Quantum Physics

912 Chapter 28 Atomic PhysicsTABLE 28.1Shell and Subshell NotationShellSubshelln Symbol Symbol1 K 0 s2 L 1 p3 M 2 d4 N 3 f5 O 4 g6 P 5 h. . . . . .Figure 28.9 A single line (A) cansplit into three separate lines (B) in amagnetic field.ABWithin a few months following the publication of Bohr’s paper, Arnold Sommerfeld(1868–1951) extended the Bohr model to include elliptical orbits. We examinehis model briefly because much of the nomenclature used in this treatment is still inuse today. Bohr’s concept of quantization of angular momentum led to the principalquantum number n, which determines the energy of the allowed states of hydrogen.Sommerfeld’s theory retained n, but also introduced a new quantum number called the orbital quantum number, where the value of ranges from 0 to n 1 in integersteps. According to this model, an electron in any one of the allowed energystates of a hydrogen atom may move in any one of a number of orbits correspondingto different values. For each value of n, there are n possible orbits corresponding todifferent values. Because n 1 and 0 for the first energy level (ground state),there is only one possible orbit for this state. The second energy level, with n 2, hastwo possible orbits, corresponding to 0 and 1. The third energy level, withn 3, has three possible orbits, corresponding to 0, 1, and 2.For historical reasons, all states with the same principal quantum number n aresaid to form a shell. Shells are identified by the letters K, L, M, . . . , which designatethe states for which n 1, 2, 3, . . . . Likewise, the states with given values of n and are said to form a subshell. The letters s, p, d, f, g, . . . are used to designate the statesfor which 0, 1, 2, 3, 4, . . . . These notations are summarized in Table 28.1.States that violate the restriction 0 n 1, for a given value of n, can’t exist.A 2d state, for instance, would have n 2 and 2, but can’t exist because thehighest allowed value of is n 1, or 1 in this case. For n 2, 2s and 2p are allowedsubshells, but 2d, 2f, . . . are not. For n 3, the allowed states are 3s, 3p, and 3d.Another modification of the Bohr theory arose when it was discovered that thespectral lines of a gas are split into several closely spaced lines when the gas is placedin a strong magnetic field. (This is called the Zeeman effect, after its discoverer.) Figure28.9 shows a single spectral line being split into three closely spaced lines. This indicatesthat the energy of an electron is slightly modified when the atom is immersed ina magnetic field. In order to explain this observation, a new quantum number, m ,called the orbital magnetic quantum number, was introduced. The theory is in accordwith experimental results when m is restricted to values ranging from to ininteger steps. For a given value of , there are 2 1 possible values of m .Finally, very high resolution spectrometers revealed that spectral lines of gasesare in fact two very closely spaced lines even in the absence of an external magneticfield. This splitting was referred to as fine structure. In 1925 Samuel Goudsmit andGeorge Uhlenbeck introduced the idea of an electron spinning about its own axisto explain the origin of fine structure. The results of their work introduced yet anotherquantum number, m s , called the spin magnetic quantum number.For each electron there are two spin states. A subshell corresponding to a givenfactor of can contain no more than 2(2 1) electrons. This number comesfrom the fact that electrons in a subshell must have unique pairs of the quantumnumbers (m , m s ). There are 2 1 different magnetic quantum numbers m ,and two different spin quantum numbers m s , making 2(2 1) unique pairs(m , m s ). For example, the p subshell ( 1) is filled when it contains 2(2 1 1) 6electrons. This fact can be extended to include all four quantum numbers, as willbe important to us later when we discuss the Pauli exclusion principle.All these quantum numbers (addressed in more detail in upcoming sections)were postulated to account for the observed spectra of elements. Only later werecomprehensive mathematical theories developed that naturally yielded the sameanswers as these empirical models.28.5 DE BROGLIE WAVES ANDTHE HYDROGEN ATOMOne of the postulates made by Bohr in his theory of the hydrogen atom was thatthe angular momentum of the electron is quantized in units of , orm e vr n