Abstract Some recent work has been done by Back and Goudsmidt on the “hyperfine” structure of the optical spectrum of bismuth,* and more recently similar work has been carried out for cæsium by Jackson. In each of these investigations the line structure was examined closely with a view to revealing a still finer structure, and it was found in both cases that the lines attributed to electronic spin were themselves composed of several distinct lines. In fact, for cæsium, each of the fine (electron spin) lines of the principal series was found to split up into two ; for bismuth the hyperfine structure was more complicated. Back and Goudsmidt attributed the structure to a nuclear spin, and working out the consequences of this on the lines of the old quantum mechanics they found that a nuclear spin of 41/2 quanta is necessary to account for the facts ; a spin of a 1/2 quantum is similarly attributed by Jackson to the nucleus of cæsium. The hypothesis explains very satisfactorily in a qualitative way the results of observation. In the work described in the present paper the methods of the new quantum mechanics have been applied to the problem. More precisely, we consider the motion of a single electron in a Coulombian field due to a nucleus possessing a 1/2 quantum of spin. It will be seen that the results can easily be extended to the case of any central field, and the principle could also be extended to the case of an atom with a nuclear spin of 1/2(nh/2π), but the detailed working out would be very heavy for n > I (at any rate, using the methods explained in this paper), owing to the large number of wave functions which would be necessary to specify any state of the atom. It will be seen that the results we obtain are substantially the same as Jackson’s so far as the energy levels are concerned, but the calculated intensities are not consistent with the observed transitions, and we deduce a combination rule which is radically different from Jackson’s.
Abstract Some recent work has been done by Back and Goudsmidt on the “hyperfine” structure of the optical spectrum of bismuth,* and more recently similar work has been carried out for cæsium by Jackson. In each of these investigations the line structure was examined closely with a view to revealing a still finer structure, and it was found in both cases that the lines attributed to electronic spin were themselves composed of several distinct lines. In fact, for cæsium, each of the fine (electron spin) lines of the principal series was found to split up into two ; for bismuth the hyperfine structure was more complicated. Back and Goudsmidt attributed the structure to a nuclear spin, and working out the consequences of this on the lines of the old quantum mechanics they found that a nuclear spin of 41/2 quanta is necessary to account for the facts ; a spin of a 1/2 quantum is similarly attributed by Jackson to the nucleus of cæsium. The hypothesis explains very satisfactorily in a qualitative way the results of observation. In the work described in the present paper the methods of the new quantum mechanics have been applied to the problem. More precisely, we consider the motion of a single electron in a Coulombian field due to a nucleus possessing a 1/2 quantum of spin. It will be seen that the results can easily be extended to the case of any central field, and the principle could also be extended to the case of an atom with a nuclear spin of 1/2(nh/2π), but the detailed working out would be very heavy for n > I (at any rate, using the methods explained in this paper), owing to the large number of wave functions which would be necessary to specify any state of the atom. It will be seen that the results we obtain are substantially the same as Jackson’s so far as the energy levels are concerned, but the calculated intensities are not consistent with the observed transitions, and we deduce a combination rule which is radically different from Jackson’s.
Electron spin In our development of quantum mechanics to this point, the behavior of a particle, usually an electron, is governed by a wave function that is dependent only on the cartesian coordinates x, y, z or, equivalently, on the spherical coordinates r , θ, ϕ. There are, however, experimental observations that cannot be explained by a wave function which depends on cartesian coordinates alone. In a quantum-mechanical treatment of an alkali metal atom, the lone valence electron may be considered as moving in the combined field of the nucleus and the core electrons. In contrast to the hydrogen-like atom, the energy levels of this valence electron are found to depend on both the principal and the azimuthal quantum numbers. The experimental spectral line pattern corresponding to transitions between these energy levels, although more complex than the pattern for the hydrogen-like atom, is readily explained. However, in a highly resolved spectrum, an additional complexity is observed; most of the spectral lines are actually composed of two lines with nearly identical wave numbers. In an alkaline-earth metal atom, which has two valence electrons, many of the lines in a highly resolved spectrum are split into three closely spaced lines. The spectral lines for the hydrogen atom, as discussed in Section 6.5, are again observed to be composed of several very closely spaced lines, with equation (6.83) giving the average wave number of each grouping.
Abstract Bohr’s theory of the atom did a fine job of explaining the absorption and em1ss10n spectra of one-electron atoms in terms of the quantum rules. Schrodinger’s wave mechanics did better in the sense that the rules of quantization were given a firm mathematical basis. However, problems still abounded. Experimental spectra revealed quite a number of problems for which wave mechanics (and matrix mechanics) appeared to have no solutions. In particular, some atomic emission lines predicted by the theory were seen to be split in the presence of a magnetic field into several quite distinct lines (a phenomenon known as the Zeeman effect). Neither wave mechanics nor matrix mechanics could explain this phenomenon without some further, largely ad hoc, assumptions. Worse still, when the spectra of atoms with two or more electrons were studied, certain predicted lines were found to be missing.
