By Hartman, E.F.; Sandia National Laboratories.; United States. Dept. of Energy.; United States. Dept. of Energy. Office of Scientific and Technical Information

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20) where A and B are arbitrary vectors, ω and q are unknown parameters; ω is known as the angular frequency of the wave and q is called the wavevector. By substituting such a form for u(t, r) into Eq. 18), we easily find that Eq. 20) is a solution of Eq. 18), if ω2 = sq2 . The relationship between ω and q = |q| is called the dispersion relation: ω = s|q|. 21) Importantly, there is no limitation to the wavevector q: a solution can be found for any q. This is valid only for infinitely extended media, for which the wavevector can be a “continuous” vector.

18). For some cases, it is convenient to operate with such a complex waveform. However, one should remember that true physical quantities always have real values. Now we consider the important wave phenomenon known as wave interference. Suppose that two sinusoidal waves of the same frequency propagate from different sources through the medium. The sources of the waves are generally at different locations, so the waves reach a point of observation r, in general, with different phase shifts ϕ1 (r) and ϕ2 (r): u1 (t, r) = B1 sin(ωt + ϕ1 (r)), u2 (t, r) = B2 sin(ωt + ϕ2 (r)).

43) that there is an equal probability of finding a particle in any point of space: | (r, t)|2 = (r, t) ∗ (r, t) = |A|2 = constant, where ∗ (r, t) is the complex conjugate of (r, t). This result appears to be in contradiction with the classical description of a particle. Just as for the previously discussed case of electromagnetic fields, this contradiction can be resolved by introducing the uncertainty principle. This principle has the form of conditions restricting the range of coordinates and the range of momenta that can be measured simultaneously for a particle.