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4. THE WAVE NATURE OF PARTICLES cou ld not be applied to ap eriodic systems, most collision and sca ttering problems were beyond its pale. Furthermore, it contained errors , contradictions, and ambiguities." It did have some virtues, however. predicted a large body of experimental results from a few simple rules, and it set th e stage for the new quantum mechanics which soon replaced it. We will now proceed to discuss the wave mechanics of de Broglie and Schrodinger. 121 c THE DIFFRACTION OF hand the de Brogl ie frequency has not been a very usefu l concept and it com e~ into play only in th e calculation of the p h a s ~ : el.ocity. H ere a dis- tin ction ap pears betwee n the relativistic and non-r: latlvistic c ~s~s. If the rest mass energy is includ ed in the total energy E (as In de Broglie s tr eatment), then the phase velocity of the wave becom es U = AY = ~ = ~ vi p2 C2 + (moc 2 ) 2 = c J I or, THE DEVELOPMENT OF WAVE MECHANICS 120 h P holds for photons as well as for both relativistic and non-relativistic materia l particles, provided that the appropriate expression for p is used. On the oth er (4.15) fik v 2m = '2' U Whether or not particles of a given momentum will exhibit their wa~( characteristics will be determined by the relative magnitude oftheir de Broglie wavelength in comparison with the physical dimensions of the environmen in which they are found. For wavelengths that are much smaller than the dimensions of apertures and obstacles, diffraction and other wave effects ~ rc not ordinarily observed. I n such cases we can assume rectilinear propag~~lOI and problems can be treated by means of ray diagrams (for .example, ~ISlb l c light in our everyday world). However, for wavelengths which a p ~ rmu m a t ' or exceed th e dimensions of objects, diffraction effects become qu ite importan and ray diagrams become meaningless (for example, audible sound i~ ou everyday world ). In order to get some insigh t into th e kinds of behavior tc 5. THE DIFFRACTION OF PARTICLES dos I dE v = dk = Ii dk . that is one-half of the particle velocity. Thus the phase velocity is not of any physical significance. The group velocity of a particle wave, however, is given by In obtaining Equation 4.I5 we have expressed the relativistic momentum as p = ymov, wh ere v = fJc is the par~icle velocity and y ~ (I -.(32) -to From our knowledge of waves" we identify the particle velocity v WIth the group velocity of th e wave packet. Since special relativity requires that v be less than c, we no te that Equation 4.15 calls for phase veloci ties greater than c. However, as no energy (that is, no signal or information) is transmitted at the phase velocity, the fact that u > c constitutes no violation of the postulates of special relativity. . . In non-relativistic quantum mechamcs the rest mass term IS neglect ed and the total energy E is merely the sum of the kinetic an d potential energies.
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