Quantum mechanical grad-<i>B</i> drift velocity operator in a weakly non-uniform magnetic field

Chan, Poh Kam; Oikawa, Shun-ichi; Kosaka, Wataru

doi:10.1063/1.4941096

Cited by 4 publications

(2 citation statements)

References 32 publications

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“…Another related topic is that the spatial non-uniformity of magnetic field can also give geometric phases [28][29][30]. A quantum treatment for gradient-B drift has been developed [34,35]. However, the construction of coherent states in inhomogeneous magnetic field requires more sophisticated techniques which are beyond the scope of this paper and will be discussed elsewhere.…”

Section: Discussionmentioning

confidence: 99%

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On the correspondence between classical geometric phase of gyro-motion and quantum Berry phase

Zhu

Qin

2017

Physics of Plasmas

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We show that the geometric phase of the gyro-motion of a classical charged particle in a uniform time-dependent magnetic field described by Newton's equation can be derived from a coherent Berry phase for the coherent states of the Schrödinger equation or the Dirac equation. This correspondence is established by constructing coherent states for a particle using the energy eigenstates on the Landau levels and proving that the coherent states can maintain their status of coherent states during the slow varying of the magnetic field. It is discovered that orbital Berry phases of the eigenstates interfere coherently to produce an observable effect (which we termed "coherent Berry phase"), which is exactly the geometric phase of the classical gyro-motion. This technique works for particles with and without spin. For particles with spin, on each of the eigenstates that makes up the coherent states, the Berry phase consists of two parts that can be identified as those due to the orbital and the spin motion. It is the orbital Berry phases that interfere coherently to produce a coherent Berry phase corresponding to the classical geometric phase of the gyro-motion.The spin Berry phases of the eigenstates, on the other hand, remain to be quantum phase factors for the coherent states and have no classical counterpart.

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Section: Discussionmentioning

confidence: 99%

“…where ρ[b] and φ[b] are defined in Eqs (34). and(35).ψ n,m (ρ, θ) = R n,m (ρ)e −i(n−m)θ are eigenstates on the Landau levels, and R n,m (ρ) are real.…”

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confidence: 99%

On the correspondence between classical geometric phase of gyro-motion and quantum Berry phase

Zhu

Qin

2017

Physics of Plasmas

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Quantum mechanical expansion of variance of a particle in a weakly non-uniform electric and magnetic field

2016

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We have solved the Heisenberg equation of motion for the time evolution of the position and momentum operators for a non-relativistic spinless charged particle in the presence of a weakly non-uniform electric and magnetic field. It is shown that the drift velocity operator obtained in this study agrees with the classical counterpart, and that, using the time dependent operators, the variances in position and momentum grow with time. The expansion rate of variance in position and momentum are dependent on the magnetic gradient scale length, however, independent of the electric gradient scale length. In the presence of a weakly non-uniform electric and magnetic field, the theoretical expansion rates of variance expansion are in good agreement with the numerical analysis. It is analytically shown that the variance in position reaches the square of the interparticle separation, which is the characteristic time much shorter than the proton collision time of plasma fusion. After this time, the wavefunctions of the neighboring particles would overlap, as a result, the conventional classical analysis may lose its validity. The broad distribution of individual particle in space means that their Coulomb interactions with other particles become weaker than that expected in classical mechanics.

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Quantum mechanical E × B drift velocity in a weakly inhomogeneous electromagnetic field

2017

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The analytical solution for the quantum mechanical drift velocity for a non-relativistic spinless charged particle of E × B drift in the presence of a weakly inhomogeneous electric and magnetic field for the magnetized plasma is presented. Using the Heisenberg equation of motion, the time evolution of the position and momentum operators for the charged particle is solved. From the time dependent operators, the analytical solution of the time dependent momenta operators and position operators is derived. The quantum mechanical expansion rates of variances are shown to agree with the numerical results. Most importantly, the quantum mechanical E × B drift velocity coincides perfectly with the classical drift velocity in the limit of Planck's constant being zero. With higher order electric field inhomogeneity, low energy particles would drift faster than what the classical drift theory predicts.

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Quantum mechanical grad-B drift velocity operator in a weakly non-uniform magnetic field

Cited by 4 publications

References 32 publications

On the correspondence between classical geometric phase of gyro-motion and quantum Berry phase

On the correspondence between classical geometric phase of gyro-motion and quantum Berry phase

Quantum mechanical expansion of variance of a particle in a weakly non-uniform electric and magnetic field

Quantum mechanical E × B drift velocity in a weakly inhomogeneous electromagnetic field

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