By analogy with the mathematical definition of degenerate systems, it is suggested that systems which are interrupted before they have traversed a full period, or which are in an external field which varies greatly in the time of one period, be called physically degenerate. It is further suggested, in harmony with the ideas of Ehrenfest and Tolman, that such systems show weak quantization, the quantization becoming less and less complete as the degeneracy increases. In the case of partially degenerate systems, the values of the related action variable and of the energy are grouped more or less closely about the mean values which they would all have if the systems were completely quantized. As examples, these ideas are applied to several problems, such as quantization in a small or rapidly changing magnetic field (including Glaser's results for the change of diamagnetic susceptibility of some gases with pressure, experiments by Breit and Ellett and by Wood and Ellett on depolarization, and the experiment of Gerlach and Stern in which atoms are projected into a strong field), absorption in the higher lines of the principal series of an alkali (including a discussion of quantization in hyperbolic and parabolic obits), and the continuous x-ray spectrum. It is then pointed out that this assumption demands the existence of quantum forces which have not been previously considered in quantum theory and whose action is to stabilize the stationary states, bringing the action variables and energy rapidly nearer and nearer to the proper values as time goes on. The suggestion is made that these forces are responsible for the large change of energy and action variables during a quantum transition. From this point of view the adiabatic theorem appears in a simple light; when the external parameters vary too rapidly, the quantization becomes poor, the quantum forces become strong and put the atom into a quantized state again. In any atom but hydrogen, quantum forces must be continuously acting to oppose the interchange of energy and momentum between electrons and to keep each electron quantized. These forces thus may prove to be important in the solution of problems of the quantum dynamics of the constitution of atoms, though they must be considered in connection with the dynamics of the oscillators.
By analogy with the mathematical definition of degenerate systems, it is suggested that systems which are interrupted before they have traversed a full period, or which are in an external field which varies greatly in the time of one period, be called physically degenerate. It is further suggested, in harmony with the ideas of Ehrenfest and Tolman, that such systems show weak quantization, the quantization becoming less and less complete as the degeneracy increases. In the case of partially degenerate systems, the values of the related action variable and of the energy are grouped more or less closely about the mean values which they would all have if the systems were completely quantized. As examples, these ideas are applied to several problems, such as quantization in a small or rapidly changing magnetic field (including Glaser's results for the change of diamagnetic susceptibility of some gases with pressure, experiments by Breit and Ellett and by Wood and Ellett on depolarization, and the experiment of Gerlach and Stern in which atoms are projected into a strong field), absorption in the higher lines of the principal series of an alkali (including a discussion of quantization in hyperbolic and parabolic obits), and the continuous x-ray spectrum. It is then pointed out that this assumption demands the existence of quantum forces which have not been previously considered in quantum theory and whose action is to stabilize the stationary states, bringing the action variables and energy rapidly nearer and nearer to the proper values as time goes on. The suggestion is made that these forces are responsible for the large change of energy and action variables during a quantum transition. From this point of view the adiabatic theorem appears in a simple light; when the external parameters vary too rapidly, the quantization becomes poor, the quantum forces become strong and put the atom into a quantized state again. In any atom but hydrogen, quantum forces must be continuously acting to oppose the interchange of energy and momentum between electrons and to keep each electron quantized. These forces thus may prove to be important in the solution of problems of the quantum dynamics of the constitution of atoms, though they must be considered in connection with the dynamics of the oscillators.
