# Adiabatic Amplification of Energy and Magnetic Moment of a Charged Particle after the Magnetic Field Inversion

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Adiabatic Evolution of the Fock States

## 3. Examples of Adiabatic Coefficients

## 4. Multiple Adiabatic Passages of Magnetic Field through Zero Value

## 5. Generalization of the Born–Fock Theorem

## 6. Mean Energy

## 7. Adiabatic Evolution of the “Invariant States” of the Magnetic Moment Operator

**r**= $(x,y)$ and $\mathbf{p}=({p}_{x},{p}_{y})$]:

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Integrals with Squares of the Laguerre Polynomials

## Appendix B. Calculation of Coefficients in the Expansion (39)

## References

- Fock, V. Bemerkung zur Quantelung des harmonischen Oszillators im Magnetfeld. Z. Phys.
**1928**, 47, 446–448. [Google Scholar] [CrossRef] - Darwin, C.G. The diamagnetism of the free electron. Math. Proc. Camb. Philos. Soc.
**1931**, 27, 86–90. [Google Scholar] [CrossRef] - Erdélyi, A. (Ed.) Bateman Manuscript Project: Higher Transcendental Functions; McGraw-Hill: New York, NY, USA, 1953. [Google Scholar]
- Gradshtein, I.S.; Ryzhik, I.M. Table of Integrals, Series, and Products, 7th ed.; Academic: Amsterdam, The Netherlands, 2007. [Google Scholar]
- Malkin, I.A.; Man’ko, V.I.; Trifonov, D.A. Coherent states and transition probabilities in a time-dependent electromagnetic field. Phys. Rev. D
**1970**, 2, 1371–1385. [Google Scholar] [CrossRef] - Agayeva, R.G. Non-adiabatic parametric excitation of oscillator-type systems. J. Phys. A Math. Gen.
**1980**, 13, 1685–1699. [Google Scholar] [CrossRef] - Abdalla, M.S. Charged particle in the presence of a variable magnetic field. Phys. Rev. A
**1988**, 37, 4026–4029. [Google Scholar] [CrossRef] [PubMed] - Jannussis, A.; Vlahos, E.; Skaltsas, D.; Kliros, G.; Bartzis, V. Squeezed states in the presence of a time-dependent magnetic field. Nuovo Cim. B
**1989**, 104, 53–66. [Google Scholar] [CrossRef] - Baseia, B.; Mizrahi, S.S.; Moussa, M.H.Y. Generation of squeezing for a charged oscillator and a charged particle in a time dependent electromagnetic field. Phys. Rev. A
**1992**, 46, 5885–5889. [Google Scholar] [CrossRef] - Dodonov, V.V.; Man’ko, O.V.; Man’ko, V.I.; Polynkin, P.G.; Rosa, L. δ-kicked Landau levels. J. Phys. A Math. Gen.
**1995**, 28, 197–208. [Google Scholar] [CrossRef] - Delgado, F.C.; Mielnik, B. Magnetic control of squeezing effects. J. Phys. A Math. Gen.
**1998**, 31, 309–320. [Google Scholar] - Menouar, S.; Maamache, M.; Choi, J.R. An alternative approach to exact wave functions for time-dependent coupled oscillator model of charged particle in variable magnetic field. Ann. Phys.
**2010**, 325, 1708–1719. [Google Scholar] [CrossRef][Green Version] - Man’ko, V.I.; Zhebrak, E.D. Tomographic probability representation for states of charge moving in varying field. Opt. Spectrosc.
**2012**, 113, 624–629. [Google Scholar] [CrossRef] - Aguiar, V.; Guedes, I. Entropy and information of a spinless charged particle in time-varying magnetic fields. J. Math. Phys.
**2016**, 57, 092103. [Google Scholar] [CrossRef] - Born, M.; Fock, V. Beweis des Adiabatensatzes. Z. Phys.
**1928**, 51, 165–180. [Google Scholar] [CrossRef] - Malkin, I.A.; Man’ko, V.I.; Trifonov, D.A. Linear adiabatic invariants and coherent states. J. Math. Phys.
**1973**, 14, 576–582. [Google Scholar] [CrossRef] - Marzlin, K.P.; Sanders, B.C. Inconsistency in the application of the adiabatic theorem. Phys. Rev. Lett.
**2004**, 93, 160408. [Google Scholar] [CrossRef][Green Version] - Amin, M.H.S. Consistency of the adiabatic theorem. Phys. Rev. Lett.
