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Matterwave

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The potential energy of the wall does not have the same value everywhere. You only know the average value. The place where you find the particle (at that particular time) could have a much lower potential than other places.

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Or, did I miss the point of the question?

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But if E, the energy of the particle, is smaller than V, then how can it be inside the wall?

Or, did I miss the point of the question?

To be inside the wall, the particle must have energy larger than V, right?

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malawi_glenn

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not in quantum mechanics, that is the point of tunneling.But if E, the energy of the particle, is smaller than V, then how can it be inside the wall?

To be inside the wall, the particle must have energy larger than V, right?

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No, I disagree. For a finite barrier there's a decaying (in space) solution inside the barrier. However, if barrier is infinitely high, the inside solution goes to zero. Afterall, thats tunneling...But if E, the energy of the particle, is smaller than V, then how can it be inside the wall?

To be inside the wall, the particle must have energy larger than V, right?

BTW, the fact that solution inside is non-zero is just a manifestation of the wave-nature (here the wavefunction). Also, in optics you can have similar behavior with so called evanescent wave.

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I have no problem with quantum tunneling. It's fine if the particle is found on either side of the wall, but not inside the wall, since the energy of the wall is V, so the particle can be on either side of it, but not inside it. If the particle is inside the wall, its energy must be larger than V, am I right?not in quantum mechanics, that is the point of tunneling.

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The question is of the form of the wavefunction psi(x) that determines whether the particle can be found inside the well. If you believe that particle can be on one side and then on the other, then connecting the solutions (through boundary conditions on psi) - particle can be found inside.I have no problem with quantum tunneling. It's fine if the particle is found on either side of the wall, but not inside the wall, since the energy of the wall is V, so the particle can be on either side of it, but not inside it

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Say a free charged particle is prepared (with specific kinetic energy E) and at some later time it is shown to be inside a wide barrier (of constant potential V>E), by a process of elimination (i.e., by operations/measurements performed only on the regions in front and behind of the barrier, so as to not disturb the particle directly).not in quantum mechanics, that is the point of tunneling.

If we just wait, then measure the particle's energy after it has left the barrier, will it now have KE V>E (consistent with the knowledge that it was definitely "in" the barrier, and if so then was energy correspondingly subtracted from the mechanism that was used to check that it wasn't outside of the barrier)?

Or, if the particle's total energy at all times remains constant (implying negative KE in the barrier), what speed will we measure for it (say, if we equalise the electric potential outside, to widen the barrier, then measure how far the particle propagates in some time period)?

Mathematically, do we just zero out the reflected/transmitted parts of a wave-packet, renormalise and wait-see what its evolution limits to?

I suppose a similar question is, if a single particle from a collimated beam passes a single slit, and is detected at a high angle of diffraction, how is momentum conserved? (And does the aperture just zero out part of the wave-function for particles that get through, or strictly does the aperture entangle with every particle that it changes the wave-function of?)

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You're right - tunneling is one form of interference. Like in the example of diffraction in the single slit its the average over space that needs to be taken in order to talk about conservation of energy/momentum. Incidentally, the momentum is conserved of course, because for the high angle you mention there's equal and opposite contribution on the other side, so the transverse (to slit) momentum cancels out leaving only components away from the slit - same as what was before the slit. You can also say, what about conservation of energy? At some fringe I can measure more than came in??? The answer is, if you integrate over all space (i.e. all the points of destructive interference) then overall energy will be conserved (assuming lossless slit). Same way in this problem... integrate over all space and you'll find that [tex]\int |\psi(x)|^2 dx[/tex]=1, even though at some point (not in this problem but in general) |psi(x=a)|^2 can be greater than 1 (you wouldn't say that you've violated conservation of energy). When you speak of waves, conservation of energy is a non-local concept.I suppose a similar question is, if a single particle from a collimated beam passes a single slit, and is detected at a high angle of diffraction, how is momentum conserved? (And does the aperture just zero out part of the wave-function for particles that get through, or strictly does the aperture entangle with every particle that it changes the wave-function of?)

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But I was speaking of just a single individual particle. We agree that energy and momentum are conserved on average, I guess the question is whether they are conservedthe momentum is conserved of course, because for the high angle you mention there's equal and opposite contribution on the other side, so the transverse (to slit) momentum cancels out [..] When you speak of waves, conservation of energy is a non-local concept.

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If we are talking about quantum mechanics, then we are talking about probabilities. We have to have enough events to build up the probability distribution (i.e. multiple measurements). If I toss a coin 5 times and get heads all the five times does that mean that the random nature of the process is violated? I have to toss the coin N (where N is aBut I was speaking of just one solitary individual quanta alone. We agree that energy and momentum are conserved on average, I guess the question is whether they are conservedonlyon average? (I'm guessing not, since otherwise I would expect independence of previous outcomes to result in a randomwalk.)

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malawi_glenn

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so here is a good question, can we measure energy and position at the same time in QM?

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how long is the measurement?so here is a good question, can we measure energy and position at the same time in QM?

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malawi_glenn

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It is quite meaningless to ask "what is the energy at position x_1?"

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sure, as [x,p] .ne. 0, and so is [x,p^2] .ne. 0H and x does not commute

you can use that mathematical construct to give the answer to this problem - i agree. there are multiple ways of saying the same thing, so perhaps this will work better?

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malawi_glenn

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hehe you answered really quick, I changed my answer, I wanted to take that point later ;-)sure, as [x,p] .ne. 0, and so is [x,p^2] .ne. 0

you can use that mathematical construct to give the answer to this problem - i agree. there are multiple ways of saying the same thing, so perhapsa this ill work better?

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i guess i saw where you were going with this :) i'm new to this forum, but would guess that this response shouldn't really go into thread discussion... so, i'll put some relevancy to the theme of the thread below (not necessarily in response to your comments):hehe you answered really quick, I changed my answer, I wanted to take that point later ;-)

tunneling is not a quantum phenomenon but a wave phenomenon. you can observe tunneling with water waves, electromagnetic waves, etc. inside the barrier all waves will be attenuated in the direction of propagation, but the energy density would still be non-zero, unless at infinity.

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malawi_glenn

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agreed about consistency! i also think that its equally important to find analogies in various branches of physics and relate them to the problem at hand. perhaps for some it is a revelation that situation discussed can be viewed totally classically?

i'm new on here and still haven't quite gotten the reply 'etiquette' if such even exists, but I guess i'm viewing the purpose of these discussions as a collective attempt to help each other understand something, and not necessarily demonstrate knowledge of for e.g. jargon (which is of course very useful to know too, don't get me wrong). so if an analogy (or a mathematical statement) can clarify the point or explain something - i don't see anything wrong with either. I can have flour and corn tortillas and switch freely between the two :P

as for paradoxes - they can 'spawn' anywhere - from having incorrect physical picture to not doing the math right...

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Vanadium 50

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Exactly. When it's in an energy eigenstate, it's not in a position eigenstate, and vice versa.It is quite meaningless to ask "what is the energy at position x_1?"

The closest thing you could ask is "If I were to then restrict it to the region between here and there, how much energy would I have to add?"

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