Adiabatic invariants

2020 ◽  
pp. 71-76
Author(s):  
Gleb L. Kotkin ◽  
Valeriy G. Serbo
Keyword(s):  
Magnetic Field ◽  
Harmonic Oscillator ◽  
Single Molecule ◽  
Uniform Magnetic Field ◽  
Adiabatic Approximation ◽  
Magnetic Trap ◽  
Simplified Model ◽  
Adiabatic Invariants ◽  
Mathematical Pendulum ◽  
Simple Systems

This chapter addresses the adiabatic invariant for a mathematical pendulum, a model of a “gas” consisting of a single molecule in a piston, adiabatic approximation, and a simplified model of an ion H2+. The chapter also discusses the connection between the volume and the pressure of a gas consisting of particles inside an elastic cube, the adiabatic invariants for a charged anisotropic harmonic oscillator in a uniform magnetic field, a magnetic trap, and the action and angle variables for the simple systems.

Adiabatic invariants

2020 ◽  
pp. 359-376
Author(s):  
Gleb L. Kotkin ◽  
Valeriy G. Serbo
Keyword(s):  
Magnetic Field ◽  
Harmonic Oscillator ◽  
Single Molecule ◽  
Uniform Magnetic Field ◽  
Adiabatic Approximation ◽  
Magnetic Trap ◽  
Simplified Model ◽  
Adiabatic Invariants ◽  
Mathematical Pendulum ◽  
Simple Systems

This chapter addresses the adiabatic invariant for a mathematical pendulum, a model of a “gas” consisting of a single molecule in a piston, adiabatic approximation, and a simplified model of an ion H2+. The chapter also discusses the connection between the volume and the pressure of a gas consisting of particles inside an elastic cube, the adiabatic invariants for a charged anisotropic harmonic oscillator in a uniform magnetic field, a magnetic trap, and the action and angle variables for the simple systems.

ENERGY LEVEL OF ELECTRON IN AN ANISOTROPIC QUANTUM DOT UNDER A MAGNETIC FIELD BY AN INVARIANT EIGENOPERATOR METHOD

2006 ◽  
Vol 20 (32) ◽  
pp. 5417-5425
Author(s):  
HONG-YI FAN ◽  
TONG-TONG WANG ◽  
YAN-LI YANG
Keyword(s):  
Magnetic Field ◽  
Energy Level ◽  
Quantum Dot ◽  
Harmonic Oscillator ◽  
Uniform Magnetic Field ◽  
Energy Levels ◽  
Oscillator Potential ◽  
Harmonic Oscillator Potential ◽  
Isotropic Harmonic Oscillator ◽  
Anisotropic Quantum Dot

We show that the recently proposed invariant eigenoperator method can be successfully applied to solving energy levels of electron in an anisotropic quantum dot in the presence of a uniform magnetic field (UMF). The result reduces to the energy level of electron in isotropic harmonic oscillator potential and in UMF naturally. The Landau diamagnetism decreases due to the existence of the anisotropic harmonic potential.

ATOMIC COHERENT STATES AS THE EIGENSTATES OF A TWO-DIMENSIONAL ANISOTROPIC HARMONIC OSCILLATOR IN A UNIFORM MAGNETIC FIELD

2009 ◽  
Vol 24 (38) ◽  
pp. 3129-3136 ◽  
Author(s):  
XIANG-GUO MENG ◽  
JI-SUO WANG ◽  
HONG-YI FAN
Keyword(s):  
Magnetic Field ◽  
Harmonic Oscillator ◽  
Coherent States ◽  
Uniform Magnetic Field ◽  
Hermite Polynomial ◽  
Entangled State ◽  
Two Dimensional ◽  
Entangled State Representation ◽  
Single Variable ◽  
State Representation

In the newly constructed entangled state representation embodying quantum entanglement of Einstein, Podolsky and Rosen, the usual wave function of atomic coherent state ∣τ〉 = exp (μJ+-μ*J-)∣j, -j〉 turns out to be just proportional to a single-variable ordinary Hermite polynomial of order 2j, where j is the spin value. We then prove that a two-dimensional time-independent anisotropic harmonic oscillator in a uniform magnetic field possesses energy eigenstates which can be classified as the states ∣τ〉 in terms of the spin values j.

scholarly journals An algebraic approach to a charged particle in a uniform magnetic field

2018 ◽  
Vol 64 (2) ◽  
pp. 127
Author(s):  
D. Ojeda-Guillén ◽  
M. Salazar-Ramírez ◽  
R.D. Mota ◽  
V.D. Granados
Keyword(s):  
Magnetic Field ◽  
Charged Particle ◽  
Harmonic Oscillator ◽  
Coherent States ◽  
Uniform Magnetic Field ◽  
Similarity Transformation ◽  
Algebraic Approach ◽  
Landau Gauge ◽  
Displacement Operator ◽  
Symmetric Gauge

We study the problem of a charged particle in a uniform magnetic field with two different gauges, known as Landau and symmetric gauges. By using a similarity transformation in terms of the displacement operator we show that, for the Landau gauge, the eigenfunctions for this problem are the harmonic oscillator number coherent states. In the symmetric gauge, we calculate the SU(1; 1) Perelomov number coherent states for this problem in cylindrical coordinates in a closed form. Finally, we show that these Perelomov number coherent states are related to the harmonic oscillator number coherent states by the contraction of the SU(1; 1) group to the Heisenberg-Weyl group.

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