**2009**, 102, 220401. [Google Scholar] [CrossRef][Green Version] - Yukalov, V.I. Adiabatic theorems for linear and nonlinear Hamiltonians. Phys. Rev. A
**2009**, 79, 052117. [Google Scholar] [CrossRef][Green Version] - Comparat, D. General conditions for quantum adiabatic evolution. Phys. Rev. A
**2009**, 80, 012106. [Google Scholar] [CrossRef][Green Version] - Rigolin, G.; Ortiz, G. Degenerate adiabatic perturbation theory: Foundations and applications. Phys. Rev. A
**2014**, 900, 022104. [Google Scholar] [CrossRef][Green Version] - Weinberg, P.; Bukov, M.; D’Alessio, L.; Polkovnikov, A.; Vajna, S.; Kolodrubetz, M. Adiabatic perturbation theory and geometry of periodically-driven systems. Phys. Rep.
**2017**, 688, 1–35. [Google Scholar] [CrossRef][Green Version] - Dodonov, V.V.; Dodonov, A.V. Adiabatic amplification of the harmonic oscillator energy when the frequency passes through zero. Entropy
**2023**, 25, 2. [Google Scholar] [CrossRef] [PubMed] - Johnson, M.H.; Lippmann, B.A. Motion in a constant magnetic field. Phys. Rev.
**1949**, 76, 828–832. [Google Scholar] [CrossRef] - Felderhof, B.U.; Raval, S.P. Diamagnetism of a confined electron gas. Physica A
**1976**, 82, 151–162. [Google Scholar] [CrossRef] - Friar, J.L.; Fallieros, S. Diamagnetism, gauge transformations, and sum rules. Am. J. Phys.
**1981**, 49, 847–849. [Google Scholar] [CrossRef] - March, N.H.; Tosi, M.P. Thermally averaged orbital diamagnetism of a localized Wigner oscillator. Nuovo Cim. D
**1985**, 6, 521–526. [Google Scholar] [CrossRef] - Stewart, A.M. General gauge independence of diamagnetism plus paramagnetism. Aust. J. Phys.
**2000**, 53, 613–629. [Google Scholar] [CrossRef] - Bliokh, K.Y.; Schattschneider, P.; Verbeeck, J.; Nori, F. Electron vortex beams in a magnetic field: A new twist on Landau levels and Aharonov–Bohm states. Phys. Rev. X
**2012**, 2, 041011. [Google Scholar] [CrossRef][Green Version] - Greenshields, C.R.; Stamps, R.L.; Franke-Arnold, S. Parallel axis theorem for free-space electron wavefunctions. New J. Phys.
**2015**, 17, 093015. [Google Scholar] [CrossRef] - Wakamatsu, M.; Kitadono, Y.; Zou, L.-P.; Zhang, P.-M. Revisiting the compatibility problem between the gauge principle and the observability of the canonical orbital angular momentum in the Landau problem. Ann. Phys.
**2021**, 434, 168647. [Google Scholar] [CrossRef] - Dodonov, V.V.; Horovits, M.B. Change of energy and magnetic moment of a quantum charged particle after a fast jump of the magnetic field in solenoids of arbitrary cross sections. Physica A
**2021**, 571, 125843. [Google Scholar] [CrossRef] - Dodonov, V.V.; Dodonov, A.V. Magnetic-moment probability distribution of a quantum charged particle in thermodynamic equilibrium. Phys. Rev. A
**2020**, 102, 042216. [Google Scholar] [CrossRef] - Dodonov, V.V. Magnetization dynamics of a harmonically confined quantum charged particle in time dependent magnetic fields inside a circular solenoid. J. Phys. A Math. Theor.
**2021**, 54, 295304. [Google Scholar] [CrossRef] - Dodonov, V.V.; Man’ko, V.I. Evolution of multidimensional systems. Magnetic properties of ideal gases of charged particles. In Invariants and the Evolution of Nonstationary Quantum Systems; Markov, M.A., Ed.; Nova Science: Commack, NY, USA, 1989; Volume 183, pp. 263–414. [Google Scholar]
- Dodonov, V.V.; Horovits, M.B. Energy and magnetic moment of a quantum charged particle in time-dependent magnetic and electric fields of circular and plane solenoids. Entropy
