p31318notes09.pdf - 1 P313 Notes 09 Fine Structure and Spin...

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1 P313 Notes 09 Fine Structure and Spin So far, we understand that the energy of an eigenfunction depends solely upon the principal quantum number n . There are n 2 eigenfunctions with a given n all with the same energy, but with different angular momenta. The angular momentum quantum number l can have any of n integral values, from 0 to n - 1, and, for a given l , the quantum number m that governs the z -component of the angular momentum, can have any integral value from - l to +l . Specifically, the energy is 2 n Rhc E n - = , the angular momentum is h ) 1 ( + l l , and its z- component is h m . We are shortly going to need another symbol m -like symbol, so, from this point I am going to use the symbol m l here, so that the z- component of the orbital angular momentum is h l m . I can think of four lines of evidence that led to the concept of spin. Two of them are spectroscopic: 1. The fine structure of spectrum lines. 2. The splitting of lines in a magnetic field (Zeeman effect) The other two are 3.. Theoretical. (Relativistic solution of the wave equation by Dirac). 4. Experimental demonstration of the existence of (and, later, precise measurement of) the magnetic moment of an electron and its spatial orientation. The lines in the Balmer series of hydrogen are not single lines, but consist of several lines very close together. The Balmer lines are very broad, for various reasons to be discussed later in the course, and are much broader than the separation between their several components. Thus the fine structure of the Balmer lines is very difficult to detect and it needs special techniques. The splitting is slightly more obvious in the spectrum of the He atom, and it is glaringly obvious in the spectra of more complex atoms. Nevertheless, we shall discuss the fine structure of the hydrogen lines to begin with, because it is the simplest case and easiest to describe. The fine structure of the lines implies that the simple energy levels that we have hitherto described are not in fact single, but the levels themselves have a fine structure. In the models we have discussed so far, all eigenfunctions with a given n had the same energy.
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2 For a given n , there may be several values of l , but these represent different values of the orbital angular momentum. This can be easily understood in that we used the nonrelativistic approximation ) 2 /( 2 m p T = in deriving the Schrödinger equation. (Here m is the mass of the particle under consideration, which, in the context of the hydrogen atom, will be its electron.) The equations need to be modified slightly to take account of relativity effects. This is part of the reason for the splitting of the energy levels. Show that to order α 2 the kinetic energy is given by .... ) 1 ( 2 2 2 1 2 + α - = m p T However, even after taking that into account, there are twice as many energy levels (eigenfunctions) as you would expect. It was proposed in 1925 by Goudsmit and Uhlenbeck that this is a consequence of an intrinsic angular momentum (spin), and hence magnetic moment, of an electron.
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