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===ფაზური და ჯგუფური სიჩქარე===
{{ითარგმნება}}
[[სურათი:Wave group.gif|მინი|frame|right|ღრმა წყალში გავრცელებადი [[ზედაპირული გრავიტაციული ტალღების]] სიხშირის დისპერსიის ილუსტრაცია. წითელი წერტილი მოძრაობს [[ფაზური სიჩქარე|ფაზური სიჩქარით]], ხოლო მწვანე წერტილი კი [[ჯგუფური სიჩქარე|ჯგუფური სიჩქარით]].]]
:<math> \psi (x, \ t) = \int_{-\infty} ^{\infty}\ dk_1 \ A(k_1)\ e^{i\left(k_1x - \omega t \right)} \ , </math>
სადაც ''A(k''<sub>1</sub>'')'' (the integral is the inverse fourier transform of A(k1)) is a function exhibiting a sharp peak in a region of wave vectors Δ''k'' surrounding the point ''k''<sub>1</sub> = ''k''. In exponential form:
:<math> A = A_o (k_1) e^ {i \alpha (k_1)} \ , </math>
with ''A''<sub>o</sub> the magnitude of ''A''. For example, a common choice for ''A''<sub>o</sub> is a [[Wave packet|Gaussian wave packet]]:<ref name=Bromley0>See, for example, Eq. 2(a) in
{{cite book |title=Quantum Mechanics: An introduction |author=Walter Greiner, D. Allan Bromley |url=http://books.google.com/books?id=7qCMUfwoQcAC&pg=PA61 |pages=60–61 |isbn=3540674586 |year=2007 |edition=2nd |publisher=Springer}}
</ref>
:<math>A_o (k_1) = N\ e^{-\sigma^2 (k_1-k)^2 / 2} \ , </math>
where σ determines the spread of ''k''<sub>1</sub>-values about ''k'', and ''N'' is the amplitude of the wave.
The exponential function inside the integral for ψ oscillates rapidly with its argument, say φ(''k''<sub>1</sub>), and where it varies rapidly, the exponentials cancel each other out, [[Interference (wave propagation)|interfere]] destructively, contributing little to ψ.<ref name=Messiah/> However, an exception occurs at the location where the argument φ of the exponential varies slowly. (This observation is the basis for the method of [[Stationary phase approximation|stationary phase]] for evaluation of such integrals.<ref name=Orland>
{{cite book |title=Quantum many-particle systems |author=John W. Negele, Henri Orland |url=http://books.google.com/books?id=mx5CfeeEkm0C&pg=PA121 |page=121 |isbn=0738200522 |year=1998 |publisher=Westview Press |edition=Reprint in Advanced Book Classics}}
</ref>) The condition for φ to vary slowly is that its rate of change with ''k''<sub>1</sub> be small; this rate of variation is:<ref name=Messiah/>
:<math>\left . \frac{d \varphi }{d k_1} \right | _{k_1 = k } = x - t \left . \frac{d \omega}{dk_1}\right | _{k_1 = k } +\left . \frac{d \alpha}{d k_1}\right | _{k_1 = k } \ ,</math>
where the evaluation is made at ''k''<sub>1</sub> = ''k'' because ''A(k''<sub>1</sub>'')'' is centered there. This result shows that the position ''x'' where the phase changes slowly, the position where ψ is appreciable, moves with time at a speed called the ''group velocity'':
:<math>v_g = \frac{d \omega}{dk} \ . </math>
The group velocity therefore depends upon the [[dispersion relation]] connecting ω and ''k''. For example, in quantum mechanics the energy ''E'' = ħω = (ħ''k'')<sup>2</sup>/(2''m''). Consequently,
:<math> \frac{d \omega}{dk}= v_g = \frac {\hbar k}{m} \ , </math>
showing that the velocity of a localized particle in quantum mechanics is its group velocity.<ref name=Messiah/> Because the group velocity varies with ''k'', the shape of the wave packet broadens with time, and the particle becomes less localized.<ref name=Fitt>
{{cite book |title=Principles of quantum mechanics: as applied to chemistry and chemical physics |author=Donald D. Fitts |url=http://books.google.com/books?id=8t4DiXKIvRgC&pg=PA15 |pages=15 ''ff'' |isbn=0521658411 |year=1999 |publisher=Cambridge University Press }}
</ref> In other words, the velocity of the constituent waves of the wave packet travel at a rate that varies with their wavelength, so some move faster than others, and they cannot maintain the same [[interference pattern]] as the wave propagates.
=== მდგარი ტალღა ===
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