**2021**, 23, 1579. [Google Scholar] [CrossRef] - Dodonov, V.V.; Dodonov, A.V. Magnetic moment invariant Gaussian states of a charged particle in a homogeneous magnetic field. Eur. Phys. J. Plus
**2022**, 137, 575. [Google Scholar] [CrossRef] - Landau, L. Diamagnetismus der Metalle. Z. Phys.
**1930**, 64, 629–637. [Google Scholar] [CrossRef] - Dulock, V.A.; McIntosh, H.V. Degeneracy of cyclotron motion. J. Math. Phys.
**1966**, 7, 1401–1412. [Google Scholar] [CrossRef] - Malkin, I.A.; Man’ko, V.I. Coherent states of a charged particle in a magnetic field. Sov. Phys.—JETP
**1969**, 28, 527–532. [Google Scholar] - Feldman, A.; Kahn, A.H. Landau diamagnetism from the coherent states of an electron in a uniform magnetic field. Phys. Rev. B
**1970**, 1, 4584–4589. [Google Scholar] [CrossRef] - Tam, W.G. Coherent states and the invariance group of a charged particle in a uniform magnetic field. Physica
**1971**, 54, 557–572. [Google Scholar] [CrossRef] - Avron, J.E.; Herbst, I.W.; Simon, B. Separation of center of mass in homogeneous magnetic fields. Ann. Phys.
**1978**, 114, 431–451. [Google Scholar] [CrossRef] - Johnson, B.R.; Hirschfelder, J.O.; Yang, K.-H. Interaction of atoms, molecules, and ions with constant electric and magnetic fields. Rev. Mod. Phys.
**1983**, 55, 109–153. [Google Scholar] [CrossRef] - von Baltz, R. Guiding center motion of two interacting n = 0 Landau electrons in two dimensions. Phys. Lett. A
**1984**, 105, 371–373. [Google Scholar] [CrossRef] - Dodonov, V.V.; Man’ko, V.I.; Polynkin, P.G. Geometrical squeezed states of a charged particle in a time-dependent magnetic field. Phys. Lett. A
**1994**, 188, 232–238. [Google Scholar] [CrossRef] - Li, C.-F.; Wang, Q. The quantum behavior of an electron in a uniform magnetic field. Physica B
**1999**, 269, 22–27. [Google Scholar] [CrossRef] - Kowalski, K.; Rembieliński, J. Coherent states of a charged particle in a uniform magnetic field. J. Phys. A Math. Gen.
**2005**, 38, 8247–8258. [Google Scholar] [CrossRef][Green Version] - Gazeau, J.P.; Baldiotti, M.C.; Gitman, D.M. Coherent states of a particle in a magnetic field and the Stieltjes moment problem. Phys. Lett. A
**2009**, 373, 1916–1920. [Google Scholar] [CrossRef][Green Version] - Mielnik, B.; Ramírez, A. Magnetic operations: A little fuzzy mechanics? Phys. Scr.
**2011**, 84, 045008. [Google Scholar] [CrossRef] - Dodonov, V.V. Coherent states and their generalizations for a charged particle in a magnetic field. In Coherent States and Their Applications: A Contemporary Panorama; Antoine, J.-P., Bagarello, F., Gazeau, J.-P., Eds.; Springer: Berlin, Germany, 2018; Volume 205, pp. 311–338. [Google Scholar]
- Champel, T.; Florens, S. A solvable model of Landau quantization breakdown. Eur. Phys. J. B
**2019**, 92, 124. [Google Scholar] [CrossRef] - Wakamatsu, M.; Kitadono, Y.; Zou, L.; Zhang, P.M. The physics of helical electron beam in a uniform magnetic field as a testing ground of gauge principle. Phys. Lett. A
**2020**, 384, 126415. [Google Scholar] [CrossRef][Green Version] - Kitadono, Y.; Wakamatsu, M.; Zou, L.; Zhang, P.M. Role of guiding centre in Landau level system and mechanical and pseudo orbital angular momenta. Int. J. Mod. Phys. A
**2020**, 35, 2050096. [Google Scholar] [CrossRef] - Fletcher, R.J.; Shaffer, A.; Wilson, C.C.; Patel, P.B.; Yan, Z.; Crépel, V.; Mukherjee, B.; Zwierlein, M.W. Geometric squeezing into the lowest Landau level. Science
**2021**, 372, 1318–1322. [Google Scholar] [CrossRef] [PubMed] - Konstantinou, G.; Moulopoulos, K. Generators of dynamical symmetries and the correct gauge transformation in the Landau level problem: Use of pseudomomentum and pseudo-angular momentum. Eur. J. Phys.
**2016**, 37, 065401. [Google Scholar] [CrossRef] - van Enk, S.J. Angular momentum in the fractional quantum Hall effect. Am. J. Phys.
**2020**, 88, 286–291. [Google Scholar] [CrossRef][Green Version] - Robertson, H.P. The uncertainty principle. Phys. Rev.
**1929**, 34, 163–164. [Google Scholar] [CrossRef] - Szegö, G. Orthogonal Polynomials, 4th ed.; American Mathematical Society: Providence, RI, USA, 1975. [Google Scholar]

