The process of simultaneous emission of two photons (of continuous probability distribution in energy) in the transition from one discrete quantum level to another is here analyzed in the correspondence principle limit in which the quantum numbers of the two states are large and the change in quantum number is small by comparison. In this limit, it proves possible to define a classical dynamical quantity [Eq. (19)], the frequency analysis of whose classical variation with time in a definite orbit determines in the given approximation the absolute transition probability and intensity distribution in the continuous spectrum in question.
The process of simultaneous emission of two photons (of continuous probability distribution in energy) in the transition from one discrete quantum level to another is here analyzed in the correspondence principle limit in which the quantum numbers of the two states are large and the change in quantum number is small by comparison. In this limit, it proves possible to define a classical dynamical quantity [Eq. (19)], the frequency analysis of whose classical variation with time in a definite orbit determines in the given approximation the absolute transition probability and intensity distribution in the continuous spectrum in question.
The correspondence principle postulates a relation between the average rate at which energy is actually being discontinuously emitted by quantum transitions and the continuous rate of emission which would be calculated from the classical theory. The principle has hitherto been used to predict the relative intensities of spectral lines of similar origin. Since, however, the classical theory permits a claculation of absolute rates of energy emission, it should be possible to calculate from the correspondence principle the absolute intensities of spectral lines. In the present article, the correspondence principle is subjected for the first time to this new kind of absolute test, using Czerny's data on the intensities of the far infra-red lines Nos. 8, 9, and 10 in the pure rotational spectrum of the dipole HCl. Tests of this kind have not previously been easy to make owing to lack of information as to the experimental values of absolute intensities or lack of knowledge of the the atomic models involved. The agreement between experimental and calculated intensities is close enough greatly to increase our confidence in the correspondence principle, justify us in subjecting it to further tests, and lead us to hope that an exact quantitative formulation of the principle may be found.
Planck formulated the correspondence principle between quantum and classical mechanics as the limit in which the Planck constant h goes to zero. Bohr formulated the correspondence principle to be the limit of large quantum numbers. For three common quantum mechanical systems it is shown that in order for eigenvalues of quantum mechanical observables to have meaningful classical limits, it is necessary to take the double limit as both the Planck constant goes to zero and the quantum number goes to infinity, subject to the constraint that their product is equal to an appropriate classical action. This synthesis of the Bohr and Planck formulations of the correspondence principle is also used to show that the quantum mechanical transition frequency between adjacent levels approaches the corresponding classical frequency. The features these systems have in common in their classical limit are explained by general considerations of the classical limit of the Schrödinger equation.
A critical examination of the theory of resonance radiation has been carried through, employing the mathematical technique of Laplace and Stieltjes transforms. Particular attention has been devoted to the question of the temporal behavior of the excited state. With no restrictions on the interaction between atom and radiation field other than that it be real, and that processes of sufficiently high frequency contribute only negligible effects, one can prove that the probability amplitude of the excited state cannot decay according to the general law ${\ensuremath{\Sigma}}_{\ensuremath{\mu},\ensuremath{\nu}}{C}_{\ensuremath{\mu}\ensuremath{\nu}}{t}^{{\ensuremath{\lambda}}_{\ensuremath{\mu}}}{e}^{\ensuremath{-}{\ensuremath{\beta}}_{\ensuremath{\nu}}t},$ where the ${\ensuremath{\lambda}}_{\ensuremath{\mu}}$ and ${\ensuremath{\beta}}_{\ensuremath{\nu}}$ are complex constants lying in the right half-plane and the ${C}_{\ensuremath{\mu}\ensuremath{\nu}}$ are arbitrary complex coefficients. The deviation from the law just cited can be termed a straggling phenomenon since exact analysis shows that the probability amplitude, for sufficiently long times, is greater than that defined according to the radioactive decay law.Rigorous analysis of the source of this apparently anomalous behavior reveals the following explanation. Although the probability amplitude of the excited state is, strictly speaking, a functional of the interaction between the transition current in the atom and the photon states of the electromagnetic field, the essential features are determined principally by those states lying near the resonance frequency. Three elementary observations are immediately apparent. First, a particularly tractable analytical approximation for the interaction can be made such that the value of the actual interaction is reproduced at the resonance frequency and such that the behavior for high and low frequency photon states is at least qualitatively correct. Exact solutions in closed form can be obtained for this interaction. Second, in terms of the preceding representation it is possible to show that, associated with the transition between two atomic states, one is presented with a concomitant picture of a damped oscillation of charge describable in purely classical terms. Third, this classical motion can be interpreted as forming a source function for a continuous stochastic process in terms of which one finally derives the representation of the probability amplitude of the excited state. The straggling phenomenon previously cited is therefore that associated with all diffusion processes. The consideration of the exact interaction leads to more involved diffusion phenomena but does not in any case permit a radioactive decay law.