Abstract The new quantum theory, based on the assumption that the dynamical variables do not obey the commutative law of multiplication, has by now been developed sufficiently to form a fairly complete theory of dynamics. One can treat mathematically the problem of any dynamical system composed of a number of particles with instantaneous forces acting between them, provided it is describable by a Hamiltonian function, and one can interpret the mathematics physically by a quite definite general method. On the other hand, hardly anything has been done up to the present on quantum electrodynamics. The questions of the correct treatment of a system in which the forces are propagated with the velocity of light instead of instantaneously, of the production of an electromagnetic field by a moving electron, and of the reaction of this field on the electron have not yet been touched. In addition, there is a serious difficulty in making the theory satisfy all the requirements of the restricted principle of relativity, since a Hamiltonian function can no longer be used. This relativity question is, of course, connected with the previous ones, and it will be impossible to answer any one question completely without at the same time answering them all. However, it appears to be possible to build up a fairly satisfactory theory of the emission of radiation and of the reaction of the radiation field on the emitting system on the basis of a kinematics and dynamics which are not strictly relativistic. This is the main object of the present paper. The theory is noil-relativistic only on account of the time being counted throughout as a c-number, instead of being treated symmetrically with the space co-ordinates. The relativity variation of mass with velocity is taken into account without difficulty. The underlying ideas of the theory are very simple. Consider an atom interacting with a field of radiation, which we may suppose for definiteness to be confined in an enclosure so as to have only a discrete set of degrees of freedom. Resolving the radiation into its Fourier components, we can consider the energy and phase of each of the components to be dynamical variables describing the radiation field. Thus if Er is the energy of a component labelled r and θr is the corresponding phase (defined as the time since the wave was in a standard phase), we can suppose each Er and θr to form a pair of canonically conjugate variables. In the absence of any interaction between the field and the atom, the whole system of field plus atom will be describable by the Hamiltonian H ═ ΣrEr + Ho equal to the total energy, Ho being the Hamiltonian for the atom alone, since the variables Er, θr obviously satisfy their canonical equations of motion Er ═ — ∂H/∂θr ═ 0, θr ═ ∂H/∂Er ═ 1.
An attempt is made to present a consistent detailed theory of optical phenomena, based on the suggestion recently made by the author in collaboration with Bohr and Kramers. This suggestion is that atoms are radiating or absorbing continuously, during their stationary states, and that transitions influence radiation only in terminating the radiation characteristic of the old state, and substituting that of the new. The atoms, on the other hand, are supposed not to change their energy while radiating, but to change it discontinuously on transition. This necessitates giving up the detailed application of the conservation of energy in interaction between atoms and radiation. In the present paper, an additional suggestion is made, namely that resonance radiation is to be identified with the radiation carried out by the spherical wavelets which, by their interference with the external field, also produce absorption. This is practically a return to the classical picture of resonance radiation. Statement of the theory consists formally of two parts: the description of the behavior of atoms, and of radiation, when the two interact. The first involves the statement of the probabilities of transition of atoms, and is taken without change from Einstein. The probability of interruption of coherent vibrations is also discussed. The second requires the specification of the fields emitted by atoms in any steady state under the influence of any external field. These consist of spherical wavelets of the frequencies of the various quantum lines which the atom can emit or absorb in the stationary state in which it is. For emission frequencies, wavelets are emitted even in the absence of an external field; for absorption frequencies, external fields induce wavelets, similar to the wavelets of linearly bound electrons in electron theory, which on the one hand interfere with the external field to produce the phenomena of dispersion, etc., and on the other carry out the resonance radiation in all directions. Detailed descriptions of these fields are given. In a discussion of the theory, it is first shown that the assumptions which have been made about the field satisfy the correspondence principle, in that for large quantum numbers the field aproaches that which would be emitted classically on account of the interaction of an external field and a multiply periodic atom. On the other hand, it is shown that the field has a character essentially like that of classical electron theory, which is known to be in general agreement with experiments. Next, the energy and momentum relations at interaction of atoms and light are investigated, and it is shown that, although these quantities are not precisely conserved, still the assumptions as to atomic and radiation processes have been so made that they are conserved on an average over a great number of atomic processes. Finally the spectral resolution of emitted and absorbed radiation is considered; the theory gives a minimum breadth for the lines, depending on the finite length of wave trains resulting from the finite life in stationary states, and it is shown that Kirchhoff's law is obeyed, the emission and absorption lines being of the same breadth. By way of illustration, specific application of the theory is made to the simple examples of emission of light by bombardment of electrons at the resonance potential, resonance radiation, its quenching by presence of foreign gases, absorption, scattering, and disperison.
We present details and expand on the framework leading to the recently introduced degenerate adiabatic perturbation theory [Phys. Rev. Lett. 104, 170406 (2010)], and on the formulation of the degenerate adiabatic theorem, along with its necessary and sufficient conditions [given in Phys. Rev. A 85, 062111 (2012)]. We start with the adiabatic approximation for degenerate Hamiltonians that paves the way to a clear and rigorous statement of the associated degenerate adiabatic theorem, where the non-Abelian geometric phase (Wilczek-Zee phase) plays a central role to its quantitative formulation. We then describe the degenerate adiabatic perturbation theory, whose zeroth-order term is the degenerate adiabatic approximation, in its full generality. The parameter in the perturbative power-series expansion of the time-dependent wave function is directly associated to the inverse of the time it takes to drive the system from its initial to its final state. With the aid of the degenerate adiabatic perturbation theory we obtain rigorous necessary and sufficient conditions for the validity of the adiabatic theorem of quantum mechanics. Finally, to illustrate the power and wide scope of the methodology, we apply the framework to a degenerate Hamiltonian, whose closed-form time-dependent wave function is derived exactly, and also to other nonexactly solvable Hamiltonians whose solutions are numerically computed.
