# Homework Help: Quantum tunneling of a car through a wall

1. Nov 29, 2012

### Lindsayyyy

Hi everyone

1. The problem statement, all variables and given/known data

I have to calculate the probablity that someone can tunnel a wall with a car. I have the mass of the car and the velocity and I know that the wall is 20 cm wide. But I have troubles estimating the potential of the wall.

2. Relevant equations

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3. The attempt at a solution

I don't know what's realistic here. Only thing I know is that the potential cant be infinte, because otherwise it would not be possible at all.

Can anyone help me out?

Thanks for the help

2. Nov 29, 2012

### Mute

What if the problem weren't about a car? What if it were about, say, electrons? Do you know how to do the problem for an electron (ignoring spin)? That is, what did you need to know to do regular problems about particles tunneling through a barrier, and what do you think you will need to calculate to apply the problem to a car?

3. Nov 29, 2012

### Lindsayyyy

I think I have to solve Schrödingers equation for the three areas in front of the wall, between it and beyond the wall. But that's all in general. The task says I shall estimate the value of the potential of the wall. But I have no idea how I can realistically estimate a potential of a wall.

Last edited: Nov 29, 2012
4. Nov 29, 2012

### Mute

Well, let's think about the particle case again. Say you have an incoming plane wave ($\psi(x) = \exp(ikx)$). What happens when it gets to the barrier? There are two possible scenarios depending on the energy of the incoming particle wave. In which one does tunneling occur?

Edit: Also, it might be helpful if you give us the full problem statement.

Last edited: Nov 29, 2012
5. Nov 29, 2012

### SteamKing

Staff Emeritus
So you think it is possible for a car to tunnel through a solid wall?

6. Nov 29, 2012

### Lindsayyyy

@ Mute I try to, but english is not my native language so the task might be a bit sloppy translated

@steamking, it's not like I made the task, it's a homework. I'm actually quite sure that the car won't tunnel through the wall ;)

task:

Someone tunneld with a Bugatti Veyron 16.4 Super Sport at maximum speed through a 20 cm wide wall.What's the probabilty? Estimate meaningful!

The probabilty is given by: $$P_T(E)=\frac{1}{{1}\ +\frac{V_0^2}{4E\left(V_0-E\right)}\sinh^2(2\kappa a)}$$

the potential is higher than the kinetic energy of the car.

7. Nov 29, 2012

### Mute

Yes, the potential energy of the wall has to be larger than the kinetic energy of the car+person. How much higher should it be? Well, you could try a few different estimates. It might be insightful to try two extreme values: a potential wall much higher than the kinetic energy of the car and one closer to the kinetic energy of the car. Which do you think is more realistic?

Once you've thought about that, we must then think about how to apply your transmission probability formula. How was that formula derived? i.e., what assumptions are you making by using that formula, and how do you apply them to this hypothetical car?

8. Nov 29, 2012

### Lindsayyyy

I was a bit worried about the 20 cm wall. I thought they gave this data for a reason like there is a mathematicl approach on how to estimate the value of the potential of the wall. I think I have no problem using the formula which is comming from the balance (what goes in comes out, reflection/transmission) as far as I know. Furthermore the potential should be much higher than the kinetic energy.

Thanks for the help

9. Nov 29, 2012

### Mute

Is the "$a$" in your transmission probabillity formula not the width of the barrier? I think this is the only place the width comes into play. Unless you were given the actual distance height of the barrier and maybe what kind of material it's made I'm not sure how you would estimate the potential energy of the barrier knowing only the width.

Last edited: Nov 29, 2012
10. Nov 30, 2012

### Lindsayyyy

ok, thank you very much. I'm very uncertain about this subject. I just try different values. You asked beforehand how much higher the potential must be. I have a question about that. Why does tunneling happen in the first place? I haven't found anything about that. My wild guess would be that it's connected to the uncertainty principle but my imagination about it is very vague.

11. Nov 30, 2012

### phinds

My understanding is that according to quantum mechanics, yes it is possible. Now "possible" can, and in this case does, simply mean that it has a non-zero probability of occurring. Keep in mind that 10E-10000000000000000, for example, is non-zero.

12. Nov 30, 2012

### Mute

Yes, one could argue that the uncertainty principle plays a role. In tunneling problems, we are not typically dealing with localized particles. Rather, the wave-function of the particle is distributed in space - the uncertainty principle basically puts limits on how much the particle's wave function (or 'wave packet' for a more physical sounding description) can be spread out in space compared to the packet's momentum.

Slight digression - have you studied fourier series and fourier transforms? I ask because the uncertainty principle pops up in fourier transforms. Basically, if you have a function of space, f(x), then the more concenrated the function is in space (i.e., the narrower the region for which f(x) is non-zero), the more spread out the fourier transform of the function is in frequency space (where the spatial frequency is basically the momentum). Or, the narrower your function of space, the more wave packets of different frequencies (momenta) it is composed of. Similarly, a wide spread of a wave packet in space is composed of only a few waves of different momenta. We can also view this the other way around: if a wave packet has a very narrow spread in momentum, then it can be decomposed into wave packets that have large spatial distributions.

What does this mean in physical terms? Well, it means that if we have a wave-packet of an electron, for example, that we are firing at the wall, it has a fairly well defined momentum (there is a slight spread in the possible momentum values). This means that the spatial distribution of the wave packet is quite wide - wide enough, even, to extend beyond the barrier in front of the electron wave packet. This means that there is a chance that we can find an electron on the other side of a barrier, even though classically we would never expect to see it there.

Now, why is the universe like this? Well, no one really knows that for sure. We've just figured out the rules and compared our guesses to experiments, and the experiments agree.

13. Nov 30, 2012

### Lindsayyyy

Well, we have no lecture about Fourier whatsoever. The physics professors sacrifice about 15 minutes to explain it. Actually this university here is a bit of a mess. Thanks for your help. I think it's a bit clearer now.

14. Nov 30, 2012

### Mute

Ok, well basically if you measure a particle's momentum, there is a large spread of possible positions it could have - including positions beyond the barrier, so if you perform a subsequent measurement it's possible to find the particle on the other side of the barrier. We interpret this result by saying that the particle "tunneled through" the barrier that classically it would have bounced off of.

15. Nov 30, 2012

### phinds

And I would add that while QM, as stated above, shows the possibility of a PARTICLE "tunneling", the odds of a whole macro object having ALL of its particles tunnel together and end up on the other side in the same order, while still not absolutely zero, it infinitesimal to say the least.

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