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Physical Review

Duration of Molecules in Upper Quantum States

Richard C. Tolman

Published Jun 1, 1924• 117 citations
Publisher

Abstract

Values of rate of decay and mean life of atoms and molecules in upper quantum states calculated from data on the intensity of absorption lines.---By combining F\"uchtbauer's method of determining from the intensity of absorption lines the probability that a molecule will absorb a quantum of energy, with Einstein's views as to the mechanism of light absorption and emission, the following equation is derived for calculating the rate at which molecules jump from upper to lower quantum states: ${A}_{21}=\left(\frac{8\ensuremath{\pi}{\ensuremath{\nu}}^{2}}{{c}^{2}{N}_{1}}\right)\left(\frac{{p}_{1}}{{p}_{2}}\right)\ensuremath{\int}{0}^{\ensuremath{\infty}}\ensuremath{\alpha}d\ensuremath{\nu}$ where ${A}_{21}$ is the chance per unit time that a molecule will jump spontaneously from quantum state 2 to quantum state 1, $\ensuremath{\nu}$ is the frequency of the light emitted in such a jump, ${p}_{1}$ and ${p}_{2}$ are the a priori probabilities of quantum states 1 and 2, and $\ensuremath{\alpha}$ is the absorption coefficient of the substance measured under conditions such that ${N}_{1}$ is the number of molecules per unit volume in the lower quantum state 1. The integral $\ensuremath{\int}\ensuremath{\alpha}d\ensuremath{\nu}$ is to be taken over the total effective width of the absorption line corresponding to the passage of molecules from quantum state 1 to quantum state 2. The mean life $\ensuremath{\tau}$ of molecules which decay from state 2 to state 1 is the reciprocal of ${A}_{21}$. Values of ${A}_{21}$ and $\ensuremath{\tau}$ are calculated from existing data for the mercury line $\ensuremath{\lambda}2537$, for a number of lines belonging to the alkali doublets, for the iodine line $\ensuremath{\lambda}5461$, and for a very considerable number of lines belonging to the rotation-oscillation spectra of the hydrogen halides. The values obtained agree with the meager data made available by other experimental methods. From these results the following conclusions are drawn.The mean life of molecules and atoms in upper quantum states may vary for different states at least over the range 1 to ${10}^{\ensuremath{-}8}$ seconds. The rate of decay is not a simple function of the frequency of the emitted light. The rate corresponding to the emission of a line of high frequency may be greater or less than that for a line of lower frequency. The data now available for the alkali doublets $1s\ensuremath{-}m{p}_{1}$ and $1s\ensuremath{-}m{p}_{2}$ indicate a higher rate of decay the smaller the change in total quantum number for the line under consideration. The rate of decay from a given $m{p}_{1}$ state is $m$ times as great as from the corresponding $m{p}_{2}$ state (already stated in another form by F\"uchtbauer and Hofmann). In the case of the rotation-oscillation spectra of the hydrogen halides the rate of decay is greater for quantum states with one unit of oscillation and many of rotation, than for those with one unit of oscillation and only a few of rotation; and in the case of different molecules but the same quantum numbers the rate of decay is greater for the molecule with the greater frequency of oscillation.Finally the possibility and method of calculating absolute values of ${A}_{21}$ from the Bohr correspondence principle is indicated.

