User:Schenad/波尔 - 范·列文定理

玻尔 - 范·列文定理（Bohr–van Leeuwen theorem）是统计力学领域中的一个定理。定理的基本阐述是：在热力学和经典力学的框架下，磁化強度的热力学平均恒等于零。这意味着固体中的磁现象完全是一种量子力学效应；经典物理学无法对抗磁性做出解释。

历史

玻尔 - 范·列文定理最初是由尼尔斯·玻尔于1911年发现的，玻尔将其写入了自己的博士论文。1919年，Hendrika Johanna van Leeuwen重新发现了这一定理，她也将此定理写入了自己的博士论文。1932年，約翰·凡扶累克在他的一本关于电极化率和磁化率的书中规范化且拓展了玻尔最初的理论。

这一发现的重要之处在于无法从經典物理學推导出順磁性、抗磁性和铁磁性。因此，上述的磁现象必须用量子力学来解释。玻尔 - 范·列文定理也许是玻尔在1913年打破传统的经典理论，提出半经典的玻尔模型的原因之一。

证明

根据直觉的初步证明

玻尔-范·列文定理适用于无法旋转的孤立系统（如果一颗孤立的恒星暴露在磁场中，该恒星可能会开始旋转）。如果在给定的温度和磁场下只有一个唯一的热平衡状态，并且在施加磁场后，我们让系统有足够的时间回归到平衡态，则磁化现象不会出现。

The probability that the system will be in a given state of motion is predicted by Maxwell–Boltzmann statistics to be proportional to $\exp(-U/k_{\text{B}}T)$, where $U$ is the energy of the system, $k_{\text{B}}$ is the Boltzmann constant, and $T$ is the absolute temperature. This energy is equal to the kinetic energy $(mv^{2}/2)$ for a particle with mass $m$ and speed $v$ and the potential energy.

The magnetic field does not contribute to the potential energy. The Lorentz force on a particle with charge $q$ and velocity $\mathbf {v}$ is

$\mathbf {F} =q\left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right),$

where $\mathbf {E}$ is the electric field and $\mathbf {B}$ is the magnetic flux density. The rate of work done is $\mathbf {F} \cdot \mathbf {v} =q\mathbf {E} \cdot \mathbf {v}$ and does not depend on $\mathbf {B}$. Therefore, the energy does not depend on the magnetic field, so the distribution of motions does not depend on the magnetic field.

在零磁场下，由于系统无法旋转，系统中带电粒子运动的统计净值为零，因此平均磁矩为零。又因为粒子运动的分布不依赖于磁场，所以任何磁场中处于热平衡状态下的磁矩仍然为零。

更加形式化的证明

So as to lower the complexity of the proof, a system with $N$ electrons will be used.

This is appropriate, since most of the magnetism in a solid is carried by electrons, and the proof is easily generalized to more than one type of charged particle.

Each electron has a negative charge $e$ and mass $m_{\text{e}}$.

If its position is $\mathbf {r}$ and velocity is $\mathbf {v}$, it produces a current $\mathbf {j} =e\mathbf {v}$ and a magnetic moment

$\mathbf {\mu } ={\frac {1}{2c}}\mathbf {r} \times \mathbf {j} ={\frac {e}{2c}}\mathbf {r} \times \mathbf {v} .$

The above equation shows that the magnetic moment is a linear function of the position coordinates, so the total magnetic moment in a given direction must be a linear function of the form

$\mu =\sum _{i=1}^{N}\mathbf {a} _{i}\cdot {\dot {\mathbf {r} }}_{i},$

where the dot represents a time derivative and $\mathbf {a} _{i}$ are vector coefficients depending on the position coordinates $\{\mathbf {r} _{i},i=1\ldots N\}$.

Maxwell–Boltzmann statistics gives the probability that the nth particle has momentum $\mathbf {p} _{n}$ and coordinate $\mathbf {r} _{n}$ as

$dP\propto \exp {\left[-{\frac {{\mathcal {H}}(\mathbf {p} _{1},\ldots ,\mathbf {p} _{N};\mathbf {r} _{1},\ldots ,\mathbf {r} _{N})}{k_{\text{B}}T}}\right]}d\mathbf {p} _{1},\ldots ,d\mathbf {p} _{N}d\mathbf {r} _{1},\ldots ,d\mathbf {r} _{N},$

where ${\mathcal {H}}$ is the Hamiltonian, the total energy of the system.

