A novel application of a Fourier integral representation of bound states in quantum mechanics

2011 ◽  
Vol 79 (2) ◽  
pp. 201-205 ◽  
Author(s):  
Guillermo Palma ◽  
Ulrich Raff
Keyword(s):  
Quantum Mechanics ◽  
Integral Representation ◽  
Bound States ◽  
Fourier Integral

scholarly journals WKB FORMALISM AND A LOWER LIMIT FOR THE ENERGY EIGENSTATES OF BOUND STATES FOR SOME POTENTIALS

2007 ◽  
Vol 22 (35) ◽  
pp. 2675-2687 ◽  
Author(s):  
LUIS F. BARRAGÁN-GIL ◽  
ABEL CAMACHO
Keyword(s):  
Quantum Mechanics ◽  
Gravitational Field ◽  
Lower Bound ◽  
Harmonic Oscillator ◽  
Approximation Method ◽  
Bound States ◽  
The Other ◽  
Homogeneous Gravitational Field ◽  
Definition Of ◽  
Ground Energy

In this work the conditions appearing in the so-called WKB approximation formalism of quantum mechanics are analyzed. It is shown that, in general, a careful definition of an approximation method requires the introduction of two length parameters, one of them always considered in the textbooks on quantum mechanics, whereas the other is usually neglected. Afterwards we define a particular family of potentials and prove, resorting to the aforementioned length parameters, that we may find an energy which is a lower bound to the ground energy of the system. The idea is applied to the case of a harmonic oscillator and also to a particle freely falling in a homogeneous gravitational field, and in both cases the consistency of our method is corroborated. This approach, together with the so-called Rayleigh–Ritz formalism, allows us to define an energy interval in which the ground energy of any potential, belonging to our family, must lie.

Nuclear Physics Methods for Problems in Relativistic Quantum Mechanics

1998 ◽  
Vol 07 (05) ◽  
pp. 559-571
Author(s):  
Marcos Moshinsky ◽  
Verónica Riquer
Keyword(s):  
Quantum Mechanics ◽  
Nuclear Physics ◽  
Bound States ◽  
Relativistic Quantum ◽  
Negative Energy ◽  
Relativistic Quantum Mechanics ◽  
Variational Parameter ◽  
Approximate Methods ◽  
Rest Energy ◽  
Quark Models

Atomic and molecular physicists have developed extensive and detailed approximate methods for dealing with the relativistic versions of the Hamiltonians appearing in their fields. Nuclear physicists were originally more concerned with non-relativistic problems as the energies they were dealing with were normally small compared with the rest energy of the nucleon. This situation has changed with the appearance of the quark models of nucleons and thus the objective of this paper is to use the standard variational procedures of nuclear physics for problems in relativistic quantum mechanics. The 4 × 4α and β matrices in the Dirac equation are replaced by 2 × 2 matrices, one associated with ordinary spin and the other, which we call sign spin, is mathematically identical to the isospin of nuclear physics. The states on which our Hamiltonians will act will be the usual harmonic oscillator ones with ordinary and sign spin and the frequency ω of the oscillator will be our only variational parameter. The example discussed as an illustration will still be the Coulomb problem as the exact energies of the relativistic bound states are available for comparison. A gap of the order of 2mc2 is observed between states of positive and negative energy, that permits the former to be compared with the exact results.

WKB method for potentials unbounded from below

2016 ◽  
Vol 30 (03) ◽  
pp. 1650003 ◽  
Author(s):  
Aleksandar Demić ◽  
Vitomir Milanović ◽  
Jelena Radovanović ◽  
Milenko Musić
Keyword(s):  
Quantum Mechanics ◽  
Bound States ◽  
Wave Functions ◽  
Wkb Method ◽  
Symmetric Potential ◽  
Overlap Integral ◽  
Entire Spectrum ◽  
One Dimensional ◽  
Model Based ◽  
Exactly Solvable

Bound states degenerated in energy (and differing in parity) may form in one-dimensional quantum mechanics if the potential is unbounded from below. We focus on symmetric potential and present quasi-exactly solvable (QES) model based on WKB method. The application of this method is limited on slow-changing potentials. We consider the overlap integral of WKB wave functions [Formula: see text] and [Formula: see text] which correspond to energies [Formula: see text] and [Formula: see text], and by setting [Formula: see text], we determine the type of spectrum depending on parameter [Formula: see text] which arises from this method. For finite value [Formula: see text], we show that the entire spectrum will consist of degenerated bound states.

scholarly journals Bound states at threshold in supersymmetric quantum mechanics

1998 ◽  
Vol 515 (1-2) ◽  
pp. 184-202 ◽  
Author(s):  
Massimo Porrati ◽  
Alexander Rozenberg
Keyword(s):  
Quantum Mechanics ◽  
Bound States ◽  
Supersymmetric Quantum Mechanics

THE MASSLESS BOUND STATE FORMALISM IN TWO-PARTICLE RELATIVISTIC QUANTUM MECHANICS

1988 ◽  
Vol 03 (05) ◽  
pp. 1235-1261 ◽  
Author(s):  
H. SAZDJIAN
Keyword(s):  
Quantum Mechanics ◽  
Bound States ◽  
Degrees Of Freedom ◽  
Relativistic Quantum ◽  
Three Dimensional ◽  
Bound State ◽  
Relativistic Quantum Mechanics ◽  
Composite Systems ◽  
Bound State Case ◽  
State Case

We develop, in the framework of two-particle relativistic quantum mechanics, the formalism needed to describe massless bound state systems and their internal dynamics. It turns out that the dynamics here is two-dimensional, besides the contribution of the spin degrees of freedom, provided by the two space-like transverse components of the relative coordinate four-vector, decomposed in an appropriate light cone basis. This is in contrast with the massive bound state case, where the dynamics is three-dimensional. We also construct the scalar product of the theory. We apply this formalism to several types of composite systems, involving spin-0 bosons and/or spin-1/2 fermions, which produce massless bound states.

scholarly journals Precision Measurement of the Position-Space Wave Functions of Gravitationally Bound Ultracold Neutrons

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Y. Kamiya ◽  
G. Ichikawa ◽  
S. Komamiya
Keyword(s):  
Quantum Mechanics ◽  
Particle Physics ◽  
Bound States ◽  
Wave Functions ◽  
Recent Measurement ◽  
Ultracold Neutrons ◽  
Neutron Guide ◽  
Position Space ◽  
Position Sensitive Detectors ◽  
Space Wave

Gravity is the most familiar force at our natural length scale. However, it is still exotic from the view point of particle physics. The first experimental study of quantum effects under gravity was performed using a cold neutron beam in 1975. Following this, an investigation of gravitationally bound quantum states using ultracold neutrons was started in 2002. This quantum bound system is now well understood, and one can use it as a tunable tool to probe gravity. In this paper, we review a recent measurement of position-space wave functions of such gravitationally bound states and discuss issues related to this analysis, such as neutron loss models in a thin neutron guide, the formulation of phase space quantum mechanics, and UCN position sensitive detectors. The quantum modulation of neutron bound states measured in this experiment shows good agreement with the prediction from quantum mechanics.

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