What is the best estimate for the penetration distance in quantum mechanics?

In summary, the conversation discusses a particle of energy E crossing a potential jump at x=0, with a wavefunction of exp(-iEt/hbar)*exp(-Kx) for x>=0. The penetration distance is estimated to be 1/kappa based on its relevance to the problem's length scale. The probability distribution also depends on this length scale, and the chosen penetration distance can vary depending on the desired threshold of probability.
  • #1
sachi
75
1
We have a particle of energy E crossing a potential jump at x=0. for x<=0, V=0, for x>=0 V=V1
We get a wavefunction for x>=0 psi(x) = exp(-iEt/hbar)*exp(-Kx)
where K = (2m(V1-E))^0.5/hbar
N.b E<V1 so classically we get no transmission

we are asked to estimate the penetration distance, and I have found a solution which says let the penetration distance equal 1/K. I can't see physically why we would pick this (it just seems like a random number that means that the wavefunction will decrease by a factor 1/e, but I can't see why this is a sensible estimate).

Thanks
 
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  • #2
Well, [tex] 1/\kappa [/tex] is the only relevant length scale in the problem, so the penetration depth has to be proportional to it. The point is that if you were to plot the wavefunction as a function of [tex] \kappa x [/tex], it would look the same no matter what the energy or barrier height were. In other words, if [tex] \kappa x < .1 [/tex] nothing much happens and if [tex] \kappa x > 10 [/tex] the wavefunction is essentially zero. Clearly, [tex] \kappa [/tex] determines the length scale over which the action happens. That being said, you have some freedom in your estimate. Maybe you would like to include a factor of [tex] \ln{2} [/tex] so you get the place where the wavefunction is one half its value at the boundary. In some other problem, it might be nice to include a factor of [tex] \pi [/tex] for convenience, for example. The convention is basically that anything within a power of ten of [tex] \kappa [/tex] is pretty much ok, but this isn't any kind of formal rule and people tend to go with the simple estimate. All you are really doing is identifying the length scale.
 
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  • #3
Consider what the probability distribution looks like.

[tex]\Psi = e^{\frac{-iE}{\hbar}t}e^{-\kappa x}[/tex]
[tex]P=\Psi^* \Psi = e^{-2\kappa x}[/tex]

Remember that the second part is completely real since V1>E.

As Physics Monkey said, this is a question of length scale. No matter what threshold of probability you choose (say, .1%), you must scale it must be a constant times 1/kappa because kappa can vary depending on what problem you're doing.

Let [tex]x=\frac{d}{\kappa}[/tex] be your chosen penetration distance, where d is just a constant.
[tex]P= e^{-2\kappa x}= e^{-2\kappa \frac{d}{\kappa}} = e^{-2d}[/tex]

Then choose d according to how close you want the probability to be zero. Since this doesn't depend on kappa, it won't matter what E and V1 are in your problem.
 
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Related to What is the best estimate for the penetration distance in quantum mechanics?

What is barrier penetration in quantum mechanics?

Barrier penetration refers to the phenomenon in which a particle with insufficient energy to overcome a potential barrier is still able to pass through it. This is possible due to the probabilistic nature of quantum mechanics, which allows for the particle to exist in a state of superposition and tunnel through the barrier.

What is the significance of barrier penetration in quantum mechanics?

Barrier penetration is a fundamental concept in quantum mechanics and is essential in understanding many phenomena, such as alpha decay and nuclear fusion. It also plays a crucial role in the development of new technologies, such as tunneling transistors and scanning tunneling microscopes.

How is barrier penetration calculated in quantum mechanics?

The calculation of barrier penetration involves solving the Schrödinger equation for the potential barrier that the particle encounters. This solution yields a probability amplitude, which can then be used to calculate the probability of the particle passing through the barrier. The probability is dependent on the energy of the particle and the height and width of the barrier.

What is the difference between classical and quantum barrier penetration?

In classical physics, a particle with insufficient energy to overcome a barrier would be completely reflected, while in quantum mechanics, there is a non-zero probability of the particle passing through the barrier. This is because in quantum mechanics, particles can exist in a state of superposition and have a probability of being found on the other side of the barrier.

Can barrier penetration violate the laws of thermodynamics?

No, barrier penetration does not violate the laws of thermodynamics. While it may seem like particles are passing through a barrier without sufficient energy, the total energy of the system is still conserved. Additionally, the probability of barrier penetration decreases exponentially with increasing barrier height, making it a rare occurrence.

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