Quantum Theory - Particle Physics Group

CHAPTER 2. WAVE MECHANICS AND THE SCHRÖDINGER EQUATION 30Using (e 2iKL − 1)(e −2iKL − 1) = 2 − e 2iKL − e −2iKL = 2(1 − cos 2Kl) = 4 sin 2 KL we determinethe reflection coefficientR = |B|2|A| 2 = [1 +and the transmission coefficient]4k 2 K 2−1 [=(k 2 − K 2 ) 2 sin 2 (KL)T = |C|2|A| 2 = [1 + (k2 − K 2 ) 2 sin 2 (KL)4k 2 K 2 ] −1=1 + 4E(E − V 0)V 20 sin 2 (KL)[1 + V 20 sin 2 (KL)4E(E − V 0 )] −1(2.88)] −1. (2.89)In general the transmission coefficient T is less than 1, in contrast to classical mechanics, wherethe particle would always be transmitted. There are two cases with perfect transmission T = 1:The first is of course when V 0 = 0 and the second is a resonance phenomenon that occurs whenKL = nπ for n = 1, 2,..., i.e. when sin KL = 0 so that the length L of the interactionregion is a half-integral multiple of the wavelength of the electrons. Conservation of probabilityR + T = 1 holds since11+X + 11+1/X = 1.As we mentioned above the case of a high barrier V 0 > E is related to the formulas forE > V 0 by analytic continuation K = iκ. For the ratios B/A and C/A we hence obtainBA = (k 2 + κ 2 )(e 2κL − 1)e 2κL (k + iκ) 2 − (k − iκ) 2, (2.90)CA =4ikκe −ikL e κLe 2κL (k + iκ) 2 − (k − iκ) 2, (2.91)which leads to the reflection and transmission coefficients[]R = |B|2|A| = 4k 2 κ 2−1 [1 += 1 + 4E(V ] −10 − E), (2.92)2 (k 2 + κ 2 ) 2 sinh 2 (κL) V0 2 sinh 2 (κL)T = |C|2|A| 2 = [1 + (k2 + κ 2 ) 2 sinh 2 (κL)4k 2 κ 2 ] −1=[1 + V 20 sinh 2 (κL)4E(V 0 − E)] −1. (2.93)For E < V 0 neither perfect transmission nor perfect reflection is possible. For large L thetransmission probability falls off exponentiallyT −→ 16E(V 0 − E)V 20e −2κL for L ≫ 1/κ. (2.94)The phenomenon that a particle has a positive probability to penetrate a classically forbiddenpotential barrier is called tunneling effect.2.2.4 Transfer matrix and scattering matrixThe wave functions u i (x) = A i e ik ix +B i e −ik ix in domains of constant potential are parametrizedby the two amplitudes A i and B i . The effect of an interaction region can therefore be regarded