scholarly journals Optimizing adiabaticity in quantum mechanics

2012 ◽  
Vol 90 (2) ◽  
pp. 187-191 ◽  
Author(s):  
R. MacKenzie ◽  
M. Pineault ◽  
L. Renaud-Desjardins
Keyword(s):  
Magnetic Field ◽  
Quantum Mechanics ◽  
Mechanical System ◽  
Evolution Operator ◽  
Time Dependent ◽  
Quantum Mechanical ◽  
Quantum Mechanical System ◽  
Adiabatic Evolution ◽  
Time Dependent Problem

A condition on the Hamiltonian of an isospectral time-dependent quantum mechanical system is derived, which, if satisfied, implies optimal adiabaticity (defined later). The condition is expressed in terms of the Hamiltonian and the evolution operator related to it. Because the latter depends in a complicated way on the Hamiltonian, it is not yet clear how the condition can be used to extract useful information about the optimal Hamiltonian analytically. The condition is tested on an exactly-soluble time-dependent problem (a spin in a magnetic field), where perfectly adiabatic evolution can be easily identified.

scholarly journals Generation of Time-Independent and Time-Dependent Harmonic Oscillator-Like Potentials Using Supersymmetric Quantum Mechanics

2018 ◽  
Vol 4 (1) ◽  
pp. 47-55
Author(s):  
Timothy Brian Huber
Keyword(s):  
Quantum Mechanics ◽  
Schrödinger Equation ◽  
Harmonic Oscillator ◽  
Mechanical System ◽  
Schrodinger Equation ◽  
Supersymmetric Quantum Mechanics ◽  
Time Dependent ◽  
Quantum Mechanical ◽  
Quantum Mechanical System ◽  
Intertwining Operator

The harmonic oscillator is a quantum mechanical system that represents one of the most basic potentials. In order to understand the behavior of a particle within this system, the time-independent Schrödinger equation was solved; in other words, its eigenfunctions and eigenvalues were found. The first goal of this study was to construct a family of single parameter potentials and corresponding eigenfunctions with a spectrum similar to that of the harmonic oscillator. This task was achieved by means of supersymmetric quantum mechanics, which utilizes an intertwining operator that relates a known Hamiltonian with another whose potential is to be built. Secondly, a generalization of the technique was used to work with the time-dependent Schrödinger equation to construct new potentials and corresponding solutions.

A quantum mechanical discussion of Rabi oscillations

2008 ◽  
Vol 86 (8) ◽  
pp. 953-960 ◽  
Author(s):  
G R Hoy ◽  
J Odeurs
Keyword(s):  
Magnetic Field ◽  
Electromagnetic Field ◽  
Quantum Mechanics ◽  
Magnetic Resonance ◽  
Spin System ◽  
Magnetic Moment ◽  
Time Dependent ◽  
Quantum Mechanical ◽  
Rabi Oscillations ◽  
Interaction Picture

In 1937, Rabi treated the problem of a magnetic moment in an applied time-dependent magnetic field. This became the well-known magnetic resonance situation. The Hamiltonian is often taken to be [Formula: see text] = – µ · [[Formula: see text]]. In this paper, the Rabi oscillations formula, describing the spin flipping, is derived in an unusual way. The method uses a modification of a method due to Heitler. In the Heitler method, one uses the Interaction Picture of quantum mechanics. Due to the time-dependence in the problem, the usual Heitler method fails. However, the solution is found after quantizing the electromagnetic field. To better understand the origin of the spin flipping, the analogous time-independent problem is also solved. It is made clear that the origin of the Rabi oscillations is not due to the time-dependent magnetic field. The spin flipping is essentially due to the fact that the spin system, when initially prepared, is not in an eigenstate of the Hamiltonian. Thus, as times progresses, the system naturally evolves through the noneigenstate basis states.PACS Nos.: 03.65.–w, 76.20.+q

Probability in quantum mechanics

1978 ◽  
Vol 10 (4) ◽  
pp. 725-729 ◽  
Author(s):  
J. V. Corbett
Keyword(s):  
Probability Theory ◽  
Hilbert Space ◽  
Quantum Mechanics ◽  
Mechanical System ◽  
Partial Order ◽  
Probability Space ◽  
Separable Hilbert Space ◽  
Mathematical Expression ◽  
Quantum Mechanical ◽  
Quantum Mechanical System

Quantum mechanics is usually described in the terminology of probability theory even though the properties of the probability spaces associated with it are fundamentally different from the standard ones of probability theory. For example, Kolmogorov's axioms are not general enough to encompass the non-commutative situations that arise in quantum theory. There have been many attempts to generalise these axioms to meet the needs of quantum mechanics. The focus of these attempts has been the observation, first made by Birkhoff and von Neumann (1936), that the propositions associated with a quantum-mechanical system do not form a Boolean σ-algebra. There is almost universal agreement that the probability space associated with a quantum-mechanical system is given by the set of subspaces of a separable Hilbert space, but there is disagreement over the algebraic structure that this set represents. In the most popular model for the probability space of quantum mechanics the propositions are assumed to form an orthocomplemented lattice (Mackey (1963), Jauch (1968)). The fundamental concept here is that of a partial order, that is a binary relation that is reflexive and transitive but not symmetric. The partial order is interpreted as embodying the logical concept of implication in the set of propositions associated with the physical system. Although this model provides an acceptable mathematical expression of the probabilistic structure of quantum mechanics in that the subspaces of a separable Hilbert space give a representation of an ortho-complemented lattice, it has several deficiencies which will be discussed later.

