1. PH 401 Dr. Cecilia Vogel

2. Review • bound vs unbound states • CA vs CF regions • Stationary States for barriers • step barrier • tunneling barrier • unbound state wavefunctions • tunneling probaility Outline

3. Recall: Step barrier • particle with energy E>Vo incident from the left • Solutions to TISE: k1>k2 l1<l2 sketch wavefunction

4. Step barrier reflection • R=[(k1-k2)/(k1+k2)]2 • R=[(sqE-sq(E-V))/(sqE+sq(E-V))]2 • R is not zero. The particle might be REFLECTED! By a CA barrier!! What??

5. Recall: Tunneling • particle with energy E<Vo incident from the left • Solutions to TISE: sketch wavefunction

6. Tunneling continuity • A1+B1=A2+B2 • ik1A1 - ik1B1 = K2A2 - K2B2 • A2eK2a +B2e-K2a = A3eik1a • K2A2eK2a - K2B2e-K2a = ik1A3eik1a

7. Tunneling probability • Tunneling into region 3: • T=|A3/A1|2 • T=[1+(V2/4E(V-E))sinh2(K2a)]-1 • If K2a>>1, then sinh(K2a) approx eK2a • T is not zero. The particle might be TUNNEL! through a CF barrier!! What??

8. Tunneling probability • Tunneling probability depends on: • particle mass – higher mass, less tunneling • particle energy – higher energy, more tunneling • barrier potential energy – higher barrier, less tunneling • thickness of barrier – thicker barrier, less tunelling

9. PAL • Find the probability for a particle with energy 10 eV tunneling through a 400-eV barrier that is 1 nm wide. • the particle is an electron, m = 0.5 MeV/c2 • the particle is an alpha, m=4Gev/c2