The spatially dependent mass Dirac equation is solved exactly for attractive scalar and repulsive vector Coulomb potentials including a tensor interaction potential under the spin and pseudospin (p-spin) symmetric limits by using the Laplace transformation method (LTM). Closed forms of the energy eigenvalue equation and wave functions are obtained for arbitrary spin-orbit quantum number
Dirac equation has become one of the most appealing relativistic wave equations for spin-1/2 particles. However, solving such a wave equation is still a very challenging problem even if it has been derived more than 80 years ago and has been utilized profusely. It is always useful to investigate the relativistic effects [
On the other hand, systems with position-dependent mass (PDM) have been found to be very useful in studying the physical properties of various microstructures [
Here, we shall attempt to solve the Dirac equation by using the Laplace transform method (LTM). The LTM is an integral transform and is comprehensively useful in physics and engineering [
The Dirac equation which describes a nucleon in repulsive vector
Equation (
Substituting (
The lower spinor wave function can be obtained via
The bound state energy eigenvalues of the Coulomb potential under the pseudo-spin symmetry limit for several values of
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1 | 1, −2 |
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0.0 | −4.86154 | −4.75676 | −4.47059 | −4.86154 |
0.2 | −4.80508 | −4.66644 | −4.36211 | −4.80508 | |||
0.4 | −4.74207 | −4.57172 | −4.26086 | −4.74207 | |||
0.5 | −4.70859 | −4.52376 | −4.21286 | −4.70859 | |||
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2 | 1, −3 |
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0.0 | −4.91089 | −4.86154 | −4.75676 | −4.75676 |
0.2 | −4.87335 | −4.80508 | −4.66644 | −4.66644 | |||
0.4 | −4.83043 | −4.74207 | −4.57172 | −4.57172 | |||
0.5 | −4.80716 | −4.70859 | −4.52376 | −4.52376 | |||
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3 | 1, −4 |
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0.0 | −4.93793 | −4.91089 | −4.86154 | −4.47059 |
0.2 | −4.91138 | −4.87335 | −4.80508 | −4.36211 | |||
0.4 | −4.88067 | −4.83043 | −4.74207 | −4.26086 | |||
0.5 | −4.86385 | −4.80716 | −4.70859 | −4.21286 | |||
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4 | 1, −5 |
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0.0 | −4.95431 | −4.93793 | −4.91089 | −4.47059 |
0.2 | −4.93461 | −4.91138 | −4.87335 | −4.36211 | |||
0.4 | −4.91166 | −4.88067 | −4.83043 | −4.26086 | |||
0.5 | −4.89903 | −4.86385 | −4.80716 | −4.21286 | |||
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1 | 2, −2 |
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−4.91089 | −4.86154 | −4.75676 | −4.91089 |
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−4.87402 | −4.80794 | −4.69083 | −4.87402 | |||
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−4.83243 | −4.75015 | −4.62526 | −4.83243 | |||
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−4.81014 | −4.72024 | −4.59287 | −4.81014 | |||
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2 | 2, −3 |
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−4.93793 | −4.91089 | −4.86154 | −4.86154 |
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−4.9116 | −4.87402 | −4.80794 | −4.80794 | |||
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−4.88134 | −4.83243 | −4.75015 | −4.75015 | |||
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−4.86487 | −4.81014 | −4.72024 | −4.72024 | |||
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3 | 2, −4 |
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−4.95431 | −4.93793 | −4.91089 | −4.75676 |
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−4.93469 | −4.9116 | −4.87402 | −4.69083 | |||
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−4.91194 | −4.88134 | −4.83243 | −4.62526 | |||
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−4.89945 | −4.86487 | −4.81014 | −4.59287 | |||
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4 | 2, −5 |
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−4.96498 | −4.95431 | −4.93793 | −4.75676 |
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−4.94984 | −4.93469 | −4.91160 | −4.69083 | |||
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−4.93218 | −4.91194 | −4.88134 | −4.62526 | |||
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−4.92244 | −4.89945 | −4.86487 | −4.59287 |
The bound state energy eigenvalues of the Coulomb potential under the pseudo-spin symmetry limit for several values of
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2 | 0, 2 |
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0.0 | −4.75676 | −4.86154 | −4.91089 | −4.96498 |
0.2 | −4.66644 | −4.80508 | −4.87335 | −4.94979 | |||
0.4 | −4.57172 | −4.74207 | −4.83043 | −4.93205 | |||
0.5 | −4.52376 | −4.70859 | −4.80716 | −4.92225 | |||
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3 | 0, 3 |
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0.0 | −4.86154 | −4.91089 | −4.93793 | −4.97231 |
0.2 | −4.80508 | −4.87335 | −4.91138 | −4.96026 | |||
0.4 | −4.74207 | −4.83043 | −4.88067 | −4.94614 | |||
0.5 | −4.70859 | −4.80716 | −4.86385 | −4.93833 | |||