Probability in quantum mechanics

1978 ◽  
Vol 10 (04) ◽  
pp. 725-729
Author(s):  
J. V. Corbett
Keyword(s):  
Probability Theory ◽  
Hilbert Space ◽  
Quantum Mechanics ◽  
Mechanical System ◽  
Partial Order ◽  
Probability Space ◽  
Separable Hilbert Space ◽  
Mathematical Expression ◽  
Quantum Mechanical ◽  
Quantum Mechanical System

Quantum mechanics is usually described in the terminology of probability theory even though the properties of the probability spaces associated with it are fundamentally different from the standard ones of probability theory. For example, Kolmogorov's axioms are not general enough to encompass the non-commutative situations that arise in quantum theory. There have been many attempts to generalise these axioms to meet the needs of quantum mechanics. The focus of these attempts has been the observation, first made by Birkhoff and von Neumann (1936), that the propositions associated with a quantum-mechanical system do not form a Boolean σ-algebra. There is almost universal agreement that the probability space associated with a quantum-mechanical system is given by the set of subspaces of a separable Hilbert space, but there is disagreement over the algebraic structure that this set represents. In the most popular model for the probability space of quantum mechanics the propositions are assumed to form an orthocomplemented lattice (Mackey (1963), Jauch (1968)). The fundamental concept here is that of a partial order, that is a binary relation that is reflexive and transitive but not symmetric. The partial order is interpreted as embodying the logical concept of implication in the set of propositions associated with the physical system. Although this model provides an acceptable mathematical expression of the probabilistic structure of quantum mechanics in that the subspaces of a separable Hilbert space give a representation of an ortho-complemented lattice, it has several deficiencies which will be discussed later.

scholarly journals CHARGED SPIN-1/2 PARTICLE IN AN ARBITRARY MAGNETIC FIELD IN TWO SPATIAL DIMENSIONS: A SUPERSYMMETRIC QUANTUM MECHANICAL SYSTEM

2000 ◽  
Vol 15 (34) ◽  
pp. 2105-2111 ◽  
Author(s):  
T. E. CLARK ◽  
S. T. LOVE ◽  
S. R. NOWLING
Keyword(s):  
Magnetic Field ◽  
Wave Function ◽  
Mechanical System ◽  
Ground State ◽  
Ground State Wave Function ◽  
Quantum Mechanical ◽  
Quantum Mechanical System ◽  
Zero Energy ◽  
State Wave ◽  
Spatial Dimensions

It is shown that the 2×2 matrix Hamiltonian describing the dynamics of a charged spin-1/2 particle with g-factor 2 moving in an arbitrary, spatially dependent, magnetic field in two spatial dimensions can be written as the anticommutator of a nilpotent operator and its Hermitian conjugate. Consequently, the Hamiltonians for the two different spin projections form partners of a supersymmetric quantum mechanical system. The resulting supersymmetry algebra can then be exploited to explicitly construct the exact zero energy ground state wave function for the system. Modulo this ground state, the remainder of the eigenstates and eigenvalues of the two partner Hamiltonians form positive energy degenerate pairs. We also construct the spatially asymptotic form of the magnetic field which produces a finite magnetic flux and associated zero energy normalizable ground state wave function.

scholarly journals PSEUDOSUPERSYMMETRIC QUANTUM MECHANICS: GENERAL CASE, ORTHOSUPERSYMMETRIES, REDUCIBILITY, AND BOSONIZATION

2003 ◽  
Vol 18 (02) ◽  
pp. 271-292 ◽  
Author(s):  
C. QUESNE ◽  
N. VANSTEENKISTE
Keyword(s):  
Magnetic Field ◽  
Quantum Mechanics ◽  
Constant Magnetic Field ◽  
Explicit Solution ◽  
Quantum Mechanical ◽  
Oscillator Algebra ◽  
Quantum Mechanical System ◽  
Irreducible Components ◽  
Deformed Oscillator ◽  
Deformed Oscillator Algebra

Pseudosupersymmetric quantum mechanics (PsSSQM), based upon the use of pseudofermions, was introduced in the context of a new Kemmer equation describing charged vector mesons interacting with an external constant magnetic field. Here we construct the complete explicit solution for its realization in terms of two superpotentials, both equal or unequal. We prove that any orthosupersymmetric quantum mechanical system has a pseudosupersymmetry and give conditions under which a pseudosupersymmetric one may be described by orthosupersymmetries of order two. We propose two new matrix realizations of PsSSQM in terms of the generators of a generalized deformed oscillator algebra (GDOA) and relate them to the cases of equal or unequal superpotentials, respectively. We demonstrate that these matrix realizations are fully reducible and that their irreducible components provide two distinct sets of bosonized operators realizing PsSSQM and corresponding to nonlinear spectra. We relate such results to some previous ones obtained for a GDOA connected with a C3-extended oscillator algebra (where C3 = ℤ3) in the case of linear spectra.

EXACT WAVE FUNCTIONS OF MULTI-SPECIES ANYON QUANTUM MECHANICS

1993 ◽  
Vol 08 (29) ◽  
pp. 5101-5113 ◽  
Author(s):  
K.H. CHO ◽  
CHAIHO RIM ◽  
D.S. SOH
Keyword(s):  
Magnetic Field ◽  
Quantum Mechanics ◽  
Angular Momentum ◽  
Wave Functions ◽  
Quantum Mechanical ◽  
Potential Well ◽  
Quantum Mechanical System ◽  
Ladder Operators ◽  
Free System ◽  
The Many

We investigate the many-anyon quantum-mechanical system of multi-species and obtain algebraically the energy and the angular momentum eigenstates in a harmonic potential well, in a constant magnetic field and of free system using ladder operators. The eigenvalues and their multiplicity of the eigenstates are given.

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