554 research outputs found
The Solution of the Relativistic Schrodinger Equation for the -Function Potential in 1-dimension Using Cutoff Regularization
We study the relativistic version of Schr\"odinger equation for a point
particle in 1-d with potential of the first derivative of the delta function.
The momentum cutoff regularization is used to study the bound state and
scattering states. The initial calculations show that the reciprocal of the
bare coupling constant is ultra-violet divergent, and the resultant expression
cannot be renormalized in the usual sense. Therefore a general procedure has
been developed to derive different physical properties of the system. The
procedure is used first on the non-relativistic case for the purpose of
clarification and comparisons. The results from the relativistic case show that
this system behaves exactly like the delta function potential, which means it
also shares the same features with quantum field theories, like being
asymptotically free, and in the massless limit, it undergoes dimensional
transmutation and it possesses an infrared conformal fixed point.Comment: 32 pages, 5 figure
Self-adjoint Extensions for Confined Electrons:from a Particle in a Spherical Cavity to the Hydrogen Atom in a Sphere and on a Cone
In a recent study of the self-adjoint extensions of the Hamiltonian of a
particle confined to a finite region of space, in which we generalized the
Heisenberg uncertainty relation to a finite volume, we encountered bound states
localized at the wall of the cavity. In this paper, we study this situation in
detail both for a free particle and for a hydrogen atom centered in a spherical
cavity. For appropriate values of the self-adjoint extension parameter, the
bound states lo calized at the wall resonate with the standard hydrogen bound
states. We also examine the accidental symmetry generated by the Runge-Lenz
vector, which is explicitly broken in a spherical cavity with general Robin
boundary conditions. However, for specific radii of the confining sphere, a
remnant of the accidental symmetry persists. The same is true for an electron
moving on the surface of a finite circular cone, bound to its tip by a 1/r
potential.Comment: 22 pages, 9 Figure
Asymptotic Freedom, Dimensional Transmutation, and an Infra-red Conformal Fixed Point for the -Function Potential in 1-dimensional Relativistic Quantum Mechanics
We consider the Schr\"odinger equation for a relativistic point particle in
an external 1-dimensional -function potential. Using dimensional
regularization, we investigate both bound and scattering states, and we obtain
results that are consistent with the abstract mathematical theory of
self-adjoint extensions of the pseudo-differential operator . Interestingly, this relatively simple system is asymptotically free. In
the massless limit, it undergoes dimensional transmutation and it possesses an
infra-red conformal fixed point. Thus it can be used to illustrate non-trivial
concepts of quantum field theory in the simpler framework of relativistic
quantum mechanics
Fate of Accidental Symmetries of the Relativistic Hydrogen Atom in a Spherical Cavity
The non-relativistic hydrogen atom enjoys an accidental symmetry,
that enlarges the rotational 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
with perfectly reflecting Robin boundary conditions, characterized by a
self-adjoint extension parameter , in general the accidental
symmetry is lifted. However, for (where is the Bohr
radius and is the orbital angular momentum) some degeneracy remains when
or . 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 and .Comment: 27 pages, 7 figure
Majorana Fermions in a Box
Majorana fermion dynamics may arise at the edge of Kitaev wires or
superconductors. Alternatively, it can be engineered by using trapped ions or
ultracold atoms in an optical lattice as quantum simulators. This motivates the
theoretical study of Majorana fermions confined to a finite volume, whose
boundary conditions are characterized by self-adjoint extension parameters.
While the boundary conditions for Dirac fermions in -d are characterized
by a 1-parameter family, , of self-adjoint extensions,
for Majorana fermions is restricted to . Based on this result,
we compute the frequency spectrum of Majorana fermions confined to a 1-d
interval. The boundary conditions for Dirac fermions confined to a 3-d region
of space are characterized by a 4-parameter family of self-adjoint extensions,
which is reduced to two distinct 1-parameter families for Majorana fermions. We
also consider the problems related to the quantum mechanical interpretation of
the Majorana equation as a single-particle equation. Furthermore, the equation
is related to a relativistic Schr\"odinger equation that does not suffer from
these problems.Comment: 23 pages, 2 figure
Harmonic Oscillator in a 1D or 2D Cavity with General Perfectly Reflecting Walls
We investigate the simple harmonic oscillator in a 1-d box, and the 2-d
isotropic harmonic oscillator problem in a circular cavity with perfectly
reflecting boundary conditions. The energy spectrum has been calculated as a
function of the self-adjoint extension parameter. For sufficiently negative
values of the self-adjoint extension parameter, there are bound states
localized at the wall of the box or the cavity that resonate with the standard
bound states of the simple harmonic oscillator or the isotropic oscillator. A
free particle in a circular cavity has been studied for the sake of comparison.
This work represents an application of the recent generalization of the
Heisenberg uncertainty relation related to the theory of self-adjoint
extensions in a finite volume.Comment: 23 pages 18 figure
Canonical quantization on the half-line and in an interval based upon a new concept for the momentum in a space with boundaries
For a particle moving on a half-line or in an interval the operator is not self-adjoint and thus does not qualify as the physical
momentum. Consequently canonical quantization based on fails. Based
upon a new concept for a self-adjoint momentum operator , we show
that canonical quantization can indeed be implemented on the half-line and on
an interval. Both the Hamiltonian and the momentum operator
are endowed with self-adjoint extension parameters that characterize the
corresponding domains and in the Hilbert space. When
one replaces Poisson brackets by commutators, one obtains meaningful results
only if the corresponding operator domains are properly taken into account. The
new concept for the momentum is used to describe the results of momentum
measurements of a quantum mechanical particle that is reflected at impenetrable
boundaries, either at the end of the half-line or at the two ends of an
interval.Comment: 19 pages, 7 figure
Alternative momentum concept for a quantum mechanical particle in a box
For a particle in a box, the operator ˆp = −i∂x is not self-adjoint. We provide an alternative construction of a momentum operator ˆpR + ipˆI, which has two self-adjoint components ˆpR and ˆpI. This leads to a description of momentum measurements performed on a particle that is strictly limited to the interior of a box of size L, which yields quantized momentum values πn/L with n ∈ Z
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