Bohr-van-Leeuwen theorem

The Bohr-van-Leeuwen theorem is a theorem from the field of solid state physics and statistical physics . It says that if classical statistics were applied, the magnetization in thermal equilibrium would be zero, since the kinetic energy of a charge does not change in the magnetic field. Accordingly, magnetism in solids is a purely quantum mechanical effect.

Heuristic classic view

The magnetization (the number of magnetic moments per unit volume) in proportion to the change in the energy of a system in a magnetic field. Since the force on a moving charge ( Lorentz force ) acts exactly perpendicular to the direction of movement of the charge, the field changes the direction of the charge, but the amount remains constant, i.e. that is, the change in energy is zero and so is the magnetization.

Mathematical proof

For one (or several) particles with charge or in magnetic fields with vector potential , the Hamilton function is defined by   . The first argument represents the so-called canonical impulse , while the kinetic impulse is.  is the kinetic energy while representing magnetic induction. In contrast to, the vector potential of the magnetic field is not unique, but you can add any gradient field to it without changing it. ${\displaystyle q}$ ${\displaystyle q_{i}}$ ${\displaystyle \mathbf {B} _{i}}$ ${\displaystyle \mathbf {A} _{i}}$ ${\displaystyle H\{(\mathbf {p} _{i}-{\frac {q_{i}}{c}}\mathbf {A} _{i}(\mathbf {r} _{i},t),\mathbf {r} _{i})\}}$ ${\displaystyle \mathbf {p} _{i}=m\mathbf {v} _{i}-{\frac {q_{i}}{c}}\mathbf {A} _{i}}$ ${\displaystyle m\mathbf {v} _{i}}$ ${\displaystyle {\frac {m}{2}}\cdot \mathbf {v} _{i}^{2}}$ ${\displaystyle \mathbf {B} _{i}={\rm {{rot\,\,}\mathbf {A} _{i}}}}$ ${\displaystyle \mathbf {A} _{i}}$ ${\displaystyle m\cdot \mathbf {v} _{i}}$ ${\displaystyle \mathbf {A} _{i}}$ ${\displaystyle \mathbf {B} _{i}}$

Nevertheless, the partition function of a system of N such (indistinguishable) particles is classically defined in statistical physics via the canonical momentum:

${\displaystyle Z={\frac {1}{h^{3N}N!}}\int _{\mathbb {R} ^{3N}}\mathrm {d} ^{3N}p\int _{V^{N}}\mathrm {d} ^{3N}r\,}$ ${\displaystyle \mathrm {e} ^{-\beta H\{(\mathbf {p} _{i}-{\frac {q_{i}}{c}}\mathbf {A} (\mathbf {r} _{i}),\mathbf {r} _{i})\}}\,,}$

treating this in three dimensions.

Now you go to kinetic momentum than by substituting: . Since all impulses are integrated over the entire three-dimensional space, the integral limits do not change. The sum of states then becomes Since this is no longer dependent on the vector potential and therefore also not dependent on the external magnetic field , the magnetization disappears, ${\displaystyle \mathbf {p} _{i}\equiv \mathbf {p} _{i}'+{\frac {q_{i}}{c}}\mathbf {A} (\mathbf {r} _{i})}$ ${\displaystyle \mathbf {p} _{i}'}$ ${\displaystyle Z={\frac {1}{h^{3N}N!}}\int _{\mathbb {R} ^{3N}}\mathrm {d} ^{3N}p'\int _{V^{N}}\mathrm {d} ^{3N}r\,\mathrm {e} ^{-\beta H\{(\mathbf {p} _{i}',\mathbf {r} _{i})\}}}$ ${\displaystyle \mathbf {A} }$ ${\displaystyle \mathbf {B} }$

${\displaystyle \mathbf {M} =-{\frac {\partial F}{\partial \mathbf {B} }}\equiv 0\,,}$where is the free energy. ${\displaystyle F=-k_{B}T\ln(Z)}$

Deviation in quantum mechanics

In quantum mechanics, the Bohr-van Leeuven theorem no longer applies because the spin of the particle has to be taken into account. As a result, the simple relationship, i.e. the independence of the kinetic energy from the vector potential , no longer applies in quantum mechanics. ${\displaystyle E_{kin}={\frac {m}{2}}v_{i}^{2}={\frac {\mathrm {p} _{i}'^{2}}{2m}}\,,}$

Instead, the Hamilton operator depends explicitly on the internal and external magnetic fields, whereby magnetism as a specific quantum mechanical phenomenon in a non-square order with respect to the magnetic field strength, e.g. B. Ferromagnetism of certain solids and paramagnetism in certain molecules can come about.

References and footnotes

1. ↑ The so-called cgs system is used here without loss of generality . In the alternative international system of units , c is replaced by 1.

See also

• Magnetism , especially the sub-chapter on the quantum mechanical explanation of the phenomenon.