Explain Tunneling Paradox: Quantum Particle & V_0 Barrier

In summary, when considering a quantum mechanical particle approaching a barrier of height V_0 and width a, with a 50% chance of reflection and a 50% chance of transmission, the sketch of the probability density would show the transmitted and reflected waves with reduced amplitudes, each being 1/2 of the incident wave's amplitude. However, if the particle's energy is less than the barrier height, the wave inside the barrier would decay exponentially before emerging as an oscillatory wave on the other side. This is known as an "evanescent wave." The probability density is obtained by multiplying the wave function by its complex conjugate.
  • #1
student1938
91
1
Consider a Quantum Mechanical particle approaching a barrier (potential) of height [tex]V_0[/tex] and width a. What will the sketch of the probability density look like if there is a 50% chance of reflection and a 50% chance of transmission? Can you explain why cause after reading Griffith' s Quantum Mechanics book I am very confused about the above case.

Regards,
student
 
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  • #2
I don't see the paradox but you do know that the reflected and transmitted waves have the same amplitude and each amplitude will be [itex]1/{\sqrt {2}}[/itex] of the incident wave.
 
  • #3
So would the probability density look like this? The transmitted wave and reflected waves have a reduced amplitude...i.e. they are 1/2 of the original amplitude (incident amplitude). This is the plot of the probability density. Does it make sense?

Looking forward to hearing from you soon.

student
 

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  • #4
That looks good! You might want to make two sketches. The one you did is appropriate if the particle's energy is greater than the barrier height. However, if the particle's energy is less than the height of the barrier then inside the barrier the wave decays exponentially (rather than being oscillatory in space). Such a wave is referred to as an "evanescent wave." Of course the wave emerges from the other side of the barrier as an ordinary oscillatory wave.
 
  • #5
Like this?

Also, just wana clarify something, I know I am probably being pedantic but anyways...Is 50 % transmission is the same as R = T = 0.5? Cause the question asks for the probability density sketches for 50% transmission.

I think 50% transmission is the same as R = T = 0.5

Thanks,
student
 

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  • #6
Not quite - when the energy is less than the barrier height what's left of the wave (transmitted) emerges as an oscillatory wave! And, yes, R = T = 0.5 is correct.
 
  • #7
So something like this where the amplitude of the transmitted wave is reduced...its intensity is reduced.
 

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  • #8
That looks good!
 
  • #9
Can you explain to me again, why what I have drawn is not the wavefunction but the probability density? I think it might be the wavefunction...I am still confused. I think it should be the continuation of exponential decay when the wave emerges from the barrier for E < V and for E > V, I think that it should be a sine wave prior to hitting the barrier and then a straight line inside the barrier and when it emerges.

student
 
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  • #10
Outside the barrier, the wave function is just the free space wave function so it must be oscillatory. Within the barrier, the wavenumber (k) is imaginary which gives the exponentially decaying solution.

You get the probability density by multiplying the wave function by its complex conjugate.
 
  • #11
So it is the probability density right? Cause that is what I want. If it isn't then how different would it be?

student
 

1. What is the tunneling paradox in quantum mechanics?

The tunneling paradox refers to the phenomenon observed in quantum mechanics where a particle is able to pass through a potential barrier, such as a wall or barrier, even though it does not have enough energy to overcome the barrier. This is in contrast to classical mechanics, where a particle would be unable to pass through the barrier unless it had enough energy to do so.

2. How does tunneling occur in quantum mechanics?

In quantum mechanics, particles are described by wave functions, which represent the probability of finding the particle at a certain position. When a particle encounters a potential barrier, there is a small probability that the particle's wave function will extend into the barrier, allowing it to "tunnel" through and appear on the other side.

3. What is the role of the V0 barrier in tunneling?

The V0 barrier, also known as the potential barrier, is a representation of the energy barrier that a particle must overcome in order to pass through it. In the tunneling paradox, the V0 barrier is lower than the energy of the particle, yet the particle is still able to pass through it due to the probabilistic nature of quantum mechanics.

4. What are the implications of the tunneling paradox in quantum mechanics?

The tunneling paradox challenges our understanding of the classical laws of physics, as it allows particles to pass through barriers that they would not be able to according to classical mechanics. This has implications for various fields, such as electronics and nuclear physics, where tunneling plays a crucial role in understanding and predicting the behavior of particles.

5. How is the tunneling paradox resolved in quantum mechanics?

The tunneling paradox is resolved in quantum mechanics by accepting the probabilistic nature of particles and their wave functions. In other words, particles do not have a definite position or energy, but rather exist as a wave of probabilities. This allows for the phenomenon of tunneling to occur, as particles can exist in multiple states simultaneously and have a small chance of passing through a potential barrier.

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