[Paper Review] Exact Solutions of the Klein-Gordon Equation for q-Deformed Manning-Rosen Potential via Asymptotic Iteration Method

Tapas Das|arXiv (Cornell University)|Sep 4, 2014

Quantum Mechanics and Non-Hermitian Physics3 citations

TL;DR

This paper presents exact analytical solutions for the one-dimensional Klein-Gordon equation with the q-deformed Manning-Rosen potential using the Asymptotic Iteration Method (AIM). It derives bound state energy eigenvalues and eigenfunctions in terms of hypergeometric functions, with results for special cases like Pöschl-Teller, Rosen-Morse, and Eckart potentials confirmed to match known literature, demonstrating AIM's accuracy and efficiency for relativistic hyperbolic potentials.

ABSTRACT

Asymptotic iteration method (AIM) is used to find the exact analytical solutions of the one dimensional Klein-Gordon equation for the q-deformed Manning-Rosen potential with equal Lorentz vector and scalar potential. The bound state eigenfunctions are obtained in terms of the hypergeometric functions. Using the present results energy eigenvalues and corresponding eigenfunctions of the special cases like Poschl-Teller potential, Rosen-Morse potential and Eckart type potential are derived before concluding the work.

Motivation & Objective

• To derive exact analytical solutions for the one-dimensional Klein-Gordon equation with the q-deformed Manning-Rosen potential under equal Lorentz scalar and vector potentials.
• To apply the Asymptotic Iteration Method (AIM) as a powerful and efficient technique for solving relativistic wave equations with hyperbolic-type potentials.
• To obtain bound state energy eigenvalues and corresponding eigenfunctions in closed form using hypergeometric functions.
• To validate the method by recovering known results for special cases such as Pöschl-Teller, Rosen-Morse, and Eckart-type potentials.
• To demonstrate the accuracy and reliability of AIM for analytically solvable relativistic quantum systems with hyperbolic potentials.

Proposed method

• Transform the one-dimensional Klein-Gordon equation into a second-order differential equation of the form $ y^{(k+2)} = ilde{ ho}_k y' + s_k y $, where $ ilde{ ho}_k $ and $ s_k $ are recursively generated.
• Apply the Asymptotic Iteration Method (AIM) by iterating the recurrence relations $ ilde{ ho}_k = ilde{ ho}_{k-1}' + s_{k-1} + ilde{ ho}_0 ilde{ ho}_{k-1} $ and $ s_k = s_{k-1}' + s_0 ilde{ ho}_{k-1} $.
• Impose the termination condition $ rac{s_k}{ ilde{ ho}_k} = rac{s_{k-1}}{ ilde{ ho}_{k-1}} = ext{const} $ to obtain a first-order differential equation for the wave function.
• Solve the resulting first-order equation to express the wave function in terms of hypergeometric functions using the wave function generator.
• Use the transformation $ z = qs $ to map the differential equation into the hypergeometric form $ z(1-z)G'' + [u - z(v+w+1)]G' - vwG = 0 $.
• Derive the unnormalized total wave function as $ ilde{ ho}(s) = N s^c (1 - qs)^ ho {}_2F_1(-n, 2(c+ ho)+n, 1+2c; qs) $, where $ N $ is the normalization constant.

Experimental results

Research questions

• RQ1 Can the Asymptotic Iteration Method (AIM) yield exact analytical solutions for the Klein-Gordon equation with the q-deformed Manning-Rosen potential?
• RQ2 What are the bound state energy eigenvalues and corresponding eigenfunctions for the q-deformed Manning-Rosen potential under equal Lorentz scalar and vector potentials?
• RQ3 How do the results for the q-deformed Manning-Rosen potential reduce to known solutions for special cases like Pöschl-Teller, Rosen-Morse, and Eckart-type potentials?
• RQ4 To what extent does the AIM provide accurate and efficient solutions for relativistic hyperbolic-type potentials compared to existing methods?
• RQ5 Can the method generate closed-form solutions for analytically solvable relativistic systems using hypergeometric functions?

Key findings

• The energy eigenvalues for the q-deformed Manning-Rosen potential are given by $ E_n^2 = m^2 - rac{eta^2}{ ho^2} ho^2 ho^2 $, where $ ho = rac{1}{2} ho ho ho ho $, and $ ho = rac{1}{2} ho ho ho ho $, with $ ho $ defined via $ ho = rac{1}{2} ho ho ho ho $.
• The bound state eigenfunctions are expressed in terms of hypergeometric functions as $ ilde{ ho}(s) = N s^c (1 - qs)^ ho {}_2F_1(-n, 2(c+ ho)+n, 1+2c; qs) $, providing a closed-form solution.
• For the Pöschl-Teller potential ($ q = -1, V_1 \to -V_1, V_2 = 0 $), the energy eigenvalues reduce to $ E_n^2 = m^2 - ho^2 ho^2 $, consistent with literature.
• For the Rosen-Morse potential ($ q = -1, V_1 \to -V_1 $), the energy eigenvalues remain $ E_n^2 = m^2 - ho^2 ho^2 $, with eigenfunctions matching known forms.
• For the Eckart-type potential ($ q = 1, V_2 \to -V_2 $), the energy eigenvalues are $ E_n^2 = m^2 - ho^2 ho^2 $, and eigenfunctions are $ ilde{ ho}(s) = N s^c (1 - s)^ ho {}_2F_1(-n, 2(c+ ho)+n, 1+2c; s) $.
• The results confirm that AIM is a reliable, accurate, and efficient method for solving exactly solvable relativistic quantum systems with hyperbolic potentials, yielding closed-form solutions.