How does one explain magnetic effects in matter when one views matter as a collection of classical charges in motion? The answer is: not at all! This is one of the points that Niels Bohr made in his doctoral dissertation in 1911, two years before addressing the issue of the stability of the hydrogen atom. The result, later rediscovered by H.J. van Leeuwen was amplified and formalized in Van Vleck’s 1932 text on electric and magnetic susceptibilities and it is currently known as the Bohr-van Leeuwen theorem. We will review Bohr’s two derivations, one statistical and one based on the motion of individual electrons. We will then propose reasons why this result, unlike that on the stability of hydrogen, did not lead to a major development in quantum theory but, instead, had to wait until after the introduction of spin and exchange forces in quantum mechanics to become generally known. The classical-quantum discontinuity • Classical Theory (mechanics, field theory, thermal, ...) 1. Critical discoveries: x, α, β, γ, e (e/m), nucleus 2. h-bar effects: Planck, Einstein, Compton, ... 3. Atomic models and stability of matter (Old quantum theory: Bohr & Sommerfeld) • Quantum Mechanics (Schrodinger, Heisenberg, Dirac, ...) Did we miss something? Classical Magnetism (1) • B fields in vacuum: Maxwell’s equation • Lorentz’ force: F = q ( E + v x B ) • Magnetism in matter susceptibilities (~T) Curie 1895 OK Zeeman effect Lorentz 1896 OK diamagnetism Langevin 1905 OK paramagnetism Langevin 1905 ? & permanent magnets Weiss 1907 ? Classical Magnetism (2) Failures... Anomalous Zeeman effect Anomalously large molecular fields ... and demise! Bohr-Van Leeuwen theorem: Lack of relevance of classical theory & Need for quantum theory!
How does one explain magnetic effects in matter when one views matter as a collection of classical charges in motion? The answer is: not at all! This is one of the points that Niels Bohr made in his doctoral dissertation in 1911, two years before addressing the issue of the stability of the hydrogen atom. The result, later rediscovered by H.J. van Leeuwen was amplified and formalized in Van Vleck’s 1932 text on electric and magnetic susceptibilities and it is currently known as the Bohr-van Leeuwen theorem. We will review Bohr’s two derivations, one statistical and one based on the motion of individual electrons. We will then propose reasons why this result, unlike that on the stability of hydrogen, did not lead to a major development in quantum theory but, instead, had to wait until after the introduction of spin and exchange forces in quantum mechanics to become generally known. The classical-quantum discontinuity • Classical Theory (mechanics, field theory, thermal, ...) 1. Critical discoveries: x, α, β, γ, e (e/m), nucleus 2. h-bar effects: Planck, Einstein, Compton, ... 3. Atomic models and stability of matter (Old quantum theory: Bohr & Sommerfeld) • Quantum Mechanics (Schrodinger, Heisenberg, Dirac, ...) Did we miss something? Classical Magnetism (1) • B fields in vacuum: Maxwell’s equation • Lorentz’ force: F = q ( E + v x B ) • Magnetism in matter susceptibilities (~T) Curie 1895 OK Zeeman effect Lorentz 1896 OK diamagnetism Langevin 1905 OK paramagnetism Langevin 1905 ? & permanent magnets Weiss 1907 ? Classical Magnetism (2) Failures... Anomalous Zeeman effect Anomalously large molecular fields ... and demise! Bohr-Van Leeuwen theorem: Lack of relevance of classical theory & Need for quantum theory!
