29 research outputs found
Spin Force and Torque in Non-relativistic Dirac Oscillator on a Sphere
The spin force operator on a non-relativistic Dirac oscillator ( in the
non-relativistic limit the Dirac oscillator is a spin one-half 3D harmonic
oscillator with strong spin-orbit interaction) is derived using the Heisenberg
equations of motion and is seen to be formally similar to the force by the
electromagnetic field on a moving charged particle . When confined to a sphere
of radius R, it is shown that the Hamiltonian of this non-relativistic
oscillator can be expressed as a mere kinetic energy operator with an anomalous
part. As a result, the power by the spin force and torque operators in this
case are seen to vanish. The spin force operator on the sphere is calculated
explicitly and its torque is shown to be equal to the rate of change of the
kinetic orbital angular momentum operator, again with an anomalous part. This,
along with the conservation of the total angular momentum, suggest that the
spin force exerts a spin-dependent torque on the kinetic orbital angular
momentum operator in order to conserve total angular momentum. The presence of
an anomalous spin part in the kinetic orbital angular momentum operator gives
rise to an oscillatory behavior similar to the \textit{Zitterbewegung}. It is
suggested that the underlying physics that gives rise to the spin force and the
\textit{Zitterbewegung} is one and the same in NRDO and in systems that
manifest spin Hall effect.Comment: 7 page
Hamiltonian for a particle in a magnetic field on a curved surface in orthogonal curvilinear coordinates
The Schr\"odinger Hamiltonian of a spin zero particle as well as the Pauli
Hamiltonian with spin-orbit coupling included of a spin one-half particle in
electromagnetic fields that are confined to a curved surface embedded in a
three-dimensional space spanned by a general Orthogonal Curvilinear Coordinate
(OCC) are constructed. A new approach, based on the physical argument that upon
squeezing the particle to the surface by a potential, then it is the physical
gauge-covariant kinematical momentum operator (velocity operator) transverse to
the surface that should be dropped from the Hamiltonian(s). In both cases,the
resulting Hermitian gauge-invariant Hamiltonian on the surface is free from any
reference to the component of the vector potential transverse to the surface,
and the approach is completely gauge-independent. In particular, for the Pauli
Hamiltonian these results are obtained exactly without any further assumptions
or approximations. Explicit covariant plug-and-play formulae for the
Schr\"odinger Hamiltonians on the surfaces of a cylinder, a sphere and a torus
are derived.Comment: 7 pages, 1 figure, references adde
Hermitian spin-orbit Hamiltonians on a surface in orthogonal curvilinear coordinates: a new practical approach
The Hermitian Hamiltonian of a spin one-half particle with spin-orbit
coupling (SOC) confined to a surface that is embedded in a three-dimensional
space spanned by a general Orthogonal Curvilinear Coordinate (OCC) is
constructed. A gauge field formalism, where the SOC is expressed as a
non-Abelian gauge field is used. A new practical approach, based on the
physical argument that upon confining the particle to the surface by a
potential, then it is the physical Hermitian momentum operator transverse to
the surface, rather than just the derivative with respect to the transverse
coordinate that should be dropped from the Hamiltonian.Doing so, it is shown
that the Hermitian Hamiltonian for SOC is obtained with the geometric potential
and the geometric kinetic energy terms emerging naturally. The geometric
potential is shown to represent a coupling between the transverse component of
the gauge field and the mean curvature of the surface that replaces the
coupling between the transverse momentum and the gauge field. The most general
Hermitian Hamiltonian with linear SOC on a general surface embedded in any 3D
OCC system is reported. Explicit plug-and-play formulae for this Hamiltonian on
the surfaces of a cylinder, a sphere and a torus are given. The formalism is
applied to the Rashba SOC in three dimensions (3D RSOC) and the explicit
expressions for the surface Hamiltonians on these three geometries are worked
out.Comment: 8 pages, 1 figur
Effective polar potential in the central force Schrodinger equation
The angular part of the Schrodinger equation for a central potential is
brought to the one-dimensional 'Schrodinger form' where one has a kinetic
energy plus potential energy terms. The resulting polar potential is seen to be
a family of potentials characterized by the square of the magnetic quantum
number m. It is demonstrated that this potential can be viewed as a confining
potential that attempts to confine the particle to the xy-plane, with a
strength that increases with increasing m. Linking the solutions of the
equation to the conventional solutions of the angular equation, i.e. the
associated Legendre functions, we show that the variation in the spatial
distribution of the latter for different values of the orbital angular quantum
number l can be viewed as being a result of 'squeezing' with different
strengths by the introduced 'polar potential'.Comment: This is an author-created, un-copyedited version of an article
accepted for publication in European Journal of Physic