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Overview
The Bohr Van Leeuwen theorem is a theorem in the field of solid state physics. The theorem posits that when applying classical statistics the magnetization in the thermal balance is zero because the kinetic energy of a pull in the magnetic field does not alter it. This makes magnetism in solids solely a quantum mechanical effect and means that classical physics cannot account for diamagnetism paramagnetism or ferromagnetism. Van Vleck describes the Bohr Van Leeuwen theorem succinctly here “At any finite temperature, and in all finite applied electrical or magnetical fields, the net magnetization of a collection of electrons in thermal equilibrium vanishes identically.”

History
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 H.J.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 subsceptibilities. The significance of this discovery is that classical physics does not allow for such things as paramagnetism, diamagnetism and ferromagnetism and thus quantum physics and relativity are needed to explain the magnetic events.

Proof
The Bohr Van Leeuwen theorem has a mathematical proof which shows the need for quantum mechanics to account for several magnetic effects.

The Maxwell-Boltzmann thermal distribution function gives the probability that the nth particle has momentum pn and coordinate rn, as the following function: $$ exp[(-1/KT)H(p1,...;..|rN)]dp1...drN $$ where

k = Boltzmann’s constant,

T = temperature,

H = Hamilton’s function, the total energy of the system.

The thermal average (TA) of any function $$F(p1, . . . ; . . . rN)$$ of these generalized coordinates is then $$TA=f FdP/f dP$$. $$jn = envn$$, is the current density created by the movement of a single charge so the integral solution is $$E, n=1->n en((Vn*Rn)/cR3n$$.where $$Rn=r-rn$$. When no magnetic field is present $$H=E,n0.5Mn(V^2)n+U(r1......rn)$$ where $$vn = pn/mn = drn/dt$$. The first term corresponding to the kinetic energy and the second to the potential energy. Newtons second law $$enE=p*n$$can then be used to derive forces from the potentials using the equation $$pn -> pn + enA(rn,t)/c$$. This gives the equation for the velocity of a particle under the effect of a vector potential m$$nvn=pn+(en/c)*A(rn,t)$$ and leads to calculation of the thermal average using $$J=det||(mn*avn)/(a*pm)||=1$$ in which $$A(r, t)$$ disappear from the Maxwell-Boltzmann factor and also the quantity H(r) when divided by the Jacobian of the transformation. This gives mathematical proof to the theorems assertion that magnetic moments caused by an outside field vanish identically.

The Bohr Van Leeuwen Theorem Applied
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.” Electromechanics and electrical engineering also see practical benefit from the Bohr van Leeuwen theorem.