Ikumi Ellis, Christoph Koutschan, Sergei Suslov
Wolfram Community, STAFF PICKS, April 12 2023
https://community.wolfram.com/groups/-/m/t/2897057