Bohr-van Leeuwen theorem

The Bohr–van Leeuwen theorem is a theorem in the field of statistical mechanics. The theorem states that when statistical mechanics and classical mechanics are applied consistently, the thermal average of the magnetization is always zero. This makes magnetism in solids solely a quantum mechanical effect and means that classical physics cannot account for diamagnetism, paramagnetism or ferromagnetism.

What is today known as the Bohr–van Leeuwen theorem was discovered by Niels Bohr in 1911 in his doctoral dissertation and was later rediscovered by Hendrika Johanna van Leeuwen in her doctoral thesis in 1919. In 1932, van Vleck formalized and expanded upon Bohr's initial theorem in a book he wrote on electric and magnetic susceptibilities.

The significance of this discovery is that classical physics does not allow for such things as paramagnetism, diamagnetism and ferromagnetism and thus quantum physics are needed to explain the magnetic events. This result, "perhaps the most deflationary publication of all time," may have contributed to Bohr's development of a quasi-classical theory of the hydrogen atom in 1913.

The Bohr–van Leeuwen theorem applies to an isolated system that cannot rotate (an isolated star could start rotating if exposed to a field). If, in addition, there is only one state of thermal equilibrium in a given temperature and field, and the system is allowed time to return to equilibrium after a field is applied, then there will be no magnetization.

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

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