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Particle in a potential Well interacts with the wall without touching it?

  1. Oct 27, 2009 #1
    Particle in a potential Well interacts with the wall without touching it???

    The ground state Energy for a particle in an infinite potential well is given by
    E = h2 / 8mL2

    Consider a particle with high mass, say 1 kg. Then the ground state Energy will be very very low. So the velocity of the particle is very very low. It takes thousands of years for such particle to hit the walls of a say, 1m3 box that confined it.

    Suppose I was able to actually setup such condition and put the particle to its lowest Energy level.
    Now, If I ask a Quantum Mechanical student, What is the Energy of this particle?
    He will calculate E = h2 / 8mL2 and tell me some value x.
    Now, without doing anything to the particle, I decrease the volume (hence L) of confinement by say 10%.
    Now, If I ask the same student about the Energy, he will use the same formula but give me another answer, say y. But y > x.

    Now I ask him, Why are you supposed to say that the Particles Energy has increased, For I have never touched it, even the slightly?

    What will he say now?????
     
  2. jcsd
  3. Oct 27, 2009 #2
    Re: Particle in a potential Well interacts with the wall without touching it???

    It's a mystery of the quantum mechanics!
     
  4. Oct 27, 2009 #3
    Re: Particle in a potential Well interacts with the wall without touching it???

    If so, can't we say that its fallacy of QM.
     
  5. Oct 27, 2009 #4
    Re: Particle in a potential Well interacts with the wall without touching it???

    There is no fallacy. The mass always interacts with the walls (it is not localized). When you decrease the volume, you perform work.
     
  6. Oct 27, 2009 #5
    Re: Particle in a potential Well interacts with the wall without touching it???

    What is "touching" at the quantum level anyway? Why is this even mysterious?

    With your numbers L=1 m, m= 1 kg, the energy difference is on the orders of h^2 ~ 1e-68 J ! ...which is zero, even exactly. Your student will say to you that he really can't measure any difference. To get a sensible shift in energy you have to confine a 1 kg ball into a nanometer cube or so, which is physically impossible, considering sizes and weights of atoms, unless you want to put a tiny black hole into the box!

    Count Iblis's argument also is very convincing, I can't even feel "a mystery" here.
     
    Last edited: Oct 28, 2009
  7. Oct 28, 2009 #6
    Re: Particle in a potential Well interacts with the wall without touching it???

    First thing is that I am not discussing here about whether the Energy Differences of the order 1e-68 J is practically measurable or not. Its measurable, at least theoretically, or say mathematically .

    Now the mystery is the result of the thought experiment I already mentioned.
    Suppose that we have got an instrument that's able to detect Energies as low as--1e-68 J.
    So, my question is as soon as I decrease the volume of confinement, do the Energy Reading of the Instrument Increase? If Yes, then I was able to increase the Energy of a Distant Particle, instantly (thus sending Information faster than light??)
     
  8. Oct 28, 2009 #7
    Re: Particle in a potential Well interacts with the wall without touching it???

    You're working with non-relativistic QM, so the speed of information is taken to be infinity anyway.

    If you take into account relativistic effects (i.e. a finite speed of light) you will have to deal with the fact that the response of the wavefunction to your change is not instantanious. There is a retardation.
     
  9. Oct 28, 2009 #8
    Re: Particle in a potential Well interacts with the wall without touching it???

    Also: The particle will only remain in the ground state with certainty if you change the volume infinitely slowly. This is the Adiabatic Theorem of quantum mechanics.

    In the opposite limit you change the volume infinitely fast and then the wavefunction of the particle stays the same . In this case this so-called "sudden approximation" does not work if you decrease the volume, only if you increase the volume.

    So, if you increase the volume fast enough, the wavefunction andthus the expectation value of the energy will stay the same. The particle will thus not be in the new ground state.
     
  10. Oct 28, 2009 #9

    f95toli

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    Science Advisor
    Gold Member

    Re: Particle in a potential Well interacts with the wall without touching it???

    The effect of time varying potential can be handled even in non-relativisti QM (for "v<<c"). If you suddently start changing the potential the particle will undero a Landau-Zener transition to a new state; the probability of this transition scales with the rate of change.
     