1.General mathematical theory.---Modifications are given in the general derivation of the Langevin-Debye formula by means of quantum mechanics published in part I which are required by the moment now being magnetic rather than electric. The main additions are the appearance of a diamagnetic term, and allowance for the fact that the magnetic moment consists of two parts arising respectively from orbital electronic motions and from the electrons' internal spins. These parts cannot always be treated as rigidly coupled to form a "permanent" resultant moment. The Hamiltonian function used for the internal spin is that of a spherical top.2.Diamagnetism.---The writer's previous claim that Pauli's formula for the diamagnetic susceptibility can be applied to molecules as well as atoms is shown to be invalid because of the anomalous fact that the square of the angular momentum, unlike the average angular momentum, does not vanish even in S states when there is more than one nucleus. Pauli's formula is instead an upper limit to the diamagnetism in non-monatomic molecules and is a good approximation when the Schr\"odinger wave function has nearly as much symmetry as in an atomic S state.3.Paramagnetism of atoms.---Limiting values for the paramagnetic susceptibilities of atoms are $\ensuremath{\chi}=N\frac{[4s(s+1)+k(k+1)]{\ensuremath{\beta}}^{2}}{3kT}$ and $\ensuremath{\chi}=[\frac{N{g}^{2}j(j+1){\ensuremath{\beta}}^{2}}{3kT}]+N\ensuremath{\alpha}$ where $\ensuremath{\beta}$ is the Bohr magneton and $\ensuremath{\alpha}$ is a constant. These two formulas are rigorously applicable when the multiplet intervals are respectively very small or very large compared to kT, and are valid regardless of whether the magnetic field is strong enough to change the quantization by producing a Paschen-Back effect. The formula for small multiplets yields susceptibilities slightly different from those given by Laporte and Sommerfeld's expression for this case, and is much simpler, as they overlooked the contribution of the portion of the magnetic moment which is perpendicular to the invariable axis.4.Paramagnetism of molecules.---The susceptibility is calculated on the basis of the Hund theory of molecular quantization. Formulas are given applicable to his couplings of type (a) and type (b) provided in the former the multiplet intervals are either very large or very small compared to kT. The experimental susceptibilities for O2 and ClO2 are in accord with the assumption that the normal states are respectively 3S and 2S terms. In the particular case of S terms the numerical results are the same in the atomic and molecular formulas, but, unlike previous theories, it is not necessary to suppose the orbits are as freely oriented in molecules as in atoms. Polyatomic molecules may have lower paramagnetic susceptibilities than diatomic ones because the dissymmetry causes large fluctuations in electronic angular momentum.5.Paramagnetism of nitic oxide.---Spectroscopists have recently found that the normal states of the NO molecule are 2P terms separated by 120.9 ${\mathrm{cm}}^{\ensuremath{-}1}$. This permits an unambigous calculation of the susceptibility of NO which agrees with the experimental value within 1.5 per cent. Deviations from Curie's law are calculated which result from the doublet interval being comparable with kT. These deviations should be detectable experimentally if the susceptibility of NO could be measured over a wide temperature range.
The entropy approach to a description of quantum processes, represented like the Markov chain, is elaborated. It is supposed that at an atomic scale the law of the conservation of entropy holds. From this principle issues the simple explanation of the different behavior of bosons and fermions in experiments with scattering particles, proved by Pauli. Introducing in consideration Markov’ chain, applied to a motion of particles, made possible to write an equation, determining the change of their states. This pure logical result, completed by physical laws, brought to a new equation, generalizing Schrödinger one. It includes electric and magnetic fields as well as the spin of particles. Analysis of its solution for a case of only electric field, led to the formula of tunnel effect. Consideration of the motion of a particle in magnetic field gave the fundamental expression of Lorentz force without any mention of electrodynamics. Addition of an angular momentum in the reasoning by including Hamiltonian, allowed deducing the result, which coincides with the spherically symmetric Schrödinger equation for hydrogen atom. In general case there is the system of equations. If the atom is put in a weak magnetic field, without a possibility of flipping, the classical formulas of Bohr radius orbits and Rydberg energy obtained. The property of intrinsic quantization of electron spin ħ/2 is explained easily as a solution of the proposed system. These equations do not repeat the known arguments only, but can predict the new phenomena. They describe the effect of spin depolarization of electrons in a strong magnetic field, earlier considered been related to vacuum polarization. The obtained solution gives its magnetic induction equal about 1012 Gs, that corresponds well with experimental data.
The origin of the spin concept was probably the most complicated of all the quantum physics. Indeed, after Bohr's triumph in 1913, Bohr and Sommerfeld's "old quantum theory'', based on the idea of quantum restrictions applied to classical magnitudes, gave a unified explanation of the spectroscopic data. Atomic physics shows that it is not possible to account for the observed effects if we assume that the electron, a point particle, and the three degrees of freedom of translation in space that we have considered up to now. In quantum relativistic mechanics, the structure of the Lorentz group reveals the kinetic momentum to any particle as an intrinsic attribute, which is defined in the same way as its electric charge and its mass. There are several possible representations of the states, accompanied by corresponding representations of the observables, whose use can be more or less convenient according to the problem under consideration.
