Bound states of the Dirac equation with vector and scalar Eckart potentials☆
Introduction
It is well known that the exact solutions of the quantum systems play an important role in quantum mechanics. There has been an increased interest in searching for analytic solutions of the Klein–Gordon and Dirac equations [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18]. In the setting of the Klein–Gordon and Dirac equations, some authors have assumed that the scalar potential is equal to the vector potential in order to solve problems which have been solved in the non-relativistic quantum mechanics. Considerable efforts have been done towards obtaining the exact solutions of the Klein–Gordon and Dirac equations with equally mixed potentials for some potentials such as the three-dimensional harmonic oscillator [9], Hulthén potential [10], pseudoharmonic oscillator [11], ring-shaped Kratzer-type potential [12], ring-shaped non-spherical oscillator [13], double ring-shaped oscillator [14], Hartmann potential [15], Rosen–Morse-type potential [16], generalized symmetrical double-well potential [17], and Scarf-type potential [18], etc.
The well-known Eckart potential, introduced by Eckart [19] in 1930, is widely used in physics [20] and chemical physics [21], [22]. In Ref. [16], the authors investigated the energy equation for the Eckart potential in the Klein–Gordon theory with equally mixed potentials by using the function analysis method. In the present Letter, we study the analytic solutions of the Dirac equation with equally mixed potentials for the Eckart potential in terms of the supersymmetric quantum mechanics method [23], shape invariance approach [24], and the function analysis method [25]. The results show that the energy equations and spinor wavefunctions for the Hulthén potential [26], generalized Morse potential [27] and attractive radial potential [28] in the Dirac theory with equally mixed potentials are included in those for the Eckart potential as special cases.
Section snippets
Bound state solutions
According to Ref. [9], the Dirac equation with both the scalar potential and vector potential is where E denotes the energy, and M denotes the mass. In the relativistic quantum mechanics, the complete set of the conservative quantities for a particle in a central field can be taken as , the spinor eigenfunctions of which are given by [29] where and are
Discussions
In this section, within the framework of the Dirac theory with equally mixed potentials, we obtain the energy equations and spinor wavefunctions for several well-known potentials by choosing appropriate parameters in the Eckart potential model.
Conclusion
In this Letter, we may conclude that the Dirac equation for equal scalar and vector Eckart potentials can be solved exactly for s-wave bound states by the method of supersymmetric quantum mechanics, shape invariance approach and the function analysis method, so the relativistic mass-energy corrections can be obtained in a non-perturbative way. There exist the same energy equations for the Eckart potentials in the setting of Dirac theory and Klein–Gordon theory with equally mixed potentials. For
Acknowledgements
The authors wish to thank the referee for his helpful comments and suggestions which greatly improved this Letter.
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Work supported by the Sichuan Province Foundation of China for Fundamental Research Projects under Grant Nos. 04JY029-062-2 and 04JY029-112.