DOI
10.1103/physrev.23.693

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Values of rate of decay and mean life of atoms and molecules in upper quantum states calculated from data on the intensity of absorption lines.---By combining F"uchtbauer's method of determining from the intensity of absorption lines the probability that a molecule will absorb a quantum of energy, with Einstein's views as to the mechanism of light absorption and emission, the following equation is derived for calculating the rate at which molecules jump from upper to lower quantum states: A21=(8\ensuremathπ\ensuremathν2c2N1)(p1p2)\ensuremath∫0\ensuremath∞\ensuremathαd\ensuremathν{A}_{21}=\left(\frac{8\ensuremath{\pi}{\ensuremath{\nu}}^{2}}{{c}^{2}{N}_{1}}\right)\left(\frac{{p}_{1}}{{p}_{2}}\right)\ensuremath{\int}{0}^{\ensuremath{\infty}}\ensuremath{\alpha}d\ensuremath{\nu}A21​=(c2N1​8\ensuremathπ\ensuremathν2​)(p2​p1​​)\ensuremath∫0\ensuremath∞\ensuremathαd\ensuremathν where A21{A}_{21}A21​ is the chance per unit time that a molecule will jump spontaneously from quantum state 2 to quantum state 1, \ensuremathν\ensuremath{\nu}\ensuremathν is the frequency of the light emitted in such a jump, p1{p}_{1}p1​ and p2{p}_{2}p2​ are the a priori probabilities of quantum states 1 and 2, and \ensuremathα\ensuremath{\alpha}\ensuremathα is the absorption coefficient of the substance measured under conditions such that N1{N}_{1}N1​ is the number of molecules per unit volume in the lower quantum state 1. The integral \ensuremath∫\ensuremathαd\ensuremathν\ensuremath{\int}\ensuremath{\alpha}d\ensuremath{\nu}\ensuremath∫\ensuremathαd\ensuremathν is to be taken over the total effective width of the absorption line corresponding to the passage of molecules from quantum state 1 to quantum state 2. The mean life \ensuremathτ\ensuremath{\tau}\ensuremathτ of molecules which decay from state 2 to state 1 is the reciprocal of A21{A}_{21}A21​. Values of A21{A}_{21}A21​ and \ensuremathτ\ensuremath{\tau}\ensuremathτ are calculated from existing data for the mercury line \ensuremathλ2537\ensuremath{\lambda}2537\ensuremathλ2537, for a number of lines belonging to the alkali doublets, for the iodine line \ensuremathλ5461\ensuremath{\lambda}5461\ensuremathλ5461, and for a very considerable number of lines belonging to the rotation-oscillation spectra of the hydrogen halides. The values obtained agree with the meager data made available by other experimental methods. From these results the following conclusions are drawn.The mean life of molecules and atoms in upper quantum states may vary for different states at least over the range 1 to 10\ensuremath−8{10}^{\ensuremath{-}8}10\ensuremath−8 seconds. The rate of decay is not a simple function of the frequency of the emitted light. The rate corresponding to the emission of a line of high frequency may be greater or less than that for a line of lower frequency. The data now available for the alkali doublets 1s\ensuremath−mp11s\ensuremath{-}m{p}_{1}1s\ensuremath−mp1​ and 1s\ensuremath−mp21s\ensuremath{-}m{p}_{2}1s\ensuremath−mp2​ indicate a higher rate of decay the smaller the change in total quantum number for the line under consideration. The rate of decay from a given mp1m{p}_{1}mp1​ state is mmm times as great as from the corresponding mp2m{p}_{2}mp2​ state (already stated in another form by F"uchtbauer and Hofmann). In the case of the rotation-oscillation spectra of the hydrogen halides the rate of decay is greater for quantum states with one unit of oscillation and many of rotation, than for those with one unit of oscillation and only a few of rotation; and in the case of different molecules but the same quantum numbers the rate of decay is greater for the molecule with the greater frequency of oscillation.Finally the possibility and method of calculating absolute values of A21{A}_{21}A21​ from the Bohr correspondence principle is indicated.
A method for obtaining the intensity distribution in a spectral line is derived from Dirac's quantum theory of the radiation field. The calculations are carried out for the case of emission from a first excited state (resonance line) and show that the intensity of frequency $\ensuremath{\nu}$ is, in this case, proportional to $\frac{1}{[4{\ensuremath{\pi}}^{2}{({\ensuremath{\nu}}_{0}\ensuremath{-}\ensuremath{\nu})}^{2}+{(A\ifmmode\cdot\else\textperiodcentered\fi{}2)}^{2}]}$ were AAA is the rate at which atoms are leaving the upper state and ${\ensuremath{\nu}}_{0}$ is the frequency of the transition. The half-width is therefore $\frac{1}{2\ensuremath{\pi}}$ times the Einstein probability coefficient for spontaneous transition to the lower state. The more general case where the lower state has a finite life time is not treated.A general method for the treatment of radiation problems, which is essentially Weyl's modification of Dirac's method, is briefly described.
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.
