Atomic physics

continuous distribution of wavelengths. In sharp contrast to this continuousdistribution spectrum is the discrete line spectrum observed when a low-pressure gas undergoes an electric discharge. Observation and analysis of these spectral lines is called emission spectroscopy.

substances is absorption spectroscopy. An absorption spectrum is obtained by passing white light from a continuous source through a gas or a dilute solution of the element being analyzed. The absorption spectrum consists of a series of dark lines superimposed on the continuous spectrum of the light source.

(1825–1898), found an empirical equation that correctly predicted the wavelengths of four visible emission lines of hydrogen: Hα (red), Hβ (bluegreen), Hγ (blue-violet), and Hδ (violet). The four visible lines occur at the wavelengths 656.3 nm, 486.1 nm, 434.1 nm, and 410.2 nm. The complete set of lines is called the Balmer series.

found following Balmer’s discovery. These spectra are called the Lyman, Paschen, and Brackett series after their discoverers. The wavelengths of the lines in these series can be calculated through the use of the following empirical equations:

No theoretical basis existed for these equations; they simply worked. The same constant RH appears in each equation, and all equations involve small integers.

model. As we previously an atom emits (and absorbs) certain characteristic frequencies of electromagnetic radiation and no others, but the Rutherford model cannot explain this phenomenon. A second difficulty is that Rutherford’s electrons are undergoing a centripetal acceleration. According to Maxwell’s theory of electromagnetism, centripetally accelerated charges revolving with frequency f should radiate electromagnetic waves of frequency f. Unfortunately, this classical model leads to a prediction of self-destruction when applied to the atom. As the electron radiates, energy is carried away from the atom, the radius of the electron’s orbit steadily decreases, and its frequency of revolution increases. This process would lead to an ever-increasing frequency of emitted radiation and an ultimate collapse of the atom as the electron plunges into the nucleus .

quantum theory, Einstein’s concept of the photon, Rutherford’s planetary model of the atom, and Newtonian mechanics to arrive at a semiclassical model based on some revolutionary ideas. The postulates of the Bohr theory as it applies to the hydrogen atom are as follows:

1. The electron moves in circular orbits around the proton under the influence of the electric force of attraction as shown in Figure.

stable. When in one of these stationary states, as Bohr called them, the electron does not emit energy in the form of radiation, even though it is accelerating. Hence, the total energy of the atom remains constant and classical mechanics can be used to describe the electron’s motion. Bohr’s model claims that the centripetally accelerated electron does not continuously emit radiation, losing energy and eventually spiraling into the nucleus, as predicted by classical physics in the form of Rutherford’s planetary model.

the electron makes a transition from a more energetic initial stationary state to a lower-energy stationary state. This transition cannot be visualized or treated classically. In particular, the frequency f of the photon emitted in the transition is related to the change in the atom’s energy and is not equal to the frequency of the electron’s orbital motion. The frequency of the emitted radiation is found from the energy-conservation expression

where Ei is the energy of the initial state, Ef is the energy of the final state,

electron orbit is determined by a condition imposed on the electron’s orbital angular momentum: the allowed orbits are those for which the electron’s orbital angular momentum about the nucleus is quantized and equal to an integral multiple of

where me is the electron mass, v is the electron’s speed in its orbit, and r is the orbital radius.

the allowed energy levels and find quantitative values of the emission wavelengths of the hydrogen atom.

the kinetic energy of the electron is

energy of the atom:

The orbit with the smallest radius, called the Bohr radius a0, corresponds to n =1 and has the value

to energy quantization. Substituting

Inserting numerical values into this expression, we find that

electron makes a transition from an outer orbit to an inner orbit:

Bohr’s Model of the Hydrogen Atom

Because the quantity measured experimentally is wavelength, it is convenient to use c=fλ to express this Equation in terms of wavelength:

Remarkably, this expression, which is purely theoretical, is identical to the general form of the empirical relationships discovered by Balmer and Rydberg:

two quantities agree to within approximately 1%, this work was recognized as the crowning achievement of his new quantum theory of the hydrogen atom. Furthermore, Bohr showed that all the spectral series for hydrogen have a natural interpretation in his theory. The different series correspond to transitions to different final states characterized by the quantum number nf.

observed in the spectra of the Sun and several other stars could not be due to hydrogen but were correctly predicted by his theory if attributed to singly ionized helium. In general, the number of protons in the nucleus of an atom is called the atomic number of the element and is given the symbol Z. To describe a single electron orbiting a fixed nucleus of charge +Ze, Bohr’s theory gives

relativity, we found that Newtonian mechanics is a special case of relativistic mechanics and is usable only for speeds much less than c. Similarly,
quantum physics agrees with classical physics when the difference between quantized levels becomes vanishingly small.

This principle, first set forth by Bohr, is called the correspondence principle.

the problem of the hydrogen atom is to substitute the appropriate potential energy function into the Schrӧdinger equation, find solutions to the equation, and apply boundary conditions. The potential energy function for the hydrogen atom is that due to the electrical interaction between the electron and the proton:

The mathematics for the hydrogen atom is more complicated than that for the particle in a box because the atom is three-dimensional and U depends on the radial coordinate r. If the time-independent Schrцdinger equation is extended to three-dimensional rectangular coordinates, the result is

with the radial function R(r) of the full wave function, is called the principal quantum number and is assigned the symbol n.

The orbital quantum number, symbolized l comes from the differential equation for f(u) and is associated with the orbital angular momentum of the electron. The orbital magnetic quantum number m, arises from the differential equation for g(f). Both l and m, are integers.

on the three parts of the full wave function leads to important relationships among the three quantum numbers as well as certain restrictions on their values:

atom depends only on the radial distance r between nucleus and electron, some of the allowed states for this atom can be represented by wave functions that depend only on r. For these states, f(u) and g(f) are constants. The simplest wave function for hydrogen is the one that describes the 1s state and is designated

radial probability density function for the hydrogen atom in its ground state: