Tunneling Phenomena

1. Tunneling Phenomena Potential Barriers

2. Tunneling • Unlike attractive potentials which traps particle, barriers repel them.

3. Tunneling • Unlike attractive potentials which traps particle, barriers repel them. • Hence we look at determining whether the incident particle is reflected or transmitted.

4. Tunneling • Unlike attractive potentials which traps particle, barriers repel them. • Hence we look at determining whether the incident particle is reflected or transmitted. • Tunneling is a purely QM effect.

5. Tunneling • Unlike attractive potentials which traps particle, barriers repel them. • Hence we look at determining whether the incident particle is reflected or transmitted. • Tunneling is a purely QM effect. It is used in field emission, radioactive decay, the scanning tunneling microscope etc.

6. Particle Scattering and Barrier Penetration Potential Barriers

7. The Square Barrier • A square barrier is represented by a potential energy U(x) in the barrier region (between x=0 and x=L). U 0 L

8. The Square Barrier • Using classical physics a particle with E<U are reflected while those with E>U are transmitted with same energy.

9. The Square Barrier • Using classical physics a particle with E<U are reflected while those with E>U are transmitted with same energy. • Therefore particles with E<U are restricted to one of the barrier.

10. The Square Barrier • However according to QM there are no forbidden regions for a particle regardless of energy.

11. The Square Barrier • However according to QM there are no forbidden regions for a particle regardless of energy. • This is because the associated matter wave is nonzero everywhere.

12. The Square Barrier • However according to QM there are no forbidden regions for a particle regardless of energy. • This is because the associated matter wave is nonzero everywhere. • A typical waveform is shown:

14. Potential Barriers E>U

15. Potential Barriers (E>U) • Consider the step potential below. E U(x)=U0 U=0 x=0

16. Potential Barriers (E>U) • Consider the step potential below. • Classical mechanics predicts that the particle is not reflected at x=0. E U(x)=U0 U=0 x=0

17. Potential Barriers (E>U) • Quantum mechanically,

18. Potential Barriers (E>U) • Quantum mechanically,

19. Potential Barriers (E>U) • Quantum mechanically,

20. Potential Barriers (E>U) • Considering the first equation,

21. Potential Barriers (E>U) • Considering the first equation, • The general solution is

22. Potential Barriers (E>U) • Considering the first equation, • The general solution is • where

23. Potential Barriers (E>U) • For the 2nd equation,

24. Potential Barriers (E>U) • For the 2nd equation, • The general solution is

25. Potential Barriers (E>U) • For the 2nd equation, • The general solution is • where

26. Potential Barriers (E>U) • However D=0 since there is no reflection as there is only a transmitted wave for x>0. We have nothing to cause reflection!

27. Potential Barriers (E>U) • However D=0 since there is no reflection as there is only a transmitted wave for x>0. We have nothing to cause reflection! • Therefore

28. Potential Barriers (E>U) • The wave equations represent a free particle of momentum p1 and p2 respectively.

29. Potential Barriers (E>U) • The behaviour is shown in the diagram below. C A U(x)=U0 B U=0 x=0

30. Potential Barriers (E>U) • The constants A, B and C must be chosen to make and continuous at x=0.

31. Potential Barriers (E>U) • The constants A, B and C must be chosen to make and continuous at x=0. • Satisfying the 1st condition we get that

32. Potential Barriers (E>U) • The constants A, B and C must be chosen to make and continuous at x=0. • Satisfying the 1st condition we get that • To satisfy the 2nd requirement, we differentiate.

33. Potential Barriers (E>U) • Substituting into we get

34. Potential Barriers (E>U) • Substituting into we get • Writing A in terms of B we get after some algebra that

35. Potential Barriers (E>U) • Substituting into we get • Writing B in terms of A we get after some algebra that • Similarly, writing C in terms of A

36. Potential Barriers (E>U) • Substituting these expressions into we have

37. Potential Barriers (E>U) • Substituting these expressions into we have • As usual we normalize to find A.

38. Potential Barriers (E>U) • The probability that the particle is reflected is given by the Reflection coefficient R.

39. Potential Barriers (E>U) • The probability that the particle is reflected is given by the Reflection coefficient R. The ratio of intensities of the reflected to incident.

40. Potential Barriers (E>U) • The probability that the particle is reflected is given by the Reflection coefficient R. The ratio of intensities of the reflected to incident.

41. Potential Barriers (E>U) • The probability that the particle is transmitted is given by the Transmission coefficient T.

42. Potential Barriers (E>U) • The probability that the particle is reflected is given by the Reflection coefficient R. The ratio of intensities of the transmitted to incident.

43. Potential Barriers (E>U) • The probability that the particle is reflected is given by the Reflection coefficient R. The ratio of intensities of the transmitted to incident.

44. Potential Barriers (E>U) • It is easy to show that

45. Potential Barriers (E>U) • A similar case to the previous example is given below A B C U(x)=U0 U=0 x=0

46. Potential Barriers (E>U) • Applying the same logic as the previous example we can show that

47. Potential Barriers (E>U) • Applying the same logic as the previous example we can show that • The solution to these equations are

48. Potential Barriers (E>U) • Where • respectively.

49. Potential Barriers (E>U) • Applying the conditions of continuity we get:

50. Potential Barriers (E>U) • Applying the conditions of continuity we get: • and