Niels Bohr’s model of the hydrogen and hydrogen-like atoms, in which the electron is allowed to revolve in certain circular orbits around the nucleus, was extended by Arnold Sommerfeld to allow for the possibility of elliptical orbits. Each elliptical orbit was constrained by two quantization conditions specified by a positive integer nφ and a non-negative integer nr. If, in the presence of the attractive Coulomb force between the electron and the nucleus, the dynamics of the atom were taken to be governed by the Newtonian laws of motion, the energy Ɛ of the two-particle system was found to be a function of the positive integer n = nr + nφ. The atomic energy levels thus exhibited an n-fold degeneracy, which would be lifted in the presence of external electric or magnetic fields, thus accounting for the fine-structure of the spectral lines emitted by the atom—phenomena observed, respectively, in the Stark effect and the Zeeman effect. Sommerfeld also examined the behavior of his elliptical orbits by allowing the revolving electron to obey Einstein’s relativistic dynamics. In this case, the energy-level degeneracies were lifted (even in the absence of external perturbations), and the elliptical orbits were found to precess around the nucleus—similarly to the precession of, say, the orbit of the planet Mercury around the Sun. What is astonishing about Sommerfeld’s relativistic expression of the energy Ɛ as a function of nφ and nr is that it is precisely the same expression as obtained for hydrogen and hydrogen-like atoms from Dirac’s equation, which came more than a decade later. Recall that the spin of the electron was not yet discovered when Sommerfeld developed his model, and that in Dirac’s fully-relativistic quantum mechanics, the spectral fine-structure arises from the coupling of the electron’s spin to its orbital motion. The present paper provides a streamlined yet detailed account of Sommerfeld’s elliptical orbits in hopes of bringing about a deeper appreciation for the miraculous coincidence of Sommerfeld’s result with that derived from Dirac’s equation.
Wave mechanics failed badly in accounting for the multiplicity of atomic energy levels. This was most conspicuous in the case of the alkali metals, lithium, sodium, potassium, and so on. It was known that an atom of any of these elements can be treated as a more-or-less inert core, consisting of the nucleus and Z − 1 inner electrons, together with a single outer electron whose transitions between energy levels are responsible for spectral lines. Since the electrostatic field felt by the outer electron is not a Coulomb field, its energy levels in the absence of external fields depend on the orbital angular momentum quantum number ℓ as well as a radial quantum number n , but because of the spherical symmetry of the atom, not on the angular momentum z -component ħ m . (See Eq. (2.1.30).) For each n , ℓ , and m there should be just one energy level. But observations of atomic spectra showed that in fact all but the s states were doubled. For instance, even a spectroscope of low resolution shows that the D line of sodium, which is produced in a 3 p → 3 s transition, is a doublet, with wavelengths 5896 and 5890 Angstroms. Pauli was led to propose that there is a fourth quantum number for electrons in such atoms, in addition to n , ℓ , and m , with the fourth quantum number taking just two values in all but s states. But the physical significance of this fourth quantum number was obscure.
Abstract Eigenstates of the square well potential are calculated and displayed. Barrier penetration and the connection to total internal reflection are explained. α–decay by barrier penetration is calculated and used to explain Geiger–Nuttall plots. Gauss–Hermite solutions to the harmonic oscillator potential are deduced and displayed. Zero point fluctuations are introduced. Hydrogen atom eigenstate wavefunctions for the Coulomb potential are calculated and displayed. Principal, orbital angular momentum and intrinsic angular momentum quantum numbers and their allowed combinations are discussed and interpreted: n, l, ml, s and ms. The Stern–Gerlach experiment and Pauli’s perception that electron spin is half-integral are presented; as are Beth’s experiment and photon spin. Dominance of electric dipole transitions and resulting selection rules discussed. Fine spectral structure and spin-orbit coupling are described. Nuclear spin and resulting hyperfine spectral structure are introduced. Landé factors introduced.
Defects determine the mechanical, electrical and optical properties in solids. Often their chemical nature is very simple, comprising for example just a simple lattice vacancy or a single atom impurity. Some defects are optically active, i.e. they show allowed optical transitions. Those defect are called colour centers and often determine the optical properties of the respective solid. Famous examples are colour centers in ruby or alkali halides. By careful control of defect center concentration in combination with optical microscopy it became recently possible to investigate single defect centers optically. One of the materials that was investigated in particular detail is diamond. Pure diamond is an optically transparent medium. However more than 100 colour centers have been identified in diamond. Owing to its energy level structure the nitrogen vacancy center in diamond is a particularly interesting one to investigate. This defect center has a paramagnetic ground and first excited state. The resulting fine structure splitting can be resolved by low temperature fluorescence excitation spectroscopy. Ground as well as first excited state are spin triplets, i.e. S=1. When spin angular momentum is conserved during optical excitation one thus expects to find three optical excitation lines owing to the three spin sub levels in each electronic state. Indeed conservation of spin angular momentum during excitation seems to be fulfilled, however only a single excitation line is found. Experimentally this is due to unfavourable photophysical properties of the other transitions. Because of low electron phonon coupling of the center the linewidth of the transition is limited by the lifetime of the optically excited state and not, as in most cases, by dephasing due to electron-phonon coupling. Due to this low electron-phonon coupling all of the oscillator strength of the optical transition is concentrated in a narrow frequency interval giving rise to a large on resonance absorption cross section. This large cross section allows for ms averaging times, which are much smaller than the spin lattice relaxation times. One thus is able to determine the specific electron spin state the defect center is in. In diamonds isotopically enriched with 13C, the electron spin of the defect center is coupled to the nuclear spins of the surrounding 13C nuclei by hyperfine interaction. The hyperfine coupling depends on the distance of the center of the defect to the nuclei. For nuclei in the first coordination shell around the defect the coupling strength is around 250 MHz for nuclei in the second shell it is roughly 60 MHz. The hyperfine coupling is thus resolvable in single defect center excitation spectra. The spin relaxation times of nuclei in diamond are even longer than those of the electron. One thus is also able to determine the nuclear spin states around the defect center. Simultaneous radiation with radio frequency generates magnetic dipole transition and thus allows to control the nuclear spin states around the center. The system provides an interesting example for readout and manipulation of single spin degrees of freedom which may be interesting as hardware in future quantum information processing application.