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Register a new userFate of Accidental Symmetries of the Relativistic Hydrogen Atom in a Spherical Cavity
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The non-relativistic hydrogen atom enjoys an accidental $SO(4)$ symmetry, that enlarges the rotational $SO(3)$ symmetry, by extending the angular momentum algebra with the Runge-Lenz vector. In the relativistic hydrogen atom the accidental symmetry is partially lifted. Due to the Johnson-Lippmann operator, which commutes with the Dirac Hamiltonian, some degeneracy remains. When the non-relativistic hydrogen atom is put in a spherical cavity of radius $R$ with perfectly reflecting Robin boundary conditions, characterized by a self-adjoint extension parameter $gamma$, in general the accidental $SO(4)$ symmetry is lifted. However, for $R = (l+1)(l+2) a$ (where $a$ is the Bohr radius and $l$ is the orbital angular momentum) some degeneracy remains when $gamma = infty$ or $gamma = frac{2}{R}$. In the relativistic case, we consider the most general spherically and parity invariant boundary condition, which is characterized by a self-adjoint extension parameter. In this case, the remnant accidental symmetry is always lifted in a finite volume. We also investigate the accidental symmetry in the context of the Pauli equation, which sheds light on the proper non-relativistic treatment including spin. In that case, again some degeneracy remains for specific values of $R$ and $gamma$.
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We discussed quantum deformations of D=4 Lorentz and Poincare algebras. In the case of Poincare algebra it is shown that almost all classical r-matrices of S. Zakrzewski classification correspond to twisted deformations of Abelian and Jordanian types
. A part of twists corresponding to the r-matrices of Zakrzewski classification are given in explicit form.
We show that the relativistic hydrogen atom possesses an SO(4) symmetry by introducing a kind of pseudo-spin vector operator. The same SO(4) symmetry is still preserved in the relativistic quantum system in presence of an U(1) monopolar vector potential as well as a nonabelian vector potential. Lamb shift and SO(4) symmetry breaking are also discussed.
We obtain the following analytical formula which describes the dependence of the electric potential of a point-like charge on the distance away from it in the direction of an external magnetic field B: Phi(z) = e/|z| [ 1- exp(-sqrt{6m_e^2}|z|) + exp(-sqrt{(2/pi) e^3 B + 6m_e^2} |z|) ]. The deviation from Coulombs law becomes essential for B > 3pi B_{cr}/alpha = 3 pi m_e^2/e^3 approx 6 10^{16} G. In such superstrong fields, electrons are ultra-relativistic except those which occupy the lowest Landau level (LLL) and which have the energy epsilon_0^2 = m_e^2 + p_z^2. The energy spectrum on which LLL splits in the presence of the atomic nucleus is found analytically. For B > 3 pi B_{cr}/alpha, it substantially differs from the one obtained without accounting for the modification of the atomic potential.
Hamiltonians describing the relativistic quantum dynamics of a particle with an arbitrary spin are shown to exhibit a supersymmetric structure when the even and odd elements of the Hamiltonian commute. For such supersymmetric Hamiltonians an exact Foldy-Wouthuysen transformation exits which brings it into a block-diagonal form separating the positive and negative energy subspaces. Here the supercharges transform between energy eigenstates of positive and negative energy. The relativistic dynamics of a charged particle in a magnetic field is considered for the case of a scalar (spin-zero) boson obeying the Klein-Gordan equation, a Dirac (spin one-half) fermion and a vector (spin-one) boson characterised by the Proca equation.
Our familiar Newtons laws allow determination of both position and velocity of any object precisely. Early nineteenth century saw the birth of quantum mechanics where all measurements must obey Heisenbergs uncertainty principle. Basically, we cannot simultaneously measure with precision, both position and momentum of particles in the microscopic atomic world. A natural extension will be to assume that space becomes fuzzy as we approach the study of early universe. That is, all the components of position cannot be simultaneously measured with precision. Such a space is called non-commutative space. In this article, we study quantum mechanics of hydrogen atom on such a fuzzy space. Particularly, we highlight expected corrections to the hydrogen atom energy spectrum due to non-commutative space.
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