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Tunneling Phenomena Potential Barriers

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

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

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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.

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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.

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Particle Scattering and Barrier Penetration Potential Barriers

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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

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

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The Square Barrier Using classical physics a particle with E U are transmitted with same energy. Therefore particles with E__
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The Square Barrier However according to QM there are no forbidden regions for a particle regardless of energy.

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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.

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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:

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Potential Barriers E>U

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

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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

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Potential Barriers (E>U) Quantum mechanically,

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Potential Barriers (E>U) Quantum mechanically,

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Potential Barriers (E>U) Quantum mechanically,

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

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

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

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

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

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

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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!

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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

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Potential Barriers (E>U) The wave equations represent a free particle of momentum p 1 and p 2 respectively.

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

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

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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

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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.

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Potential Barriers (E>U) Substituting into we get

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

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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

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

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

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

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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.

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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.

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

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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.

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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.

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

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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

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

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

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Potential Barriers (E>U) Where respectively.

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

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

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

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Potential Barriers E__
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Potential Barriers (E__
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Potential Barriers (E__0.
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Potential Barriers (E__0. Consider what happens quantum mechanically.
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Potential Barriers (E__
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Potential Barriers (E__
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Potential Barriers (E__
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Potential Barriers (E__
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Potential Barriers (E__0, the solution to SE is
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Potential Barriers (E__0, the solution to SE is where
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Potential Barriers (E__
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Potential Barriers (E__
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Potential Barriers (E__
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Potential Barriers (E__
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Potential Barriers (E__
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Potential Barriers (E__
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Potential Barriers (E__
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Potential Barriers (E__
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Potential Barriers (E__
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Potential Barriers (E__
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Potential Barriers (E__
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Potential Barriers (E__
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Potential Barriers (E__
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Potential Barriers (E__
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Potential Barriers (E__
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Potential Barriers (E__
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Potential Barriers (E__
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Potential Barriers (E__
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Potential Barriers (E__
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Potential Barriers (E__
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Potential Barriers (E__
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Potential Barriers (E__
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Potential Barriers (E__
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Potential Barriers (E__
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Potential Barriers (E__
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Potential Barriers (E__
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Potential Barriers (E__
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Potential Barriers (E__
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Square Barrier General Case

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Potential Barriers (E__
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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

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

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

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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

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Wide Barriers The transmission curve is shown on the handout.

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

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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

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Barrier Penetration (Applications) We briefly at one case…

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Barrier Penetration (Applications) Ammonia inversion

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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.

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Barrier Penetration (Applications) Ammonia inversion

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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.

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

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Barrier Penetration (Applications) Ammonia inversion

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