By Flügge S. (ed.)

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D. M. Lifshitz, Quantum Mechanics, Theory (Pergamon Press, London 1958) Chap. 7. M. Morse and H. Feshbach, Methods of Theoretical Physics, (McGraw-Hill Book Company, New York, NY, 1953) vol 2. p. 1092. B. Dingle, Asymptotic Expansions: Their Derivation and Interpretation (Academic Press, New York, NY, 1973) Chapter XIII. S. G. K. Park, The WKB approximation with phase loss and anharmonicity effect, Phys. Rev. 58, 3443 (1998). [6] C. Lanczos, Linear Differential Operators , (D. Van Nostrand, London 1961) Chap.

We take x\ to be the root with positive imaginary part and X2 the other root. We write Eq. 74) in which F is a positive quantity. 75) Sl = iVF, s2 = -iVF if F > 0. 76) Using the method just described we can write the approximate solution for ip as ^ik^- 1 ^ 5 6 "^)' (3-77) V dx and *™-mD$(-iF-i){^Sei*)> ( 3 - 78 ) V dx respectively, where Dv{z) is the parabolic cylinder function of order v [9] , and rS{x) . / rx 1 {F + G2)ida= / JS\ , jQ{x)dx. 79) Jxi The parameter F is chosen in such a way that it satisfies the equality (X2 \/Q(x)dx = [ \ Jx1 v JSi F + o2)l*do = -UTTF.

37) Let us consider the specific problem where a wave packet is trapped behind a barrier and then by tunneling it escapes to infinity [16] [17] [18]. suppose that a particle of mass m is initially confined to a segment 0 < x < a behind a rectangular potential of height VQ and width b — a. The wave function is obtained from the solution of the Schrodinger equation and is { • ^ y sin(pr) Ae^r + Be-^ Ce*pr + De-ipr for 0 < r < a for a < r < b , for & < (2-38) r where p2 < 2mVo and 7 = yj2mV0 -p2. 39) The normalization constant N(p) is chosen so as to satisfy the relation / JO rp(r)%>(r)dr = 5(p-p).

### Encyclopedia of Physics. The Non-Linear Field Theories of Mechanics by Flügge S. (ed.)

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