This chapter discusses the prime cause of the magnetic moment within an individual atom. In the classical model, the angular momentum of the electrons can be used to determine the magnetic moments of the electrons by invoking the concept of electrical charge in motion. There are two contributions to the electronic magnetic moment: an orbital magnetic moment due to orbital angular momentum and a spin magnetic moment due to electron spin. The component of orbital angular momentum along the axis of a magnetic field is restricted to discrete values by quantum mechanics. Electrons have spin angular momentum, which can be represented by the spin quantum number. As the magnetic field is increased and the field splitting becomes greater than the multiple splitting, the anomalous Zeeman effect changes over to a normal Zeeman effect. The chapter looks at experimental evidence confirming the quantization of magnetic moments, specifically the Zeeman effect and the Stern-Gerlach experiment.
In the 1920s, the old quantum theory was gradually being superseded by the new quantum theory. The cornerstone of the old theory was Bohr’s model of the hydrogen atom, which predicted that an electron cannot orbit the proton in the hydrogen atom in any arbitrary fashion. Orbits are “quantized,” meaning that only certain sizes, shapes, and magnetic properties are allowed. The principal quantum number n determined the allowed radii of the orbits, the orbital quantum number l determined the allowed shapes, and the magnetic quantum number m determined the magnetic behavior. Additionally, there is a fourth quantum number s which denotes the fact that the electron has an additional angular momentum, loosely associated with self rotation about its own axis, and that is quantized in units of (1/2)~. The old quantum theory was useful to infer the existence of discrete energy levels in atoms, calculate energy spacings between these levels, and therefore allowed one to interpret atomic spectra. The new quantum theory appeared to be more revolutionary and more powerful. It was triggered by Heisenberg’s discovery of matrix mechanics and Schro¨dinger’s discovery of wave mechanics. These two formalisms would not only predict the quantization of energy and provide a prescription to determine the energy difference between the levels (and thus explain the multiplicity of atomic spectra), but also allow one to calculate easily probabilities of transitions between different quantized energy states. At first, matrix mechanics and wave mechanics looked entirely different in their mathematical appearance and physical meaning. However, Schro¨dinger and Eckart [1] independently showed that the two theories are mathematically equivalent. Toward the end of 1926, Dirac unified the two theories using the concept of state vector and thus established the transformation theory of quantum mechanics. This ultimately had a profound implication for the quantum mechanical (mathematical) recipe to treat spin, as we will show in this chapter. The transformation theory is the mathematical recipe to handle modern quantum mechanics. In Heisenberg’s matrix mechanics, a physical quantity is expressed by a matrix, whereas in Schro¨dinger’s wave mechanics, a physical quantity is expressed by a linear operator. In the unified transformation theory, physical quantities are represented by abstract linear operators called Dirac’s q-numbers, which are linear operators in an infinite-dimensional linear space. Depending upon which types of orthogonal coordinate systems are used in this linear space, either matrix mechanics or wave mechanics emerges.
The Bohr formulation—the original one—is now largely a matter of history. The usual career of a theory was its fate. It explained much, but it failed at the first signs of complication, even for the simplest molecules as hydrogen and helium. Some results were either only approximate in a loose or incomplete fashion, e.g. in the application of the correspondence principle to intensities, the only reliable predictions being only for an absence of certain lines, or they quite disagreed with experiment, e.g. in the case of the hydrogen molecule and the hydrogen molecule ion, the normal and excited states of helium, the anomalous Zeeman effect, the Ramsauer effect (the phenomenon of the deflection of electrons at low speeds being less than at higher speeds in atomic collisions), and in the treatment of problems where more than one electron was involved. In the last mentioned cases the perturbation method, so successful in solving “many-body” problems in celestial mechanics, failed completely because systems of several electrons, possessing frequencies of the order of light waves, cannot interact classically. A bit of much forced pragmatism was practiced in treating the many electron atom as a variation of the hydrogen atom. Besides, there were technical oddities, like ½ quantum numbers, and impenetrated anologies between visible and X-ray spectra.