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Classically a particle cant exist in a region where V>E

  1. Jan 25, 2010 #1
    classically a particle cant exist in a region where V>E, v=pot energy, E=total energy, but quantum mechanically its possible. We can explain it(or draw an analogy) with concepts of optics,when light falls from a denser medium to rarer medium.but that wave is a real one,the quantum mechanical wave is a probabilistic one which relates to finding of particle.
    If particle can't be found how can its probability wave be found?

    thanks for any answer
     
  2. jcsd
  3. Jan 25, 2010 #2
    Re: tunneling

    Wherever the wave function is different from zero, there is some probability the particle can be found there. So it can be found in a classically forbidden zone. One evidence of this behavior is the existence of transistors where electrons can be found beyond a potential barrier.
     
  4. Jan 25, 2010 #3

    SpectraCat

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    Re: tunneling


    Even though there is a finite probability density inside the classically forbidden region (CFR), it can be shown that the particle cannot ever be measured inside that region. Since the momentum is imaginary inside the classical region, that would correspond to a negative kinetic energy, which is nonsensical. What saves this is the uncertainty principle, using which one can demonstrate that the uncertainty in position is always larger than the penetration depth at which one tries to measure, so no measurement of the position could ever confirm that the particle is inside the CFR. A nice workup of this can be found here:
    http://books.google.com/books?id=3r...lity in classically forbidden region&f=false"
     
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  5. Jan 25, 2010 #4
    Re: tunneling

    alpha particles show tunelling phenomenon... what about it??
     
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  6. Jan 25, 2010 #5

    SpectraCat

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    Re: tunneling

    Nothing in what I wrote rules out tunneling, which involves detecting a particle in another classically allowed region of space. My post was about why the particle can never be detected *inside* a classically forbidden region. Did you read the link I posted?

    Take the example of a particle tunneling through a 1-D potential. It is possible to observe the particle in the classically allowed region on either side of the barrier, but it is impossible to observe the particle in the CFR represented by the inside of the potential barrier. In other words, the probabilities of observing a reflection event or a transmission event sum to one. This is true even though the particle has a non-zero probability amplitude inside the CFR. You can find a description of this in any elementary Q.M. text.
     
  7. Jan 25, 2010 #6
    Re: tunneling

    I don't believe this. The whole probability interpretation of QM would be wrong if particles can never be measured in a classically forbidden region where the wavefn is nonzero.
     
  8. Jan 25, 2010 #7

    SpectraCat

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    Re: tunneling

    I don't think this is true. In any case, there are no known experiments (at least as far as I am aware), that measure a particle while it is inside a CFR. Did you look at the reference I posted? Basically, the exponentially decaying wavefunction inside the CFR ensures that the uncertainty in position is always greater than the penetration depth into the CFR. This means that one can never conduct an observation that conclusively shows a particle located inside the CFR.
     
  9. Jan 25, 2010 #8
    Re: tunneling

    Well I don't know about experiments, but it is a theoretical objection. Assume a particle is in an energy eigenstate of a potential well. The integral of the squared wavefunction within the classically allowed region of the well is always a bit less than 1. But you are saying that a measurement of position will find it inside the classically allowed region with probability 1, which sounds like you are denying the probability interpretation of the squared wavefunction.
     
  10. Jan 25, 2010 #9

    SpectraCat

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    Re: tunneling

    I am certainly not denying the Born interpretation of the squared wf. What I am saying is that any measurement of the position will correspond to one of the following cases:

    1) it will return a mean value that lies within the classically allowed region (CAR),

    2) it will return a mean value that lies within the CFR for a given quantum state, but with an uncertainty so large that one cannot exclude the possibility that the act of measurement excited the particle to a higher lying state where that measured value *does* correspond to the CAR.

    So, from a certain point of view, I guess that one could say that the probability of *measuring* the position (or any other physical property) in the CAR is indeed one, but that says nothing about the probability density of the wavefunction. I don't think there is such a thing as a "position space probability operator", so I don't think it is possible to measure the probability density directly.

    Can you concoct a theoretical experiment whereby the a quantum particle is measured inside a CFR without violating the HUP? I have tried to do this, but I have not been able to come up with anything.

    One mitigating factor here that is getting glossed over is that what I have been talking about so far is formulated from the point of view of stationary states, whereas any meaningful measurement is necessarily time-dependent (there must be a "before" and "after"). Thus, one should take into account that any real measurement will amount to a time-dependent perturbation of the quantum state of the system, I have never worked out such a treatment, nor have I seen it done.
     
  11. Jan 25, 2010 #10
    Re: tunneling

    First of all, thanks for the discussion that got me thinking about this problem.

    Now, my point is: can't we find a particle in a classically forbidden region? I'm not really convinced by the argument in the book. Actually I perfer this explanation http://www.chem.uci.edu/undergrad/applets/dwell/uncertainty.htm [Broken]

    Another mathematical point is that if the momentum is imaginary in the forbidden region, that means that the momentum operator is not hermitian there. Then how can we use the uncertainty relation?
     
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  12. Jan 25, 2010 #11
    Re: tunneling

     
  13. Jan 26, 2010 #12
    Re: tunneling

    Now THAT makes sense.
     
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  14. Jan 26, 2010 #13
    Re: tunneling

    As is so often the case, the explanation is clear using the http://www.tcm.phy.cam.ac.uk/~mdt26/pilot_waves.html" [Broken] of QM. There, the wave is not a 'wave of probability' as you state - it is an objectively existing wave which exists in addition to the electrons and pushes them around (this picture logically follows from the ordinary equations of QM - the only assumption that differs from normal is that particles continue to exist when you don't look at them, i.e. when you don't measure where they are)..