For the paradigm of the quantum double-slit experiment (DSE), we apply Ockham's Razor to interpret quantum observations and to evaluate terminology associated with wave-particle duality. One finds that the Correspondence Principle (CP), combined with classical wave DSEs, e.g., Young [1804], is sufficient to separate and anticipate the observed quantum particle and wave phenomena. The empirical approach of Ockham infers that individual quanta are only whole particles during transit from source to detector; an individual quantum never acts as a wave. The wave nature of quanta emerges only in the distribution of large numbers of single-quantum observation events. That is, the "measurement problem" is no problem at all; "particle" and "wave" derive from separate and different aspects of a set of observations. Such artificial constructs as wave function collapse are irrelevant to the observation of individual quanta, each of which acts as a whole "particle" from emission to measurement. The histogram of detected events is a collective property identical to a classical wave interference pattern in the limit of large numbers, as the CP decrees. For a specific quantum, Ockham's Razor renders irrelevant any hypothesis regarding wave- or particle-like behavior of the quantum in the region between emission and detection. To analyze actual data sets consisting of a large number of identical observation events and to predict future DSEs, the CP of standard quantum (wave) mechanics is sufficient: in the limit, the distribution of observations approaches the continuous density defined by the standard quantum mechanical wave function. Scientific progress beyond this picture requires new, relevant experiments. From the DSE, Ockham's Razor infers that a theoretical quantum system consists of at least one quantum particle plus a wave function specifying the distribution of a large number of such particles.
Radiation can change the energy state of matter via three optical processes: spontaneous emission, absorption and stimulated emission. In spontaneous emission, an excited atomic or molecular state decays to a lower energy state with the emission of a photon of energy equal to the difference in the quantum state energies. Photon absorption is the reverse process, with a photon causing excitation from a lower to an excited quantum state. Stimulated emission was first proposed by Einstein in a paper published in 1917. He attempted to balance spontaneous emission and photon absorption, but could not get the correct balance without the postulated process of stimulated emission (see Section 4.2). With stimulated emission, an excited quantum state is stimulated to decay to a lower quantum state by an impinging photon. The existing photon survives with another identical photon being created (so that energy is conserved). The rates at which radiative transitions occur can be calculated using quantum mechanics. The time-dependent Schrodinger equation is used with the system wavefunction comprising time-varying linear combinations of the two wavefunctions for the lower and upper quantum states. In semi-classical treatments, an atom is treated quantum mechanically with the radiation field treated classically. There is a perturbation in energy associated with, say, the electric field of the radiation interacting with the electric dipole of the atom, though other interactions can be significant if the electric dipole interaction energy is small. In absorption, for example, a differential equation representing the change in ‘weighting’ of the wavefunction from the lower to upper state is obtained. A ‘time constant’ for the speed of change represents the rate of the radiative process. We examine the quantum mechanical nature of radiation absorption in Section 10.1 before discussing (Section 10.3) another important parameter arising with radiative transitions between discrete bound quantum states: the lineshape function. Quantum Theory of the Atom–Radiation Interaction An expression for the Einstein B-coefficient for photo-absorption is derived using quantum mechanics in this section. Einstein determined the detailed balance relationship between the rates of spontaneous emission, photo-absorption and stimulated emission when light interacts with an atom by considering an atomic system of two energy levels which are in equilibrium with a black-body radiation field. We use a semi-classical treatment where the atom is treated quantum mechanically and the radiation field is treated as a classical oscillating electric field.
The theory of double-resonance phenomena, or the simultaneous action of optical transitions and rf perturbations between the Zeeman sublevels of an atom, is extended to the induced emission and absorption processes effected by the monochromatic radiation and the discrete resonances involved in the gas laser. Expressions for the probability amplitudes of the Zeeman sublevels under such conditions are derived from time-dependent perturbation theory, the lifetimes of the states being introduced in a phenomenological way. Zeeman splittings of both upper and lower states involved in the laser transition are considered, and an effective term diagram used to discuss the results. Small signal approximations are applied to consider some specific cases, which show that the light beats seen in spontaneous emission when the rf frequency ${\ensuremath{\omega}}_{0}$ equals the Larmor frequency, and in other instances, do not occur in induced emission unless such frequencies coincide with axial resonances of the laser cavity. Variations in the intensity and resonance effects are indicated at Zeeman splittings corresponding to ${\ensuremath{\omega}}_{0}$, with similar variations in the amplitude of any beat frequencies due to multiple axial resonances. Additional coherence effects occur where levels cross, which will give rise to changes in the intensity and polarization of the induced emission at the corresponding frequency.