scholarly journals Nonperturbative approach for a time-dependent quantum mechanical system

2004 ◽  
Vol 69 (2) ◽  
Author(s):  
Hyeong-Chan Kim ◽  
Jae Hyung Yee
Keyword(s):  
Mechanical System ◽  
Time Dependent ◽  
Quantum Mechanical ◽  
Quantum Mechanical System

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.

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.

Theory of moments, Hückel rule and stability

1988 ◽  
Vol 53 (8) ◽  
pp. 1607-1612 ◽  
Author(s):  
Štěpán Pick
Keyword(s):  
Wave Function ◽  
Mechanical System ◽  
Present Approach ◽  
Quantum Mechanical ◽  
Quantum Mechanical System ◽  
Conjugated Systems ◽  
And Topology ◽  
Hückel Rule ◽  
Simple Chemical

The connection between moments of the electronic Hamiltonian and topology of a quantum mechanical system is studied. Based on simplifications similar to those usually employed in simple chemical and physical theories, criteria resembling the Hückel rule for cyclic conjugated systems are suggested. Several examples of interest in chemistry and solid physics are discussed. No information on the wave function is necessary in the present approach.

scholarly journals BMN gauge theory as a quantum mechanical system

2003 ◽  
Vol 558 (3-4) ◽  
pp. 229-237 ◽  
Author(s):  
N. Beisert ◽  
C. Kristjansen ◽  
J. Plefka ◽  
M. Staudacher
Keyword(s):  
Gauge Theory ◽  
Mechanical System ◽  
Quantum Mechanical ◽  
Quantum Mechanical System

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.

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