Backward, turn backward, O time in your flight. Make me a child again, just for tonight. –Elizabeth Akers Allen Since the time of Kirchhoff it has been known that, when light is passed through an atomic gas, those wavelengths are observed that would be emitted if the gas were incandescent. If the gas is sufficiently cold, then the wavelengths observed are limited to ground-state transitions. As the temperature of the sample is elevated, wavelengths corresponding to transitions between excited states become absorbing, and balances between emission and absorption occur. The study of the central wavelengths of emission lines or absorption notches is known as first-order spectroscopy, and it provides information on the energy-level structure of the atom. The study of the shape of these lines in frequency space is known as second-order spectroscopy, and provides information on the lifetime of the level and the collision rates and temperature of the gas. Thus, whereas first-order spectroscopy shows that emission and absorption measurements yield the same central wavelengths, second-order spectroscopy shows that the natural linewidth for emission and absorption are both specified by the level lifetime, and that the intensity of emission and absorption features both involve the linestrength factor (through the emission transition probability rate and the absorption oscillator strength). The connection between the lifetime and the linewidth can be made plausible by a simple semiclassical model. When an electron is excited to a specific orbit in an atom, its binding energy is established through the exchange of virtual photons with the effective central core. If the perturbations that eventually cause the electron to make a transition from that orbit are weak, the meanlife will be long.
Most introductory courses on physics mention the basics of atomic physics: we learn about the ground state and excited states, and that the excited states have a finite life-time decaying eventually to the ground state while emitting radiation. You might have hoped for a better understanding of these decay processes when you started studying quantum mechanics. If this was the case, you must have been disappointed. Courses on basic quantum mechanics in fact assume that all quantum states have an infinitely long life-time. You learn how to employ the Schrödinger equation to determine the wave function of a particle in various setups, for instance in an infinitely deep well, in a parabolic potential, or in the spherically symmetric potential set up by the nucleus of a hydrogen atom. These solutions to the Schrödinger equation are explained as being stationary, the only time-dependence of the wave function being a periodic phase factor due to the energy of the state. Transitions between different quantum states are traditionally not discussed and left for courses on advanced quantum mechanics. If you have been disappointed by this, then this chapter will hopefully provide some new insight: we learn how matter and radiation interact, and, in particular, how excited atomic states decay to the ground state. Our main tool to calculate emission and absorption rates for radiative processes is Fermi’s golden rule (see Section 1.6.1). We thus start by refreshing the golden rule and specifying its proper use. Before we turn to explicit calculations of transition rates, we use the first part of the chapter for some general considerations concerning emission and absorption, those being independent of the details of the interaction between matter and electromagnetic radiation. In this context we briefly look at master equations that govern the probabilities to be in certain states and we show how to use those to understand properties of the equilibrium state of interacting matter and radiation. We show that it is even possible to understand so-called black-body radiation without specifying the exact interaction Hamiltonian of matter and radiation.
The mean life of the mercury line $\ensuremath{\lambda}2537$ was measured by the alternating voltage method previously described by Webb. The radiation was excited in a quartz tube by impact with electrons from a hot cathode. The photoelectric system was contained in a separate tube also of quartz. The photoelectric surface was zinc evaporated on a nickel plate. Measurements of the apparent life were made with diminishing mercury vapor pressures until further reduction of pressure produced no change in the measured rate of decay of the radiation. At these low pressures (below 3\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}4}$ mm, for the geometry used) the absorption and reemission of the radiation was negligible and the true life of a single excitation process was measured. This was found to be 1.08\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}7}$ sec. with an estimated precision of one percent.Absorption coefficient of $\ensuremath{\lambda}1849$. An absorption cell with a one mm thick absorbing layer of mercury vapor was mounted so as to move in and out between the excitation and photoelectric tubes used in the above measurements. The absorption of $\ensuremath{\lambda}1849$ was measured at pressures such that the absorption of $\ensuremath{\lambda}2537$ was negligible. From this the atomic absorption coefficient for the main component of the line was found to be 6.2\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}11}$ from which the mean life of the 21P12^{1}P_{1}21P1​ state was calculated to be 2.9\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}10}$ sec. with an estimated precision of 25 percent.
This paper asks an experimental question about photon absorption and emission in a two-state electronic system where the incident frequency of exciting light is below the mean frequency of the emission line but still within the line width. Since it is well known that the absorption spectrum and the emission spectrum are identical [1]-[3], then quantum mechanics as well as astronomical observations say that excitations will still occur. Once a quantum state excites, then it is expected to decay back with a typical time constant and the usual emission spectrum. The problem is that the mean frequency of that emission spectrum will then be higher than the frequency of the exciting photon, suggesting that more energy is emitted than being received on average. A search of quantum mechanical references finds no discussion of this issue, so this paper considers the options available to resolve this inconsistency, including an experiment.