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 Cl
O2 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.
By a canonical transformation on the Dirac Hamiltonian for a free particle, a representation of the Dirac theory is obtained in which positive and negative energy states are separately represented by two-component wave functions. Playing an important role in the new representation are new operators for position and spin of the particle which are physically distinct from these operators in the conventional representation. The components of the time derivative of the new position operator all commute and have for eigenvalues all values between $\ensuremath{-}c$ and
c. The new spin operator is a constant of the motion unlike the spin operator in the conventional representation. By a comparison of the new Hamiltonian with the non-relativistic Pauli-Hamiltonian for particles of spin \textonehalf{}, one finds that it is these new operators rather than the conventional ones which pass over into the position and spin operators in the Pauli theory in the non-relativistic limit. The transformation of the new representation is also made in the case of interaction of the particle with an external electromagnetic field. In this way the proper non-relativistic Hamiltonian (essentially the Pauli-Hamiltonian) is obtained in the non-relativistic limit. The same methods may be applied to a Dirac particle interacting with any type of external field (various meson fields, for example) and this allows one to find the proper non-relativistic Hamiltonian in each such case. Some light is cast on the question of why a Dirac electron shows some properties characteristic of a particle of finite extension by an examination of the relationship between the new and the conventional position operators.
The canonical partition function Q(H, N, s) for a spin-s anisotropic Heisenberg model with N sites, in external magnetic field H, is examined in the limit s → ∞. The coupling coefficients and the magnetic moment per spin are taken to be inversely proportional to s2 and s, respectively. It is proved that lim lim s→∞s−NQ(H,N,s)=(2π)−NQ(H,N),where Q(H, N) is the partition function for a corresponding classical Heisenberg model with spins of unit magnitude. This theorem makes precise the widely believed, but heretofore unproved equivalence between the classical Heisenberg model and the infinite-spin limit of the quantum Heisenberg model. No appeal to the thermodynamic limit is necessary.
Bell-like inequalities have been used in order to distinguish non-local quantum pure states by various authors. The behavior of such inequalities under Lorentz transformation (LT) has been a source of debate and controversies in the past. In this paper, we consider the two most commonly studied three-particle pure states, that of W and Greenberger–Horne–Zeilinger (GHZ) states which exhibit distinctly different types of entanglement. We discuss the various types of three-particle inequalities used in previous studies and point to their corresponding shortcomings and strengths. Our main result is that if one uses Czachor’s relativistic spin operator and Svetlichny’s inequality as the main measure of non-locality and uses the same angles in the rest frame (S) as well as the moving frame ([Formula: see text]), then maximally violated inequality in S will decrease in the moving frame, and will eventually lead to lack of non-locality (i.e. satisfaction of inequality) in the [Formula: see text] limit. This is shown for both the GHZ and W states and in two different configurations which are commonly studied (Cases 1 and 2). Our results are in line with a more familiar case of two particle case. We also show that the satisfaction of Svetlichny’s inequality in the [Formula: see text] limit is independent of initial particles’ velocity. Our study shows that whenever we use Czachor’s relativistic spin operator, results draws a clear picture of three-particle non-locality making its general properties consistent with previous studies on two-particle systems regardless of the W state or the GHZ state is involved. Throughout the paper, we also address the results of using Pauli’s operator in investigating the behavior of [Formula: see text] under LT for both of the GHZ and W states and two cases (Cases 1 and 2). Our investigation shows that the violation of [Formula: see text] in moving frame depends on the particle’s energy in the lab frame, which is in agreement with some previous works on two and three-particle systems. Our work may also help us to classify the results of using Czachor’s and Pauli’s operators to describe the spin entanglement and thus the system spin in relativistic information theory.
In these continuation papers (VI and VII) we are interested in approach the problem of spin from a classical point of view. In this first paper we will show that the spin is neither basically relativistic nor quantum but reflects just a simmetry property related to the Lie algebra to which it is associated. The classical approach will be paraleled with the usual quantum one to stress their formal similarities and epistemological differences. The important problem of Einstein-Bose condensation for fermions will also be addressed.