By L. D. Landau, E. M. Lifshitz
This version has been thoroughly revised to incorporate a few 20% of latest fabric. very important fresh advancements resembling the idea of Regge poles are actually incorporated. Many issues of options were extra to these already inside the booklet.
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Extra info for Quantum Mechanics Non-Relativistic Theory (3rd Edition) (Course on Theoretical Physics, Volume 3)
8) should hold for an arbitrary nor malized function T f a ) , it is necessary (cf. §3) that an arbitrary function can be expanded in t e r m s of the functions H-"n(^). T h i s means that the functions H-*n(#) form a complete set of normalized and orthogonal functions. If t h e initial wave function of t h e electron coincides with one of t h e func tions Ψ Λ( ί ) , then the corresponding constant an is evidently equal to unity, while all t h e others are zero. I n other words, a measurement m a d e on an electron in t h e state Ψ Λ( ϊ ) gives with certainty the n t h result.
I t is natural to define t h e derivative / of a quantity / as t h e quantity whose mean value is equal to t h e derivative, with respect to time, of t h e mean value / . T h u s we have t h e definition (9-1) / - / Starting from this definition, it is easy to obtain an expression for the q u a n t u m - m e c h a n i c a l operator / corresponding to t h e quantity / : • . d r . Γ df CW, f ,3Ψ H e r e dfjdt is t h e operator obtained by differentiating t h e operator / with respect to t i m e ; / may depend on t h e time as a parameter.
I n geometrical optics, however, t h e propagation of light along definite paths, or rays, is considered. Such an analogy enables u s to see that t h e passage from q u a n t u m mechanics t o t h e limit of classical mechanics occurs similarly to t h e passage from wave optics to geometrical optics. L e t us recall how this latter transition is m a d e mathematically (see Fields, §53). L e t u be any of t h e field c o m p o n e n t s in t h e electromagnetic wave. It can b e written in t h e form u = ae** (with a a n d φ real), w h e r e a is called t h e amplitude a n d φ t h e phase of t h e wave (called in geometrical optics t h e eikonal).