Quantum Theory - Particle Physics Group

CHAPTER 2. WAVE MECHANICS AND THE SCHRÖDINGER EQUATION 28✻V(x)V 0✛✎☞✍✌ I✎☞0 ✍✌ IIL✎☞✍✌ III✲x❄Figure 2.5: Potential barrierdescribe right-moving and left-moving electron rays with velocities v = ±k/m, respectively.Indeed, inserting into the formula (2.14) for the probability current density we findj right = k m |A|2 and j left = − k m |B|2 . (2.72)As a concrete example we again consider the square potential. For V 0 > 0 we have a potentialbarrier and for V 0 < 0 a potential well. Classically all electrons would be transmitted as longas E > V 0 and all electrons would be reflected by the potential barrier if E < V 0 . Quantummechanically we generically expect to find a combination of reflection and transmission, likein optics. For a high barrier V 0 > E we will find an exponentially suppressed but non-zeroprobability for electrons to be able to penetrate the classically forbidden region, which is calledtunneling effect. Our ansatz for the stationary wave function in the potential of figure 2.5 isu I = Ae ikx + Be −ikx for x < 0 with k =⎧⎪⎨ Fe −κx + Ge +κx for E < V 0 with κ =u II =⎪⎩ Fe iKx + Ge −iKx for E > V 0 with K =√2mE 2 , (2.73)√2m(V 0 −E), 2√2m(E−V 0 ) 2= iκ,(2.74)u III = Ce ikx + De −ikx for x > L. (2.75)Since for tunneling E < V 0 and for the case E > V 0 the ansätze in the interaction region II aswell as the resulting continuity equations are formally related by K = iκ, both cases can betreated in a single calculation. Moreover, the ansatz for E > V 0 covers scattering at a potentialbarrier V 0 > 0 as well as the scattering at a potential well V 0 < 0.Considering the physical situation of an electron source at x ≪ 0 and detectors measuringthe reflected and the transmitted particles we observe that A is the amplitude for the incomingray, B is the amplitude for reflection, C is the amplitude for transmission and we have to set