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4 | 0, 4 |
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0.0 | −4.91089 | −4.93793 | −4.95431 | −4.97756 |
0.2 | −4.87335 | −4.91138 | −4.93461 | −4.96777 | |||
0.4 | −4.83043 | −4.88067 | −4.91166 | −4.95628 | |||
0.5 | −4.80716 | −4.86385 | −4.89903 | −4.94991 | |||
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5 | 0, 5 |
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0.0 | −4.93793 | −4.95431 | −4.96498 | −4.98144 |
0.2 | −4.91138 | −4.93461 | −4.94979 | −4.97334 | |||
0.4 | −4.88067 | −4.91166 | −4.93205 | −4.96381 | |||
0.5 | −4.86385 | −4.89903 | −4.92225 | −4.95852 | |||
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2 | 1, 2 |
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0.0 | −4.86154 | −4.91089 | −4.93793 | −4.97231 |
0.2 | −4.80794 | −4.87402 | −4.91160 | −4.96028 | |||
0.4 | −4.75015 | −4.83243 | −4.88134 | −4.94621 | |||
0.5 | −4.72024 | −4.81014 | −4.86487 | −4.93843 | |||
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3 | 1, 3 |
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0.0 | −4.91089 | −4.93793 | −4.95431 | −4.97756 |
0.2 | −4.87402 | −4.91160 | −4.93469 | −4.96778 | |||
0.4 | −4.83243 | −4.88134 | −4.91194 | −4.95632 | |||
0.5 | −4.81014 | −4.86487 | −4.89945 | −4.94997 | |||
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4 | 1, 4 |
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0.0 | −4.93793 | −4.95431 | −4.96498 | −4.98144 |
0.2 | −4.91160 | −4.93469 | −4.94984 | −4.97757 | |||
0.4 | −4.88134 | −4.91194 | −4.93218 | −4.96383 | |||
0.5 | −4.86487 | −4.89945 | −4.92244 | −4.95856 | |||
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5 | 1, 5 |
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0.0 | −4.95431 | −4.96498 | −4.97231 | −4.98440 |
0.2 | −4.93469 | −4.94984 | −4.96028 | −4.97758 | |||
0.4 | −4.91194 | −4.93218 | −4.94621 | −4.96957 | |||
0.5 | −4.89945 | −4.92244 | −4.93843 | −4.96512 |
The bound state energy eigenvalues of the Coulomb potential for several states under the pseudo-spin symmetry limit with values of
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1 |
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−1.00 | −4.75676 | −4.52376 | −4.96498 | −4.92225 |
−1.25 | −4.76351 | −4.53699 | −4.96595 | −4.92441 | |||
−1.50 | −4.77027 | −4.55022 | −4.96693 | −4.92657 | |||
−1.75 | −4.77703 | −4.56345 | −4.96790 | −4.92873 | |||
−2.00 | −4.78378 | −4.57668 | −4.96887 | −4.93089 | |||
−2.25 | −4.79054 | −4.58991 | −4.96984 | −4.93305 | |||
−2.50 | −4.79730 | −4.60313 | −4.97082 | −4.93521 | |||
−2.75 | −4.80405 | −4.61636 | −4.97179 | −4.93736 | |||
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3 |
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−1.00 | −4.91089 | −4.80543 | −3.20000 | −2.90681 |
−1.25 | −4.91337 | −4.81083 | −3.25000 | −2.96496 | |||
−1.50 | −4.91584 | −4.81624 | −3.30000 | −3.0231 | |||
−1.75 | −4.91832 | −4.82164 | −3.35000 | −3.08124 | |||
−2.00 | −4.92079 | −4.82705 | −3.40000 | −3.13939 | |||
−2.25 | −4.92327 | −4.83245 | −3.45000 | −3.19753 | |||
−2.50 | −4.92574 | −4.83786 | −3.50000 | −3.25568 | |||
−2.75 | −4.92822 | −4.84326 | −3.55000 | −3.31382 | |||
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1 |
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−1.00 | −4.86154 | −4.70859 | −4.86154 | −4.70859 |
−1.25 | −4.86538 | −4.71669 | −4.86538 | −4.71669 | |||
−1.50 | −4.86923 | −4.72478 | −4.86923 | −4.72478 | |||
−1.75 | −4.87308 | −4.73288 | −4.87308 | −4.73288 | |||
−2.00 | −4.87692 | −4.74097 | −4.87692 | −4.74097 | |||
−2.25 | −4.88077 | −4.74907 | −4.88077 | −4.74907 | |||
−2.50 | −4.88462 | −4.75716 | −4.88462 | −4.75716 | |||
−2.75 | −4.88846 | −4.76526 | −4.88846 | −4.76526 | |||
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3 |
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−1.00 | −4.93793 | −4.86385 | −4.47059 | −4.21286 |
−1.25 | −4.93966 | −4.86764 | −4.48529 | −4.23472 | |||
−1.50 | −4.94138 | −4.87142 | −4.50000 | −4.25659 | |||
−1.75 | −4.9431 | −4.8752 | −4.51471 | −4.27845 | |||
−2.00 | −4.94483 | −4.87898 | −4.52941 | −4.30032 | |||
−2.25 | −4.94655 | −4.88276 | −4.54412 | −4.32218 | |||
−2.50 | −4.94828 | −4.88654 | −4.55882 | −4.34405 | |||
−2.75 | −4.95000 | −4.89033 | −4.57353 | −4.36591 |
To avoid repetition in the solution of (
In Tables
Finally, we plot the relativistic energy eigenvalues under spin and p-spin symmetry limitations in Figures
The variation of the energy levels as a function
The variation of the energy levels as a function
The variation of the energy levels as a function
The variation of the energy levels as a function
In this paper, the relativistic equation for particles with spin 1/2 was solved exactly with both spatially-dependent mass and tensor interaction for attractive scalar and repulsive vector Coulomb potentials under the spin symmetry limit via the Laplace transformation method. Some numerical results are given for specific values of the model parameters. Effects of the tensor interaction on the bound states were presented that tensor interaction removes degeneracy between two states in spin doublets. We also investigated the effects of the spatially-dependent mass on the bound states under the conditions of the spin symmetry limits for