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Then (53) [§16] Revision of Quantum Mechanics 25 which is, of course, different from the classical value of the velocity as p/m. This shows at once that we must be careful when dealing with velocities. Furthermore, we may expect that at least for a free particle p/m still represents correctly its particle velocity vg (which will be shown to be true at the end of this section) so that vp and vg must be expected to be different. We must now try to define more precisely what we mean by wave and particle velocities, which are called phase velocity vp and group velocity vg respectively.

Ii) The most general case, when fx and f2 in the expression f = cxfx c a re + ifi complex, has been considered in this exercise. The variational function / contains therefore four functions and the most general combination of them should contain four coefficients. However, since we have taken cx and c2 to be complex, we had exactly this number of coefficients. In the exercise, we had really four independent variables to vary, namely the real and imaginary parts of c1 and c2. It is simpler, however, as we have done, to replace these four variables by cl9 c29 c*9 and c29 treated as independent variables.

33) We want to estimate v(E)9 for which we must study the degeneracy arising from the possible combinations of κχ9 Ky9 and κζ (see Table 1). (31)]. 2 If we take for the sake of an example s = 16 we can represent graphically the possible states, as shown in Fig. 17, where κχ and Ky are given along respective perpendicular axes. e. all pairs of values 2 2 κΧ9 Ky)9 which are possible, that is which satisfy the condition κ\+κ < 4 . 2 2 As an example, for the point A9 /c + /c = 13. Actually, since we can Β 8] Free-electron Theory of Metals 53 also take negative values of κχ and Ky, the figure should be extended symmetrically into all four quadrants.