Quantization of spin and orbital angular momenta in standard quantum mechanics is rooted in the fundamental rotational symmetry of space in conjunction with the results of experiments on atomic and subatomic particles conducted with the Stern-Gerlach apparatus. We explain some of the most important features of spin and orbital angular momenta as inescapable consequences of elementary Stern-Gerlach experiments viewed in light of the rotational symmetry of space. The rotation operator is first derived for spin-½ particles, then extended to spin-1 particles by treating the case of a hydrogen atom in its ground state, where the spin of the proton combines with that of the electron to create a spin-1 triplet state. Along the way, we examine the evolution of a particle’s spin state under an applied magnetic field, where the interaction energy arises from the existence of a magnetic dipole moment for each particle in proportion to its spin angular momentum. Whereas a constant magnetic field B0 causes the particle’s magnetic moment to gyrate around the direction of B0, it will be seen that the presence of a small oscillating magnetic field perpendicular to B0 brings about the so-called Rabi oscillations, which involve periodic flippings of the particle’s spin orientation. In the case of a hydrogen atom in its ground state, the interaction between the magnetic dipole moments of the proton and the electron splits the ground state between a lower energy singlet having a net angular momentum of zero (spin-0 state), and a higher energy triplet having a total angular momentum of ħ (spin-1 state). This hyperfine splitting is the source of the well-known 21-cm line observed in the spectrum of the hydrogen atom, which arises from transitions between the atom’s singlet and triplet states. In the final section, we put forward an argument (originally made by Richard Feynman) that ties the rotational symmetry of space to the conservation of angular momentum in quantum mechanics. The argument is, in fact, quite general, enabling one to deduce other conservation laws from various inherent symmetries of quantum mechanical systems.
In the first two chapters we have examined the basic principles – superposition (p. 18), complementarity (p. 19), quantization (p. 44), statistical algorithm (p. 57), and correspondence (p. 72) (see also Subsec. 2.3.4) – and the basic entities, observables and states, of quantum mechanics, as well as the main differences with respect to classical mechanics. While what we have discussed so far is rather a static picture of observables and states, in this chapter we shall deal with quantum dynamics, i.e. with the time evolution of quantum-mechanical systems. Historically, after Bohr had provided a quantized description of the atom (see Subsec. 1.5.4), Einstein showed the quantized nature of photons (see Subsec. 1.2.1), and de Broglie hypothized the wave-like nature of matter (see Subsec. 1.5.5), the first building block of quantum mechanics was provided by the commutation relations, proposed by Heisenberg in 1925, whose consequence is represented by the uncertainty relation (see Subsec. 2.2.7 and Sec. 2.3). This was the subject of the previous chapters. The dynamical part of the theory was proposed by Schrödinger in 1926, and is known as the Schrödinger equation. It is also known as wave mechanics (see Subsec. 1.5.7). In this chapter we shall show that Heisenberg's and Schrödinger's formulations are only two different aspects of the same theory. We shall also come back to this point in Sec. 8.1.1. Here, first we shall derive the fundamental equation which rules quantum dynamics (Sec. 3.1), and, in Sec. 3.2, we shall summarize the main properties of the Schrödinger equation.
The behavior of a system under the action of a timedependent perturbation is considered in the adiabatic approximation, degeneracy being assumed in the absence of the perturbation. The state of the system is characterized by the density matrix in space of eigenfunctions belonging to a given degenerate level. A unitary operator is set up that relates the density matrix before the perturbation is turned on to that after the perturbation is turned off. The case when there is no exact degeneracy in the absence of a perturbation but rather a set of closely spaced levels (almost degenerate states) is also considered. As an illustration of application of the theory, the behavior of spin 1 and 1/2 particles in a variable magnetic field is examined. (auth)