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6) tThe rate of change of the condition must be so slow that the relevant frequencies are all small compared to the frequencies involved in transitions between stationary states. 46 T H E O L D Q U A N T U M THEORY which, in many cases, is just the total energy, ^Λ-ϋ, expressed in terms of the pk and qk. 5) are often difficult to solve, and we can use the following method to simplify them. If we introduce a function S{qk\o-k) of the qk and of s new variables α*, and if we perform a transformation from the Pk and qk to a new set of variables a* and ßk through the equations dS „ dS the equations of motion are again in canonical form, = ^'--e^k^ (^•«> where is the Hamiltonian expressed in terms of the α* and ßk.
Let us now consider the spectrum of an alkali metal atom, such as sodium. The sodium atom consists of a nucleus of charge 4-11^ surrounded by 11 electrons. 42)) that a single electron is responsible for the spectrum, we must consider the motion of one electron in the combined field of the nucleus and the 10 other electrons. A theoretical justification is afforded by the shell model of the atom which we shall brieñy ATOMIC SPECTRA I N THE O L D Q U A N T U M THEORY 53 discuss later in this chapter.
47). It turns out that the energies in the F I G . 6. Grotrian diagram for sodium. OQT 56 THE OLD QUANTUM THEORY column marked " i " correspond to nl orbits, those in the column marked to n2 orbits, in the "rf" column to «3 orbits, and in the column to n4 orbits. The nomenclature s, p, d, f was introduced because these levels led to the fundamental, sharp, diffuse, and fundamental series. The fact that only energy-levels in adjacent columns combine is due to so-called selection rules, one of which states that in a transition k must either increase or decrease by unity.