1. PH 401 Dr. Cecilia Vogel

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

3. Recall: Step barrier   particle with energy E>Vo incident from the left  Solutions to TISE: k1>k2 1< 2 sketch wavefunction

4. Step barrier reflection  R=[(k 1 -k 2 )/(k 1 +k 2 )] 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  A 1 +B 1 =A 2 +B 2  ik 1 A 1 - ik 1 B 1 = K 2 A 2 - K 2 B 2  A 2 e K2a +B 2 e -K2a = A 3 e ik1a  K 2 A 2 e K2a - K 2 B 2 e -K2a = ik 1 A 3 e ik1a

7. Tunneling probability  Tunneling into region 3:  T=|A3/A1| 2  T=[1+(V 2 /4E(V-E))sinh 2 (K 2 a)] -1   If K 2 a>>1, then sinh(K 2 a) approx e K2a  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. a)the particle is an electron, m = 0.5 MeV/c 2 b)the particle is an alpha, m=4Gev/c 2