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Register a new userAny l-state solutions of the Woods-Saxon potential in arbitrary dimensions within the new improved quantization rule
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The approximated energy eigenvalues and the corresponding eigenfunctions of the spherical Woods-Saxon effective potential in $D$ dimensions are obtained within the new improved quantization rule for all $l$-states. The Pekeris approximation is used to deal with the centrifugal term in the effective Woods-Saxon potential. The inter-dimensional degeneracies for various orbital quantum number $l$ and dimensional space $D$ are studied. The solutions for the Hulth{e}n potential, the three-dimensional (D=3), the $% s$-wave ($l=0$) and the cases are briefly discussed.
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Approximate bound state solutions of the Dirac equation with -deformed Woods-Saxon plus a new generalized ring-shaped potential are obtained for any arbitrary L-state. The energy eigenvalue equation and corresponding two-component wave function are calculated by solving the radial and angular wave equations within a shortcut of the Nikiforov-Uvarov method. The solutions of the radial and polar angular parts of the wave function are expressed in terms of the Jacobi polynomials. A new approximation being expressed in terms of the potential parameters is carried out to deal with the strong singular centrifugal potential term L(L+1)/r^2. Under some limitations, we can obtain solution for the ring-shaped Hulthen potential and the standard usual spherical Woods-Saxon potential (q=1).
It was shown recently that Birkhoffs theorem for doubly stochastic matrices can be extended to unitary matrices with equal line sums whenever the dimension of the matrices is prime. We prove a generalization of the Birkhoff theorem for unitary matrices with equal line sums for arbitrary dimension.
The scattering solutions of the one-dimensional Schrodinger equation for the Woods-Saxon potential are obtained within the position-dependent mass formalism. The wave functions, transmission and reflection coefficients are calculated in terms of Heuns function. These results are also studied for the constant mass case in detail.
The Dirac equation is solved approximately for the Hulthen potential with the pseudospin symmetry for any spin-orbit quantum number $kappa$ in the position-dependent mass background. Solutions are obtained reducing the Dirac equation into a Schr{o}dinger-like differential equation by using an appropriate coordinate transformation. The Nikiforov-Uvarov method is used in the calculations to get energy eigenvalues and the corresponding wave functions.
In 2017, Lienert and Tumulka proved Borns rule on arbitrary Cauchy surfaces in Minkowski space-time assuming Borns rule and a corresponding collapse rule on horizontal surfaces relative to a fixed Lorentz frame, as well as a given unitary time evolution between any two Cauchy surfaces. Here, we prove Borns rule on arbitrary Cauchy surfaces from a different, but equally reasonable, set of assumptions. The conclusion is that if detectors are placed along any Cauchy surface $Sigma$, then the observed particle configuration on $Sigma$ has distribution $|Psi_Sigma|^2$, suitably understood. The main different assumption is that the Born and collapse rules hold on any spacelike hyperplane, i.e., at any time coordinate in any Lorentz frame. Heuristically, this follows if the dynamics of the detectors is Lorentz invariant. In addition, we assume, as did Lienert and Tumulka, that there is no interaction faster than light and that there is no propagation faster than light.
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