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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 L

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

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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.

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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. E U(x)=U0 U=0 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. E U(x)=U0 U=0 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 2nd equation,

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**Potential Barriers (E>U)**

For the 2nd equation, The general solution is

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**Potential Barriers (E>U)**

For the 2nd 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 p1 and p2 respectively.

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**Potential Barriers (E>U)**

The behaviour is shown in the diagram below. C A U(x)=U0 B U=0 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.

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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 1st 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 1st condition we get that To satisfy the 2nd 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 A B C U(x)=U0 U=0 x=0

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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<U

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**Potential Barriers (E<U)**

Consider a stream of particles incident to barrier below. Total energy E is constant. U(x)=U0 E U=0 x=0

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**Potential Barriers (E<U)**

Classically a particle can not enter the region x>0.

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**Potential Barriers (E<U)**

Classically a particle can not enter the region x>0. Consider what happens quantum mechanically.

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**Potential Barriers (E<U)**

The time independent Schrödinger's equation for the regions are:

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**Potential Barriers (E<U)**

The time independent Schrödinger's equation for the regions are:

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**Potential Barriers (E<U)**

Again in the region x<0, the solution to SE is

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**Potential Barriers (E<U)**

Again in the region x<0, the solution to SE is where

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**Potential Barriers (E<U)**

For the region x>0, the solution to SE is

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**Potential Barriers (E<U)**

For the region x>0, the solution to SE is where

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**Potential Barriers (E<U)**

Choose A, B, C and D so that conditions are satisfied.

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**Potential Barriers (E<U)**

Choose A, B, C and D so that conditions are satisfied. 1st note that C=0 otherwise the TISE diverges as

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**Potential Barriers (E<U)**

Choose A, B, C and D so that conditions are satisfied. 1st note that C=0 otherwise the TISE diverges as Considering continuity conditions at x=0 leads to:

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**Potential Barriers (E<U)**

Writing the constants A and B in terms of D we get:

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**Potential Barriers (E<U)**

Writing the constants A and B in terms of D we get:

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**Potential Barriers (E<U)**

Therefore the waveform for each region can be rewritten in terms of one constant. And we get,

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**Potential Barriers (E<U)**

Therefore the waveform for each region can be rewritten in terms of one constant. And we get, where we can normalize to find D.

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**Potential Barriers (E<U)**

The ratio of the reflected probability flux to the incident probability flux gives

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**Potential Barriers (E<U)**

The ratio of the reflected probability flux to the incident probability flux gives

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**Potential Barriers (E<U)**

The ratio of the reflected probability flux to the incident probability flux gives

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**Potential Barriers (E<U)**

The ratio of the reflected probability flux to the incident probability flux gives

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**Potential Barriers (E<U)**

The ratio of the reflected probability flux to the incident probability flux gives

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**Potential Barriers (E<U)**

That is the probability of being reflected is unity.

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**Potential Barriers (E<U)**

That is the probability of being reflected is unity. These results are in agreement with classical physics.

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**Potential Barriers (E<U)**

Step potential

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**Potential Barriers (E<U)**

U(x)=U0 B C U=0 x=0

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**Potential Barriers (E<U)**

The TISEs are

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**Potential Barriers (E<U)**

The TISEs are

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**Potential Barriers (E<U)**

The TISEs are The solution to these equations are

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**Potential Barriers (E<U)**

Choose A, B, C and D so that conditions are satisfied.

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**Potential Barriers (E<U)**

Choose A, B, C and D so that conditions are satisfied. In this case D=0 otherwise the TISE diverges as

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**Potential Barriers (E<U)**

Choose A, B, C and D so that conditions are satisfied. In this case D=0 otherwise the TISE diverges as Considering continuity conditions at x=0 leads to:

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**Potential Barriers (E<U)**

Eliminating B we get

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**Potential Barriers (E<U)**

Eliminating B we get

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**Potential Barriers (E<U)**

Eliminating B we get Dividing the 1st equation by B we get

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**Potential Barriers (E<U)**

Eliminating B we get Dividing the 1st equation by B we get

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**Potential Barriers (E<U)**

Therefore we calculated the reflection coefficient.

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**Potential Barriers (E<U)**

Therefore we calculated the reflection coefficient. Therefore an infinitely wide barrier reflects all incident particles!

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Square Barrier General Case

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**Potential Barriers (E<U)**

Recall:

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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 L

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**Potential Barriers (E<U)**

The infinitely wide step potential is a special case of the square barrier where the width L is infinite.

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**Potential Barriers (E<U)**

The infinitely wide step potential is a special case of the square barrier where the width L is infinite. To complete our look at various potential barriers we summarize a few ideas using the square barrier.

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

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**Potential Barriers (E<U)**

On the LHS of the barrier, the particle is free. The waveform for free particles is The part of the wavefunction A is interpreted as a wave incident on the barrier. B is the wave reflected. The squared-amplitude of intensity of the reflected wave relative to the incident is

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**Potential Barriers (E<U)**

In wave terminology, R is the wave intensity. For particles, R is the probability that that the particle incident on the barrier is reflected.

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**Potential Barriers (E<U)**

On the RHS of the barrier, the particle is free. The waveform for the free particle The part of the wavefunction F is interpreted as wave traveling to the right. G has no interpretation. The squared-amplitude of intensity of the transmitted relative to the incident is

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**Potential Barriers (E<U)**

The transmission coefficient measures the likelihood that a particle incident on the barrier penetrates and emerges on the other side.

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**Potential Barriers (E<U)**

The sum of the two probabilities (the probability of transmission and the probability of reflection) is one.

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**Potential Barriers (E<U)**

The region of the barrier (0<x<L) where U=U0 the TISE can be written in the form, With E<U, the expression in the braces is a positive constant. The solution to which are in the real exp form.

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**Potential Barriers (E<U)**

The parameter is the penetration depth. The distance in the barrier where the waveform and hence the probability of finding the particle remains appreciable.

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**Potential Barriers (E<U)**

Therefore the waveform in the barrier is

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**Potential Barriers (E<U)**

The arbitrary constants are determined by applying the continuity conditions at the boundaries x=0 and x=L.

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**Potential Barriers (E<U)**

Applying these conditions gives the following equations, Where terms must be written as ratios. eg

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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: In field emissions Alpha decay Josephson Junction Ammonia inversion 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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