The classical spin of a particle is essentially its angular momentum of rotation about an axis passing through its center of mass. The classical Poisson brackets are generalized to include spin for any system for which the rate of change of spin is given by $\frac{d\stackrel{\ensuremath{\rightarrow}}{s}}{\mathrm{dt}}=\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{s}\ifmmode\times\else\texttimes\fi{}(\frac{\ensuremath{\partial}H}{\ensuremath{\partial}\stackrel{\ensuremath{\rightarrow}}{s}})$ where
H is the Hamiltonian. In a previous paper, we showed that this equation is satisfied by the Breit-Pauli Hamiltonian for a molecular system of electrons and nuclei interacting with an electromagnetic field. This Hamiltonian contains all of the fine-structure terms with the exception of the Lamb shift. Thus, our generalized Poisson brackets should apply to almost all molecular problems. Our generalized Poisson brackets are the same as those which Sudarshan and Mukunda derived by group-theoretic arguments for the direct product of a pure spin group with the group of transformations of generalized coordinates and their conjugate momenta.
In the first part of this series we have shown how the Schrodinger equation for a single particle and the corresponding non relativistic quantum observables can be obtained from a purely classical phonon model through the Newtonian equations of motion. In this work we focus instead on how the classical Hamiltonian formalism applied to the same phonon system allows to recover the feature of relativistic quantum mechanics for a single spinless particle. Using the classical nature of the phonon model, we naturally define continuous Poisson brackets between classical observables, which allows to recover the dynamics of such observables, i.e. the Ehrenfest relations associated to real-valued Klein-Gordon fields. The Poisson brackets also permits to obtain the generic form of constants of motions, thus generalizing the concept of inner products and momentum on Klein-Gordon fields. We then connect the formalism of real-valued classical functionals with that of hermitian operators and complex-valued wave functions. This is done through the introduction of a non-local complex-valued change of variables which allows to rewrite the real-valued Klein-Gordon equation in a form akin to the Schrodinger equation, and the classical observables as quantum expectation values. Then, we show how this change of variables allows to rewrite the classical Poisson brackets as commutators of hermitian operators. This points out the strict equivalence between the Heisenberg formalism and the formalism of classical Poisson bracket. Eventually, we illustrate how the Poisson brackets allows to recover the transformations of Poincare group in 1+1 dimension together with its algebra. The latter makes the link between the Lorentz invariant inner product of Mostafazadeh and the Casimir invariant associated to the mass of particle.
The position operator in the Heisenberg representation obeys the usual classical equation of motion. The similarity between the Heisenberg equations of motion and the classical Hamiltonian equations of motion arises from the fact that the commutator of the operators for the physical quantities plays a similar role in quantum mechanics, to that of the Poisson bracket of the classical quantities A and B, in classical mechanics. Generally, the Heisenberg equations of motion are more difficult to solve than the corresponding classical equations because of the lack of commutation of quantum mechanical operators. The chapter explains the notion of a time-ordered product of operators, and then can be written much more compactly.
A simple prescription for quantizing fields is developed and applied to the sytsem of interacting spins in an electromagnetic field. The classical equations of motion are written in the standard form $\ensuremath{-}i(\frac{d}{\mathrm{dt}})|V〉+\mathfrak{M}|V〉=0$, where the state vector
∣V〉 has the state variables, such as the generalized coordinates and momenta, as its components, and the time development operator
M contains derivative, integral, and matrix operators in general. By diagonalizing
M and replacing the resulting classical variables with operators obeying boson commutation relations, the system is quantized, and the Hamiltonian and Heisenberg equations of motion are automatically cast into the harmonic-oscillator form. Thus the quantization is reduced to the simple problem of diagonalizing a matrix. This avoids the laborious process of finding a Lagrangian density (by trial and error in general), determining the canonical variables, forming the Hamiltonian density, and inventing transformations to creation and annihilation operators. Problems with zero-frequency modes, which can be annoying in the standard treatment, are automatically eliminated by the prescription. The bookkeeping of forward and backward waves, different branches of dispersion curves, etc., is simplified. In addition to affording a straightforward and simple method of quantizing fields, the method also gives the necessary and sufficient conditions for a Hamiltonian to be transformable into harmonic-oscillator form, gives a direct relation between the classical equations of motion and the Heisenberg quantum-mechanical equations, and gives a simple method of determining the constants of the motion.
Abstract The years ago Kramers and pauli quantized the diatomic molecular model which bears their names, obtaining the formula which has placed so large a part in hand spectroscopy. It is sometimes pointed out in modern text-books that these results, founded on the old mechanics, do not offer a very far-reaching analogy to those which the wave mechanics give for the real diatomic molecule. I think the reason will be seen on closely examining the classical calculation, either in the original papers, or in Born's "Atom-mechanik," Without adequate cause, all types of motion save one were rejected in the quantization; all that remained was the simple type of motion which, apart from the gyroscope's axial spin consists merely of a steady precession of the body as a whole about the invariable axis. It will here be shown that in reality we can have Kramers and Pauli models performing a very different type of motion, while completely satisfying the old-quantum equations (which were determined in the original papers). In these motions the gyroscope presesses not only about the invariable axis but also about the nuclear axis. Kemble proposed that in a modified model, having an elastically mounted gyroscope, notions of this more complex type should be possible as quantized States, though his actual treatment is confined to motions of the simpler type. As stated above, we shall see that the original model can have the more complex type of motion, in strictly quantized states. Kemble also proposed an energy formula for the corresponding motions of the real molecule; it is with this and with the well-established molecular formulæ of the wave mechanics that results obtained from the model are to be compared. For specifying orders of magnitude it is convenient to think what happens when the mass of the nuclei is made very great, their angular momentum retaining the same order of magnitude as that of the gyroscope. Then the existence of quantized motions other than those prescribed by Kramers and Pauli is not surprising; for by analogy with the vectorial conception of the real molecular (described, notably, by Hund and Weizel), we should expext, in these circumstances, to find models in whose quantized states the nuclei, if of great mass, will rotate about the invariable axis very s;lowly in comparison with the gyroscope's precession about the nuclear axis. For the real molecule, wave mechanics equations such as Kronig's (22) will hold with an accuracy increasing with the nuclear mass, in the sense that though the main rotational terms will became smaller, the neglected perturbations (if so they may be called) with diminish far more rapidly.