    Consider tunneling through a square barrier - the case where the initial energy of the particle E is less than the barrier height V. Initially the particle - traveling along with a [tex]\Psi[/tex]-wave packet - moves towards the barrier. The interaction of the packet with the barrier leads to the formation of reflected and transmitted packets of diminished amplitude, perhaps together with a small packet persisting inside the barrier. The particle ends up in one of these (which one depends upon its initial starting position within the wave packet)..

    Tunneling arises from the modification of the total energy of the particle due to the rapid spacetime fluctuations of the [tex]\Psi[/tex]-wave in the vicinity of the barrier (the force [tex]-\nabla Q[/tex] exerted by the wave on the particle turns out to be proportional to the curvature of the wave). The effective 'barrier' encountered by the particle is not V but V+Q - where Q is the potential energy function of the wave field. This may be higher or lower than V and may vary outside the 'true' barrier. For tunneling you require only that the energy of the particle [tex]\geq V + Q[/tex] then the particle may enter or cross the barrier region.

    In effect, the additional 'force' exerted by the wave field on the particle just shoves the particle over the barrier.

    As far as I know it is impossible to explain this effect consistently using an interpretation of QM involving the wave function alone (all you can do is predict the probabilities of the final outcomes). Cue much wailing and gnashing of teeth and people telling you that I'm not allowed to explain this to you because the interpretation is non-standard.
     
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  15. Jan 26, 2010 #14
    Re: tunneling

    Now why doesn't that surprise me? :rofl: Of your explanation of tunneling is correct however, why wouldn't say, an electron tunnel EVERY time? If the energy is there in the pilot wave to "push" it over the energy differential, why doesn't it happen every time?

    That said... as far as I know you're correct, and while its necessity is explained and predicted by QM, there is no single explanation involving the [tex]\Psi[/tex] alone. Bah.
     
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  16. Jan 26, 2010 #15
    Re: tunneling

    Hidden variables, mate. Repeat the experiment 1 million times with the same state preparation procedure (i.e. the same starting wave function). However, the individual members of the ensemble are not the same because of the initial positions of the electrons (which are distributed according to the square of the wave field). As the system time evolves, different starting points evolve to different finishing points i.e. the electrons follow a different trajectory each time. If the probabilty of tunneling through the barrier is 0.3, then 30 per cent of initial trajectories will end up on the other side.

    See the attached picture.
    Really annoying isn't it.. :approve:
     

    Attached Files:

  17. Jan 26, 2010 #16
    Re: tunneling

    Ahhhh... I see.

    and... YES!!!! lol.
     
  18. Jan 26, 2010 #17

    SpectraCat

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    Re: tunneling

    ok ... but that is just your preference. The two explanations are formally identical.

    This is the same argument as in the book you like, except I didn't put the word "logically" negative in there. IOW, you can describe a CFR as a region where the kinetic energy is "logically" negative, or as a region where the momentum is "logically" imaginary. Either concept is non-sensical, and I think they are identical, since a negative KE implies an imaginary momentum.

    However, I do see how the non-Hermetian point is a little more serious ... but isn't that a general problem for solving the Schrodinger equation inside a CFR as well? If one of the postulates of Q.M. is that the eigenvalues of physical observables is real, then how can we use the mechanics of Q.M. to calculate a probability density in a region where a relevant observable is imaginary?
     
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  19. Jan 26, 2010 #18

    SpectraCat

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    Re: tunneling

    Maybe I am being dense, but how does this explain the original question about detecting the particle inside a CFR?

    Also, while I like dBB and I am impressed by it every time I learn more, I don't really find this explanation any more clear than the TCI-probability based one, which I think I understand (uh oh .. that means I don't, according to Feynman).
     
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  20. Jan 26, 2010 #19
    Re: tunneling

    Talking about the mean of a set of measurements is quite different from the result of a single measurement. Furthermore, we can assume we have very good experimentalists so there is no uncertainty in any direct measurements they make.

    Put an electron in an energy eigenstate of a potential well.
    Turn off the potential well and simultaneously turn on a huge electric field which accelerates the electron downwards into a piece of photographic paper. Sometimes, the dot on the paper will be found outside of the classically allowed areas on the paper. The HUP is built into the formalism of quantum mechanics, so it can't be violated by anything I do.

    Also, I don't know what it means to say momentum is imaginary in the CFR. Momentum is a hermitian operator, or a real number.
     
  21. Jan 26, 2010 #20

    SpectraCat

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    Re: tunneling

    That doesn't change the situation at all, but perhaps my choice of terms was poor. The upshot of this is that any time that you measure a position observable that corresponds to the CFR for a particular quantum state, the conditions of the measurement will be such that you cannot be certain that the act of conducting the measurement did not perturb the system so that it was a higher lying state, where the measured position corresponds to a CAR.

    Are you sure about this? Has the experiment been done? What are the characteristics of the trap? What are the characteristics of the "huge" field? In order for this to work, you would have to be sure that the transverse distance dx traveled by the electron during its acceleration by the huge field is smaller than the penetration depth of the wf into the CFR, let's see some numbers for those conditions. Note also that it would need to be a 3D-trap, which means that the electronic wavefunction would have non-zero components along the direction of the accelerating field, which is a serious complication in this case, because it means that the wavefunction will couple to your "huge" field, and therefore have the chance to be perturbed by it. This then means that you are in the same boat as above, you cannot ever be sure that the measured position is in the CFR for the quantum state of the system, because you cannot independently know the quantum state of the system from the same single measurement. That is the consequence of the HUP.

    I don't know either ... but isn't that a problem for any measurement in the CFR? But I guess you agree that the K.E. would have to be negative in the CFR, and the K.E. is given by p^2/2m, so what other conclusion could one draw than that the momentum is imaginary? Did you see my earlier post on this question?
     
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