**Figure 1.**The probability distribution (44) of finding the initial Fock state $|n,m\rangle $ in the Fock state $|q,m\rangle $ after the frequency slowly passes through zero value, as a function of q for a fixed parameter m, in the case of $|{u}_{-}|=1$ and $|{u}_{+}|=\sqrt{2}$.

**Figure 2.**The probability distribution (44) of finding the initial Fock state $|n,m\rangle $ in the Fock state $|q,m\rangle $ after the frequency slowly passes through zero value, as a function of q for a fixed parameter n, in the case of $|{u}_{-}|=1$ and $|{u}_{+}|=\sqrt{2}$.

**Figure 3.**The probability distribution (44) of finding the initial Fock state $|n,m\rangle $ in the Fock state $|q,m\rangle $ after the frequency slowly passes through zero value, as a function of q for a fixed parameter m, in the case of $|{u}_{-}|=\sqrt{3}$ and $|{u}_{+}|=2$.

**Figure 4.**The probability distribution (44) of finding the initial Fock state $|n,m\rangle $ in the Fock state $|q,m\rangle $ after the frequency slowly passes through zero value, as a function of q for a fixed parameter n, in the case of $|{u}_{-}|=\sqrt{3}$ and $|{u}_{+}|=2$.

**Figure 5.**The probability of finding the initial Fock state $|n,m\rangle $ in the same Fock state after the frequency slowly passes through zero value, as a function of n for different fixed values of the angular moment quantum number $\left|m\right|$, for $|{u}_{-}|=1$ and $|{u}_{+}|=\sqrt{2}$.

**Figure 6.**The probability of finding the initial Fock state $|n,m\rangle $ in the same Fock state after the frequency slowly passes through zero value, as a function of n for different fixed values of the angular moment quantum number $\left|m\right|$, for $|{u}_{-}|=\sqrt{3}$ and $|{u}_{+}|=2$.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Dodonov, V.V.; Dodonov, A.V. Adiabatic Amplification of Energy and Magnetic Moment of a Charged Particle after the Magnetic Field Inversion. *Entropy* **2023**, *25*, 596.
https://doi.org/10.3390/e25040596

**AMA Style**

Dodonov VV, Dodonov AV. Adiabatic Amplification of Energy and Magnetic Moment of a Charged Particle after the Magnetic Field Inversion. *Entropy*. 2023; 25(4):596.
https://doi.org/10.3390/e25040596

**Chicago/Turabian Style**

Dodonov, Viktor V., and Alexandre V. Dodonov. 2023. "Adiabatic Amplification of Energy and Magnetic Moment of a Charged Particle after the Magnetic Field Inversion" *Entropy* 25, no. 4: 596.
https://doi.org/10.3390/e25040596