The Klein-Gordon (KG) equation for the two-dimensional scalar-vector harmonic oscillator plus Cornell potentials in the presence of external magnetic and Aharonov-Bohm (AB) flux fields is solved using the wave function ansatz method. The exact energy eigenvalues and the wave functions are obtained in terms of potential parameters, magnetic field strength, AB flux field, and magnetic quantum number. The results obtained by using different Larmor frequencies are compared with the results in the absence of both magnetic field (
The exact solution of Schrödinger equation (SE) and the relativistic wave equations for some physical potentials are very important in many fields of physics and chemistry since they contain all the necessary information for the quantum system under investigation. The hydrogen atom and the harmonic oscillator are usually given in textbooks as two of several exactly solvable problems in both classical and quantum physics [
Recently, some authors have studied the bound states of the
It is well-known that the non-relativistic quantum mechanics is an approximate theory of the relativistic one. However, when a particle moves in a strong potential field, the relativistic effects must be considered which give the corrections for non-relativistic quantum mechanics [
Recently, the Schrödinger equation is solved exactly for its bound states (energy spectrum and wave functions) [
Very recently, we studied the scalar charged particle exposed to relativistic scalar-vector Killingbeck potentials in presence of magnetic and Aharonov-Bohm flux fields and obtained its energy eigenvalues and wave functions using the analytical exact iteration method [
The structure of this paper is as follows. We study effect of external uniform magnetic and AB flux fields on a relativistic spinless particle (antiparticle) under equal mixture of scalar and vector Killingbeck potentials in Section
The Klein-Gordon atom for the spinless particle with mass
The positive energy states require that
In the non-relativistic limit, when
As shown in Figure
The KG effective potential function for (a)
In this case of
In this section, we discuss some special cases of interest from our general solution.
(i) If we set
(ii) Setting
In Figure
The Schrödinger effective potential function and corresponding bound state energy levels (
In Tables
The KG energy eigenvalues (
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1.83929 | 2.04353 | 2.40325 | 2.75615 | 3.08137 | 3.38066 | 3.65815 | 3.91752 | 4.16166 |
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3.09625 | 3.4289 | 4.03986 | 4.65113 | 5.21808 | 5.74094 | 6.22602 | 6.67941 | 7.10609 | |
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4.12383 | 4.51488 | 5.26359 | 6.03138 | 6.75146 | 7.4193 | 8.04087 | 8.62301 | 9.17158 | |
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5.03104 | 5.45806 | 6.30283 | 7.18744 | 8.02542 | 8.80667 | 9.53605 | 10.2205 | 10.8663 | |
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2.50976 | 3.16597 | 3.89307 | 4.55265 | 5.14478 | 5.68284 | 6.17802 | 6.63858 | 7.0706 |
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3.62919 | 4.27262 | 5.12444 | 5.93706 | 6.68087 | 7.36315 | 7.99437 | 8.58337 | 9.13707 | |
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4.58916 | 5.22885 | 6.16824 | 7.09549 | 7.95634 | 8.7516 | 9.49036 | 10.1815 | 10.8323 | |
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5.45354 | 6.09216 | 7.09891 | 8.11893 | 9.0768 | 9.96689 | 10.7966 | 11.5745 | 12.3081 |
The KG energy eigenvalues (
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1.83929 | 2.50976 | 3.09625 | 3.62919 | 4.12383 | 4.58916 | 5.03104 | 5.45354 | 5.85966 |
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3.09625 | 3.62919 | 4.12383 | 4.58916 | 5.03104 | 5.45354 | 5.85966 | 6.25166 | 6.6313 | |
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4.12383 | 4.58916 | 5.03104 | 5.45354 | 5.85966 | 6.25166 | 6.63137.0 | 7.0 | 7.35892 | |
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5.03104 | 5.45354 | 5.85966 | 6.25166 | 6.6313 | 7.0 | 7.35892 | 7.70901 | 8.05108 | |
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2.50976 | 3.09625 | 3.62919 | 4.12383 | 4.58916 | 5.03104 | 5.45354 | 5.85966 | 6.25166 |
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3.62919 | 4.12383 | 4.58916 | 5.03104 | 5.45354 | 5.85966 | 6.25166 | 6.6313 | 7.0 | |
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4.58916 | 5.03104 | 5.45354 | 5.85966 | 6.25166 | 6.6313 | 7.0 | 7.35892 | 7.70901 | |
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5.45354 | 5.85966 | 6.25166 | 6.6313 | 7.0 | 7.35892 | 7.70901 | 8.05108 | 8.38582 |
The nonrelativistic energy eigenvalues (
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1.0 | 1.41421 | 2.23607 | 3.16228 | 4.12311 | 5.09902 | 6.08276 | 7.07107 | 8.06226 |
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3.0 | 4.24264 | 6.7082 | 9.48683 | 12.3693 | 15.2971 | 18.2483 | 21.2132 | 24.1868 | |
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5.0 | 7.07107 | 11.1803 | 15.8114 | 20.6155 | 25.4951 | 30.4138 | 35.3553 | 40.3113 | |
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7.0 | 9.89949 | 15.6525 | 22.1359 | 28.8617 | 35.6931 | 42.5793 | 49.4972 | 56.4358 | |
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2.0 | 3.82843 | 6.47214 | 9.32456 | 12.2462 | 15.198 | 18.1655 | 21.1421 | 24.1245 |
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4.0 | 6.65685 | 10.9443 | 15.6491 | 20.4924 | 25.3961 | 30.3311 | 35.2843 | 40.249 | |
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6.0 | 9.48528 | 15.4164 | 21.9737 | 28.7386 | 35.5941 | 42.4966 | 49.4264 | 56.3735 | |
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8.0 | 12.3137 | 19.8885 | 28.2982 | 36.9848 | 45.7922 | 54.6621 | 63.5685 | 72.4981 |
The nonrelativistic energy eigenvalues (
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1.0 | 2.0 | 3.0 | 4.0 | 5.0 | 6.0 | 7.0 | 8.0 | 9.0 |
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3.0 | 4.0 | 5.0 | 6.0 | 7.0 | 8.0 | 9.0 | 10.0 | 11.0 | |
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5.0 | 6.0 | 7.0 | 8.0 | 9.0 | 10.0 | 11.0 | 12.0 | 13.0 | |
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7.0 | 8.0 | 9.0 | 10.0 | 11.0 | 12.0 | 13.0 | 14.0 | 15.0 | |
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2.0 | 3.0 | 4.0 | 5.0 | 6.0 | 7.0 | 8.0 | 9.0 | 10.0 |
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4.0 | 5.0 | 6.0 | 7.0 | 8.0 | 9.0 | 10.0 | 11.0 | 12.0 | |
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6.0 | 7.0 | 8.0 | 9.0 | 10.0 | 11.0 | 12.0 | 13.0 | 14.0 | |
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8.0 | 9.0 | 10.0 | 11.0 | 12.0 | 13.0 | 14.0 | 15.0 | 16.0 |
To sum up, in this paper, we have studied the solution of two-dimensional KG and Schrödinger equations with the Killingbeck potential for low vibrational and rotational energy levels without and with a constant magnetic field having arbitrary Larmor frequeny and AB flux field. We have applied the wave function ansatz method for
One of the authors (Sameer M. Ikhdair) would like to thank the president of An-Najah National University (ANU), Professor Rami Alhamdallah, and the acting president Professor Maher Natsheh for their continuous support during his present work at ANU. The authors acknowledge the partial support of the Scientific and Technological Research Council of Turkey.