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

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## Abstract

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## 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

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**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 $|{u}_{+}|=2$.

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**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