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Tunneling Phenomena Potential Barriers. Tunneling Unlike attractive potentials which traps particle, barriers repel them.

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Presentation on theme: "Tunneling Phenomena Potential Barriers. Tunneling Unlike attractive potentials which traps particle, barriers repel them."— Presentation transcript:

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 L0

8 The Square Barrier Using classical physics a particle with E U are transmitted with same energy.

9 The Square Barrier Using classical physics a particle with E U are transmitted with same energy. Therefore particles with E { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/3913311/13/slides/slide_8.jpg", "name": "The Square Barrier Using classical physics a particle with E U are transmitted with same energy.", "description": "Therefore particles with E

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:

13

14 Potential Barriers E>U

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

16 Potential Barriers (E>U) Consider the step potential below. Classical mechanics predicts that the particle is not reflected at x=0. U(x)=U 0 U=0 E 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 2 nd equation,

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

25 Potential Barriers (E>U) For the 2 nd 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 p 1 and p 2 respectively.

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

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 1 st 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 1 st condition we get that To satisfy the 2 nd 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 U(x)=U 0 U=0 x=0 A B C

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

51 Potential Barriers (E>U) As an exercise show that the transmission and reflection coefficients are the same.

52 Potential Barriers E { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/3913311/13/slides/slide_51.jpg", "name": "Potential Barriers E

53 Potential Barriers (E { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/3913311/13/slides/slide_52.jpg", "name": "Potential Barriers (E

54 Potential Barriers (E0.

55 Potential Barriers (E0. Consider what happens quantum mechanically.

56 Potential Barriers (E { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/3913311/13/slides/slide_55.jpg", "name": "Potential Barriers (E

57 Potential Barriers (E { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/3913311/13/slides/slide_56.jpg", "name": "Potential Barriers (E

58 Potential Barriers (E { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/3913311/13/slides/slide_57.jpg", "name": "Potential Barriers (E

59 Potential Barriers (E { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/3913311/13/slides/slide_58.jpg", "name": "Potential Barriers (E

60 Potential Barriers (E0, the solution to SE is

61 Potential Barriers (E0, the solution to SE is where

62 Potential Barriers (E { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/3913311/13/slides/slide_61.jpg", "name": "Potential Barriers (E

63 Potential Barriers (E { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/3913311/13/slides/slide_62.jpg", "name": "Potential Barriers (E

64 Potential Barriers (E { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/3913311/13/slides/slide_63.jpg", "name": "Potential Barriers (E

65 Potential Barriers (E { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/3913311/13/slides/slide_64.jpg", "name": "Potential Barriers (E

66 Potential Barriers (E { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/3913311/13/slides/slide_65.jpg", "name": "Potential Barriers (E

67 Potential Barriers (E { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/3913311/13/slides/slide_66.jpg", "name": "Potential Barriers (E

68 Potential Barriers (E { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/3913311/13/slides/slide_67.jpg", "name": "Potential Barriers (E

69 Potential Barriers (E { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/3913311/13/slides/slide_68.jpg", "name": "Potential Barriers (E

70 Potential Barriers (E { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/3913311/13/slides/slide_69.jpg", "name": "Potential Barriers (E

71 Potential Barriers (E { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/3913311/13/slides/slide_70.jpg", "name": "Potential Barriers (E

72 Potential Barriers (E { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/3913311/13/slides/slide_71.jpg", "name": "Potential Barriers (E

73 Potential Barriers (E { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/3913311/13/slides/slide_72.jpg", "name": "Potential Barriers (E

74 Potential Barriers (E { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/3913311/13/slides/slide_73.jpg", "name": "Potential Barriers (E

75 Potential Barriers (E { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/3913311/13/slides/slide_74.jpg", "name": "Potential Barriers (E

76 Potential Barriers (E { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/3913311/13/slides/slide_75.jpg", "name": "Potential Barriers (E

77 Potential Barriers (E { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/3913311/13/slides/slide_76.jpg", "name": "Potential Barriers (E

78 Potential Barriers (E { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/3913311/13/slides/slide_77.jpg", "name": "Potential Barriers (E

79 Potential Barriers (E { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/3913311/13/slides/slide_78.jpg", "name": "Potential Barriers (E

