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As is known, the semiclassical approximation applies to the solution of the Schrödinger equation - ℏ/2m d2 ψ (χ)/dχ2 + U (χ)ψ(χ) = Eψ(χ) (5.1) if the potential U(χ) changes slowly at the distances on the order of the de Broglie wavelength of the particle, i.e., at those distances where the solution ψ (χ) itself varies essentially. Equation (5.1) can be rewritten somewhat otherwise: d2ψ(χ)/dχ2 + κ2 (chi;)ψ(χ) = 0, κ2(χ = 2m/ ℏ2 [E - U(χ)]. (5.2) Now the condition of applicability of the semiclassical approximation is that κ(χ) should vary slowly at the distances on the order of 1/κ(χ). Equation (5.2) describes in fact a sufficiently wide class of phenomena, in no way confined to quantum mechanics. One such problem will be considered below.
Original language | English |
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Title of host publication | Theoretical Kaleidoscope |
Editors | I.B. Khriplovich |
Pages | 53-66 |
Number of pages | 14 |
DOIs | |
State | Published - 11 Jan 2008 |
Name | Lecture Notes in Physics |
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Volume | 748 |
ISSN (Print) | 0075-8450 |
ID: 36641962