81,488 research outputs found
Semiclassical energy formulas for power-law and log potentials in quantum mechanics
We study a single particle which obeys non-relativistic quantum mechanics in
R^N and has Hamiltonian H = -Delta + V(r), where V(r) = sgn(q)r^q. If N \geq 2,
then q > -2, and if N = 1, then q > -1. The discrete eigenvalues E_{n\ell} may
be represented exactly by the semiclassical expression E_{n\ell}(q) =
min_{r>0}\{P_{n\ell}(q)^2/r^2+ V(r)}. The case q = 0 corresponds to V(r) =
ln(r). By writing one power as a smooth transformation of another, and using
envelope theory, it has earlier been proved that the P_{n\ell}(q) functions are
monotone increasing. Recent refinements to the comparison theorem of QM in
which comparison potentials can cross over, allow us to prove for n = 1 that
Q(q)=Z(q)P(q) is monotone increasing, even though the factor Z(q)=(1+q/N)^{1/q}
is monotone decreasing. Thus P(q) cannot increase too slowly. This result
yields some sharper estimates for power-potential eigenvlaues at the bottom of
each angular-momentum subspace.Comment: 20 pages, 5 figure
Energy bounds for the spinless Salpeter equation: harmonic oscillator
We study the eigenvalues E_{n\ell} of the Salpeter Hamiltonian H =
\beta\sqrt(m^2 + p^2) + vr^2, v>0, \beta > 0, in three dimensions. By using
geometrical arguments we show that, for suitable values of P, here provided,
the simple semi-classical formula E = min_{r > 0} {v(P/r)^2 + \beta\sqrt(m^2 +
r^2)} provides both upper and lower energy bounds for all the eigenvalues of
the problem.Comment: 8 pages, 1 figur
Coulomb plus power-law potentials in quantum mechanics
We study the discrete spectrum of the Hamiltonian H = -Delta + V(r) for the
Coulomb plus power-law potential V(r)=-1/r+ beta sgn(q)r^q, where beta > 0, q >
-2 and q \ne 0. We show by envelope theory that the discrete eigenvalues
E_{n\ell} of H may be approximated by the semiclassical expression
E_{n\ell}(q) \approx min_{r>0}\{1/r^2-1/(mu r)+ sgn(q) beta(nu r)^q}.
Values of mu and nu are prescribed which yield upper and lower bounds.
Accurate upper bounds are also obtained by use of a trial function of the form,
psi(r)= r^{\ell+1}e^{-(xr)^{q}}. We give detailed results for
V(r) = -1/r + beta r^q, q = 0.5, 1, 2 for n=1, \ell=0,1,2, along with
comparison eigenvalues found by direct numerical methods.Comment: 11 pages, 3 figure
Relativistic N-Boson Systems Bound by Oscillator Pair Potentials
We study the lowest energy E of a relativistic system of N identical bosons
bound by harmonic-oscillator pair potentials in three spatial dimensions. In
natural units the system has the semirelativistic ``spinless-Salpeter''
Hamiltonian H = \sum_{i=1}^N \sqrt{m^2 + p_i^2} + \sum_{j>i=1}^N gamma |r_i -
r_j|^2, gamma > 0. We derive the following energy bounds: E(N) = min_{r>0} [N
(m^2 + 2 (N-1) P^2 / (N r^2))^1/2 + N (N-1) gamma r^2 / 2], N \ge 2, where
P=1.376 yields a lower bound and P=3/2 yields an upper bound for all N \ge 2. A
sharper lower bound is given by the function P = P(mu), where mu =
m(N/(gamma(N-1)^2))^(1/3), which makes the formula for E(2) exact: with this
choice of P, the bounds coincide for all N \ge 2 in the Schroedinger limit m
--> infinity.Comment: v2: A scale analysis of P is now included; this leads to revised
energy bounds, which coalesce in the large-m limi
Asymptotic iteration method for eigenvalue problems
An asymptotic interation method for solving second-order homogeneous linear
differential equations of the form y'' = lambda(x) y' + s(x) y is introduced,
where lambda(x) \neq 0 and s(x) are C-infinity functions. Applications to
Schroedinger type problems, including some with highly singular potentials, are
presented.Comment: 14 page
FearNot! An Anti-Bullying Intervention: Evaluation of an Interactive Virtual Learning Environment
Original paper can be found at: http://www.aisb.org.uk/publications/proceedings.shtm
Transient modeling of the thermohydraulic behavior of high temperature heat pipes for space reactor applications
Many proposed space reactor designs employ heat pipes as a means of conveying heat. Previous researchers have been concerned with steady state operation, but the transient operation is of interest in space reactor applications due to the necessity of remote startup and shutdown. A model is being developed to study the dynamic behavior of high temperature heat pipes during startup, shutdown and normal operation under space environments. Model development and preliminary results for a hypothetical design of the system are presented
General energy bounds for systems of bosons with soft cores
We study a bound system of N identical bosons interacting by model pair
potentials of the form V(r) = A sgn(p)r^p + B/r^2, A > 0, B >= 0. By using a
variational trial function and the `equivalent 2-body method', we find explicit
upper and lower bound formulas for the N-particle ground-state energy in
arbitrary spatial dimensions d > 2 for the two cases p = 2 and p = -1. It is
demonstrated that the upper bound can be systematically improved with the aid
of a special large-N limit in collective field theory
Semirelativistic stability of N-boson systems bound by 1/r pair potentials
We analyze a system of self-gravitating identical bosons by means of a
semirelativistic Hamiltonian comprising the relativistic kinetic energies of
the involved particles and added (instantaneous) Newtonian gravitational pair
potentials. With the help of an improved lower bound to the bottom of the
spectrum of this Hamiltonian, we are able to enlarge the known region for
relativistic stability for such boson systems against gravitational collapse
and to sharpen the predictions for their maximum stable mass.Comment: 11 pages, considerably enlarged introduction and motivation,
remainder of the paper unchange
An effective singular oscillator for Duffin-Kemmer-Petiau particles with a nonminimal vector coupling: a two-fold degeneracy
Scalar and vector bosons in the background of one-dimensional nonminimal
vector linear plus inversely linear potentials are explored in a unified way in
the context of the Duffin-Kemmer-Petiau theory. The problem is mapped into a
Sturm-Liouville problem with an effective singular oscillator. With boundary
conditions emerging from the problem, exact bound-state solutions in the spin-0
sector are found in closed form and it is shown that the spectrum exhibits
degeneracy. It is shown that, depending on the potential parameters, there may
or may not exist bound-state solutions in the spin-1 sector.Comment: 1 figure. arXiv admin note: substantial text overlap with
arXiv:1009.159
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