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

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.

FICTON THEORY OF DYNAMICAL SYSTEMS WITH NOISY PARAMETERS

1965 ◽  
Vol 43 (4) ◽  
pp. 619-639 ◽  
Author(s):  
R. C. Bourret
Keyword(s):  
Magnetic Field ◽  
Exact Solution ◽  
Dynamical Systems ◽  
Approximate Solutions ◽  
Frequency Parameter ◽  
Time Dependent ◽  
Quantum Mechanical ◽  
Graphical Comparison ◽  
The One ◽  
Random Fluctuations

The theory of randomly perturbed waves described previously (Bourret 1962a, b) is presented in a form applicable to purely time-dependent systems, classical or quantum mechanical. It is then applied to the problem of a spin-[Formula: see text] dipole in a magnetic field with random fluctuations. One- and two-ficton processes are taken into account and a "renormalization" approximation is given also. Graphical comparison of the approximate solutions with the exact solution is presented. As a classical example, the harmonic oscillator with a noisy frequency parameter is analyzed in both the one- and two-ficton approximations.

scholarly journals Полностью оптический магнитометрический датчик для задач магнитоэнцефалографии и томографии сверхслабого поля

2020 ◽  
Vol 46 (17) ◽  
pp. 43
Author(s):  
А.К. Вершовский ◽  
А.С. Пазгалев ◽  
М.В. Петренко
Keyword(s):  
Magnetic Field ◽  
Magnetic Resonance ◽  
Shot Noise ◽  
Magnetic Moment ◽  
Resonance Line ◽  
Optical Pumping ◽  
Resonance Excitation ◽  
A Cell ◽  
Resonance Line Width ◽  
The Magnetic Field

A version of the scheme of an atomic cesium vapour magnetometric sensor using magnetic resonance excitation by modulated radiation transverse to the magnetic field of hyperfine optical pumping is proposed and experimentally studied. It is shown that when using a cell with a volume of 0.125 cm3, the variational sensitivity of the sensor, estimated from the ratio of the steepness of the signal at the center of the magnetic resonance to the shot noise of the detecting radiation, reaches a level of less than 10 fT/Hz1/2 in the frequency band determined by the magnetic resonance line width (of the order of 800 Hz). The sensor, which does not use and does not emit resonant radio-frequency fields, is designed to operate in magnetoencephalographic complexes. Possible ways to increase the frequency response of the circuit for detecting relatively fast (~ 4.2 kHz in a field of 0.1 mT) proton magnetic moment precession signals in promising ultralow field tomography schemes are considered.

scholarly journals A Quantum Charged Particle under Sudden Jumps of the Magnetic Field and Shape of Non-Circular Solenoids

2019 ◽  
Vol 1 (2) ◽  
pp. 193-207 ◽  
Author(s):  
Viktor V. Dodonov ◽  
Matheus B. Horovits
Keyword(s):  
Magnetic Field ◽  
Charged Particle ◽  
Magnetic Fields ◽  
Electric Fields ◽  
Cross Sections ◽  
Magnetic Moment ◽  
Landau Gauge ◽  
Time Dependent ◽  
The Mean ◽  
The Magnetic Field

We consider a quantum charged particle moving in the x y plane under the action of a time-dependent magnetic field described by means of the linear vector potential of the form A = B ( t ) − y ( 1 + β ) , x ( 1 − β ) / 2 . Such potentials with β ≠ 0 exist inside infinite solenoids with non-circular cross sections. The systems with different values of β are not equivalent for nonstationary magnetic fields or time-dependent parameters β ( t ) , due to different structures of induced electric fields. Using the approximation of the stepwise variations of parameters, we obtain explicit formulas describing the change of the mean energy and magnetic moment. The generation of squeezing with respect to the relative and guiding center coordinates is also studied. The change of magnetic moment can be twice bigger for the Landau gauge than for the circular gauge, and this change can happen without any change of the angular momentum. A strong amplification of the magnetic moment can happen even for rapidly decreasing magnetic fields.

EXACT SOLUTIONS OF SCHRODINGER’S EQUATION FOR SPIN SYSTEMS IN A CLASS OF TIME-DEPENDENT MAGNETIC FIELDS

1991 ◽  
Vol 05 (29) ◽  
pp. 1919-1924 ◽  
Author(s):  
M.J. TAHMASEBI ◽  
Y. SOBOUTI
Keyword(s):  
Magnetic Field ◽  
Magnetic Fields ◽  
Exact Solutions ◽  
Spin System ◽  
Schrodinger Equation ◽  
Unitary Transformation ◽  
Variable Magnetic Field ◽  
Spin Systems ◽  
Time Variable ◽  
Time Dependent

A spin system in a time variable magnetic field is considered. For certain fields there exists a frame in which the Hamiltonian becomes static. The criterion for such fields is derived. The unitary transformation that accomplishes this task is obtained and the underlying Schrodinger equation is solved exactly.

scholarly journals Representation of population exchange at level anti-crossings

2020 ◽  
Vol 1 (2) ◽  
pp. 347-365
Author(s):  
Bogdan A. Rodin ◽  
Konstantin L. Ivanov
Keyword(s):  
Magnetic Field ◽  
Nuclear Magnetic Resonance ◽  
Magnetic Resonance ◽  
Spin System ◽  
Evolution Operator ◽  
Singlet State ◽  
Spin Dynamics ◽  
Theoretical Framework ◽  
Cross Polarization ◽  
Population Exchange

Abstract. A theoretical framework is proposed to describe the spin dynamics driven by coherent spin mixing at level anti-crossings (LACs). We briefly introduce the LAC concept and propose to describe the spin dynamics using a vector of populations of the diabatic eigenstates. In this description, each LAC gives rise to a pairwise redistribution of eigenstate populations, allowing one to construct the total evolution operator of the spin system. Additionally, we take into account that in the course of spin evolution a “rotation” of the eigenstate basis case take place. The approach is illustrated by a number of examples, dealing with magnetic field inversion, cross-polarization, singlet-state nuclear magnetic resonance and parahydrogen-induced polarization.

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