80 Potential Barriers (E { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/3913311/13/slides/slide_79.jpg", "name": "Potential Barriers (E

81 Potential Barriers (E { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/3913311/13/slides/slide_80.jpg", "name": "Potential Barriers (E

82 Potential Barriers (E { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/3913311/13/slides/slide_81.jpg", "name": "Potential Barriers (E

83 Potential Barriers (E { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/3913311/13/slides/slide_82.jpg", "name": "Potential Barriers (E

84 Potential Barriers (E { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/3913311/13/slides/slide_83.jpg", "name": "Potential Barriers (E

85 Potential Barriers (E { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/3913311/13/slides/slide_84.jpg", "name": "Potential Barriers (E

86 Potential Barriers (E { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/3913311/13/slides/slide_85.jpg", "name": "Potential Barriers (E

87 Potential Barriers (E { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/3913311/13/slides/slide_86.jpg", "name": "Potential Barriers (E

88 Potential Barriers (E { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/3913311/13/slides/slide_87.jpg", "name": "Potential Barriers (E

89 Potential Barriers (E { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/3913311/13/slides/slide_88.jpg", "name": "Potential Barriers (E

90 Square Barrier General Case

91 Potential Barriers (E { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/3913311/13/slides/slide_90.jpg", "name": "Potential Barriers (E

92 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 L0

93

94 Potential Barriers (E { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/3913311/13/slides/slide_93.jpg", "name": "Potential Barriers (E

95 Potential Barriers (E { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/3913311/13/slides/slide_94.jpg", "name": "Potential Barriers (E

96 The Square Barrier For the general case the stationary wave may be decomposed into incident, reflected and transmitted waves. L0

97 Potential Barriers (E { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/3913311/13/slides/slide_96.jpg", "name": "Potential Barriers (E

98 Potential Barriers (E { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/3913311/13/slides/slide_97.jpg", "name": "Potential Barriers (E

99 Potential Barriers (E { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/3913311/13/slides/slide_98.jpg", "name": "Potential Barriers (E

100 Potential Barriers (E { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/3913311/13/slides/slide_99.jpg", "name": "Potential Barriers (E

101 Potential Barriers (E { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/3913311/13/slides/slide_100.jpg", "name": "Potential Barriers (E

102 Potential Barriers (E { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/3913311/13/slides/slide_101.jpg", "name": "Potential Barriers (E

103 Potential Barriers (E { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/3913311/13/slides/slide_102.jpg", "name": "Potential Barriers (E

104 Potential Barriers (E { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/3913311/13/slides/slide_103.jpg", "name": "Potential Barriers (E

105 Potential Barriers (E { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/3913311/13/slides/slide_104.jpg", "name": "Potential Barriers (E

106 Potential Barriers (E { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/3913311/13/slides/slide_105.jpg", "name": "Potential Barriers (E

107 Wide Barriers The solution to the previous set of equations are tedious and generally too complicated to be useful.

108 Wide Barriers The solution to the previous set of equations are tedious and generally too complicated to be useful. A simplification is possible where In these cases the wide barrier approximates to the infinitely wide barrier. This leads to the approximation that

109 Wide Barriers The transmission curve is shown on the handout.

110 Barrier Penetration (Applications) We consider a few practical applications of potential barriers.

111 Barrier Penetration (Applications) We consider a few practical applications of potential barriers. Some interesting cases are: a. In field emissions b. Alpha decay c. Josephson Junction d. Ammonia inversion e. Decay of Black Holes

112 Barrier Penetration (Applications) We briefly at one case…

113 Barrier Penetration (Applications) Ammonia inversion

114 Barrier Penetration (Applications) The inversion of ammonia is an example of tunneling. There are two configurations with the same energy. As shown, the two configurations are on either side of the hydrogen plane.

115 Barrier Penetration (Applications) Ammonia inversion

116 Barrier Penetration (Applications) The inversion of ammonia is an example of tunneling. There are two configurations with the same energy. As shown, the two configurations are on either side of the hydrogen plane. The repulsive Coulomb force creates a barrier.

117 Barrier Penetration (Applications) The nitrogen must overcome this force to get from one configuration to the other.

118 Barrier Penetration (Applications) Ammonia inversion


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