Spectrum of the hydrogen atom

The hydrogen atom presented a unique opportunity in the development of quantum mechanics. The single electron moves in a coulombic field, free from fhe effecfs of infer-elecfron repulsions. This has fwo imporfanf consequences fhaf do nof apply fo any atom wifh fwo or more elecfrons  

The Schrddinger equation (Equation 1.28) is exacfly soluble wifh fhe hamilfonian of Equation (f.30). 

In 1947 Lamb and Retherford observed the 2 P j2 transition using microwave 

The hydrogen atom and its spectrum are of enormous importance in astrophysics because of the large abundance of hydrogen atoms both in stars, including the sun, and in the interstellar medium. 

Question. Calculate, to three significant figures, the wavelength of the first member of each of the series in the spectrum of atomic hydrogen with the quantum number (see Section f.2) n = 90 and 166. In which region of the electromagnetic spectrum do these transitions appear  

Answer. With dimensions of wavenumber, rather than frequency, Equation (1.11) becomes 

---

The hydrogen atom and one-electron ions are the simplest systems in the sense that, having only one electron, there are no inter-electron repulsions. However, this unique property leads to degeneracies, or near-degeneracies, which are absent in all other atoms and ions. The result is that the spectrum of the hydrogen atom, although very simple in its coarse structure (Figure 1.1) is more unusual in its fine structure than those of polyelectronic atoms. For this reason we shall defer a discussion of its spectrum to the next section. 

Whereas the emission spectrum of the hydrogen atom shows only one series, the Balmer series (see Figure 1.1), in the visible region the alkali metals show at least three. The spectra can be excited in a discharge lamp containing a sample of the appropriate metal. One series was called the principal series because it could also be observed in absorption through a column of the vapour. The other two were called sharp and diffuse because of their general appearance. A part of a fourth series, called the fundamental series, can sometimes be observed. 

Fig. 15-3. The spectrum of the hydrogen atom. isibte region UhravioleT region...

Plainly, the agreement between calculation and experiment is too good to be accidental. Our notched beam is a useful basis for interpreting the spectrum of the hydrogen atom. 

Prior to the development of quantum mechanics, the spectrum of the hydrogen atom posed quite a dilemma. To see the problem, and how it was resolved, let s go back about fifty or sixty years and trace the history of this problem. This is a valuable example because it shows how science advances. 

Actually Schrddinger s original paper on quantum mechanics already contained a relativistic wave equation, which, however, gave the wrong answer for the spectrum of the hydrogen atom. Due to this fact, and because of problems connected with the physical interpretation of this equation, which is of second order in the spaoe and time variables, it was temporarily discarded. Dirac took seriously the notion of first... 

The atomic spectra of most elements are complex and show little regularity. However, the emission spectrum of the hydrogen atom is sufficiently simple to be described by a single formula ... 

The ESR spectrum of the hydrogen atom can be interpreted as transitions between Ei and E3 levels and bet ween Ez and 2 4 levels. Resonances occur at a frequency... 

Brackett series Progression in the spectrum of the hydrogen atom starting with n = 4 and present in the infrared. 

Although Dirac s equation does not directly admit of a completely self-consistent single-particle interpretation, such an interpretation is physically acceptable and of practical use, provided the potential varies little over distances of the order of the Compton wavelength (h/mc) of the particle in question. It allows, for instance, first-order relativistic corrections to the spectrum of the hydrogen atom and to the core-level densities of many-electron atoms. The latter aspect is of special chemical importance. The required calculations are invariably numerical in nature and this eliminates the need to investigate central-field solutions in the same detail as for Schrodinger s equation. A brief outline suffices. 

Figure 1.2. An image produced by exciting hydrogen gas and separating the outgoing light with a prism, reprinted from [Her. Fig. 1. p. 5]. Specifically, this is the emission spectrum of the hydrogen atom in the visible and near ultraviolet region. The label marks the position of the limit of the series of wavelengths.

J. P. Vigier, The energy spectrum of the hydrogen atom with magnetic spin-orbit and spin-spin interactions, Phys. Lett. A 237(6), 349-353 (1998). 

Aqueous glasses can only be obtained from solutions containing high concentrations of acid or alkali. Trapped electrons are not detected in acid glasses although the esr spectrum of the hydrogen atom is observed62. Presumably the reaction... 

In 1885 Balmer was able to fit the discrete wavelengths X of part of the emission spectrum of the hydrogen atom, now called the Balmer series and illustrated in Figure 1.1, to the empirical formula... 

Starting somewhat later, Dirac applied his version of the new quantum mechanics to the hydrogen atom. Dirac s approach to the hydrogen atom was very different and more general than PauH s method and it too was successful in explaining the Balmer spectrum of the hydrogen atom. Dirac s paper on the hydrogen... 

To explain the spectrum of the hydrogen atom, all the facts had to be known and the power of both quantum mechanics and relativity theory were required. Neither version of quantum mechanics—Heisenberg s nor Schrodinger s— incorporated the theory of relativity. Neither Heisenberg s nor Schrodinger s quantum mechanics embraced the spin of the electron—a basic property of the electron discovered after Heisenberg and before Schrodinger did their seminal work. 

In 1928, Paul Dirac brought quantum mechanics and relativity together and, in the process, provided a solid theoretical basis for understanding the spectrum of the hydrogen atom. He made no assumptions, working from the firm footings of quantum mechanics and relativity theory. His work not only gave a complete... 

The hydrogen atom is the simplest one in existence, and the only one for which essentially exact theoretical calculations can be made on the basis of the fairly well confirmed Coulomb law of interaction and the Dirac equation for the electron. Such refinements as the motion of the proton and the magnetic interaction with the spin of the proton are taken into account in rather approximate fashion. Nevertheless, the experimental situation at present is such that the observed spectrum of the hydrogen atom does not provide a very critical test either of the theory or of the Coulomb law of interaction between point charges. A critical test would be obtained from a measurement of the fine structure of the n = 2 quantum state. 

Just about the time Rydberg composed his simple mathematical formula, he also discovered Balmer s paper on the spectrum of the hydrogen atom. Rydberg realized immediately that he could... 

Figure 2.4 The relative energies of the circular orbits of the Bohr model of the hydrogen atom. The electronic energies that give rise to the line spectrum of the hydrogen atom...

However, although the Bohr theory, involving a single quantum number n, was adequate to explain the line spectrum of the hydrogen atom with a single valence electron (Figures 2.4 and 2.5, respectively), it was inadequate to explain, in detail, the line spectrum of elements with more than one electron. To do this, it was found necessary to introduce the idea of three further quantum numbers, in addition to the principal quantum number, n. These arise from the wave nature of the electron. 

ON THE CONTINUOUS SPECTRUM OF THE HYDROGEN ATOM By Paul S. Epstein and Morris Muskat Norman Bridge Laboratory of Physics, California Institute of Technology Communicated March 23, 1929... 

Now tjb.6 relations observed in tie line spectrum of the hydrogen atom are known very exactly (fig. 1). It was Balmer (1885) who first showed that the lines situated in the visible region—all that were then known—can be represented by the formula... 

---