The thermal average of any function $f(\mathbf {p} _{1},\ldots ,\mathbf {p} _{N};\mathbf {r} _{1},\ldots ,\mathbf {r} _{N})$ of these generalized coordinates is then

$\langle f\rangle ={\frac {\int fdP}{\int dP}}.$

In the presence of a magnetic field,

${\mathcal {H}}={\frac {1}{2m_{\text{e}}}}\sum _{i=1}^{N}\left(\mathbf {p} _{i}-{\frac {e}{c}}\mathbf {A} _{i}\right)^{2}+e\phi (\mathbf {q} ),$

where $\mathbf {A} _{i}$ is the magnetic vector potential and $\phi (\mathbf {q} )$ is the electric scalar potential. For each particle the components of the momentum $\mathbf {p} _{i}$ and position $\mathbf {r} _{i}$ are related by the equations of Hamiltonian mechanics:

${\begin{aligned}{\dot {\mathbf {p} }}_{i}&=-\partial {\mathcal {H}}/\partial \mathbf {r} _{i}\\{\dot {\mathbf {r} }}_{i}&=\partial {\mathcal {H}}/\partial \mathbf {p} _{i}.\end{aligned}}$

Therefore,

${\dot {\mathbf {r} }}_{i}\propto \mathbf {p} _{i},$

so the moment $\mu$ is a linear function of the momenta $\mathbf {p} _{i}$.

The thermally averaged moment,

$\langle \mu \rangle ={\frac {\int \mu dP}{\int dP}},$

is the sum of terms proportional to integrals of the form

$\int _{-\infty }^{\infty }pdp,$

where $p$ represents one of the moment coordinates.

The integrand is an odd function of $p$, so it vanishes.

Therefore, $\langle \mu \rangle =0$.

玻尔-范·列文定理的应用

The Bohr–van Leeuwen theorem is useful in several applications including plasma physics, "All these references base their discussion of the Bohr–van Leeuwen theorem on Niels Bohr's physical model, in which perfectly reflecting walls are necessary to provide the currents that cancel the net contribution from the interior of an element of plasma, and result in zero net diamagnetism for the plasma element."

Diamagnetism of a purely classical nature occurs in plasmas but is a consequence of thermal disequilibrium, such as a gradient in plasma density. Electromechanics and electrical engineering also see practical benefit from the Bohr–van Leeuwen theorem.

参考资料

• Aharoni, Amikam. Introduction to the Theory of Ferromagnetism. Clarendon Press. 1996. ISBN 0-19-851791-2. （原始内容存档于2011-06-29）. 
• Bohr, Niehls. The Doctor's Dissertation (Text and Translation). Rosenfeld, L.; Nielsen, J. Rud (编). Early Works (1905-1911). Niels Bohr Collected Works 1. Elsevier. 1972: 163, 165–393 [originally published as "Studier over Metallernes Elektrontheori", Københavns Universitet (1911)]. ISBN 978-0-7204-1801-9. doi:10.1016/S1876-0503(08)70015-X. 
• Feynman, Richard P.; Leighton, Robert B.; Sands, Matthew. The Feynman Lectures on Physics 2. 2006. ISBN 0-8053-9045-6. 
• Roth, Reece. Plasma Stability and the Bohr-Van Leeuwen Theorem (PDF). NASA. 1967 [2008-10-27]. 
• van Leeuwen, Hendrika Johanna. Problèmes de la théorie électronique du magnétisme. Journal de Physique et le Radium. 1921, 2 (12): 361–377. 
• van Vleck, J. H. The theory of electric and magnetic susceptibilities. Clarendon Press. 1932. ISBN 0-19-851243-0. 
• van Vleck, J. H. Quantum mechanics: The key to understanding magnetism (Nobel lecture, 8 December 1977). Lundqvist, Stig (编). Nobel Lectures in Physics 1971-1980. World Scientific. 1992. ISBN 981-02-0726-3. 

外部链接

• The early 20th century: Relativity and quantum mechanics bring understanding at last

[[Category:经典力学]] [[Category:物質內的電場和磁場]] [[Category:物理定理]] [[Category:统计力学]]