  11. Oct 28, 2009 #10
    Re: Particle in a potential Well interacts with the wall without touching it???

    Whatever be the case, (relativistic / non-relativistic),
    Does QM implies that--,
    if I put a small marble in a big box near the center at almost rest, the marble is always interacting with the wall, though the wall may be kilometers away, So that, any changes in the potential energy or distance between wall results in the change of marble's wave function.

    If so the wall should be constantly sending Signal to the marble. What type of signal is it.
    I mean, it can't surely be EM signal???
     
  12. Oct 28, 2009 #11
    Re: Particle in a potential Well interacts with the wall without touching it???

    The crucial point is this:

    In practice this means that that the whole system has to be cooled to a very low temperature (Exercise: estimate this temperature). Only then will the system be in the ground state and become "de-localized".
     
  13. Oct 28, 2009 #12
    Re: Particle in a potential Well interacts with the wall without touching it???

    Are you trying to tell me that we can't setup such condition practically or are you trying to tell that there is no way to do that even in theoretical basis.
     
    Last edited: Oct 28, 2009
  14. Oct 28, 2009 #13

    Dale

    Staff: Mentor

    Re: Particle in a potential Well interacts with the wall without touching it???

    I don't think it is correct to think of it as "the marble's wave function". The wave function describes the state of the system, and the system includes both the marble and the box. As you change the box you are changing the system so of course the state and therefore the wavefunction must also change. In relativistic QM it does so in a way that respects relativity.
     
  15. Oct 28, 2009 #14
    Re: Particle in a potential Well interacts with the wall without touching it???

    Hi.
    Pressure ∂E/∂L works on the wall. You cannot but change energy by moving the wall.
    Regards.
     
    Last edited: Oct 29, 2009
  16. Oct 29, 2009 #15
    Re: Particle in a potential Well interacts with the wall without touching it???


    No, what I'm saying is that in the given problem, you cannot picture the marble as being in the middle of the box. If you just put the marble in the middle of the box, it will definitely not be in the ground state.

    You can write down a wavefunction for the marble's center of mass motion that does describe what will go on in reality if you just put it in the middle of the box. Then you can show that changing the volume does not have any effects.
     
    Last edited: Oct 29, 2009
  17. Oct 29, 2009 #16
    Re: Particle in a potential Well interacts with the wall without touching it???

    Let us consider a light QM particle in an infinitely high well of some finite size L. The wave function is zero at the well walls. Does it mean the particle does not touch the walls?
     
  18. Oct 29, 2009 #17
    Re: Particle in a potential Well interacts with the wall without touching it???

    The wavefunction goes to zero near the walls. This means the derivative, and also the second derivative of the wavefunction is not zero. The kinetic energy of the wavefunction is determined through the momentum operator squared, which is precisely the second derivative of the wavefunction: [itex]\nabla^2/2m[/itex]. So the fact that the wavefunction tends to zero gives the particle some momentum and kinetic energy.

    If you move the wall, you force the wavefunction to go faster to zero. This is a consequence of the fact that wavefunction stays normalized. The second derivative increases, and so does the kinetic energy.
     
  19. Oct 29, 2009 #18
    Re: Particle in a potential Well interacts with the wall without touching it???

    What a use of the kinetic energy if the particle does not touch the walls?
     
  20. Oct 29, 2009 #19
    Re: Particle in a potential Well interacts with the wall without touching it???

    If you're gonna say that the wavefunction or particle doesn't touch the walls, then the only way to realize this is by making the wall an infinitely large and infinitely steep barrier. The typical textbook problem. It's not a very realistic model, but fine, let's just move on from here. In this case there is no overlap with the barrier since the wavefunction cannot have some value in the regime of infinite potential, so indeed, there is no "interaction energy" present.

    That doesn't mean the barrier doesn't do anything. On the contrary, the barrier imposes very strict boundary conditions on the wavefunction since it needs to go to zero at the boundary, it needs to stay normalized and it needs to stay continious. In effect, the kinetic energy term goes up.
     
  21. Oct 29, 2009 #20
    Re: Particle in a potential Well interacts with the wall without touching it???

    I am joking. The stationary wave function is zero because it is a superposition of two running waves - the incident and reflected. Of course, the walls participate and experience some pressure.
     
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