Explain Tunneling Paradox: Quantum Particle & V_0 Barrier

[student1938]

Consider a Quantum Mechanical particle approaching a barrier (potential) of height $$V_0$$ 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

 

[Tide]

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 $1/{\sqrt {2}}$ of the incident wave.

 

[student1938]

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

 

[Tide]

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.

 

[student1938]

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

 

[Tide]

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.

 

[student1938]

So something like this where the amplitude of the transmitted wave is reduced...its intensity is reduced.

 

[Tide]

That looks good!

 

[student1938]

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

 

[Tide]

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.

 

[student1938]

So it is the probability density right? Cause that is what I want. If it isn't then how different would it be?

student