Quantum tunneling

  • #26
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Your case B is the correct one. And yes, there's lots of evidence. As in: Every single experiment ever done on an entangled state. An entangled state, in its simplest form, is two particles describable by one wave function, not a sum of the wave functions of two independent particles.

If an entangled state as you describe it involves two particles, then I don't see that we're dealing with that in the present example. It's just one electron fired at a barrier.

In the case of a single particle that's either reflected or not, if you insist on looking it as two wave functions, then it's two wave functions entangled with respect to location. And measuring the existence of the particle at one location or the other will cause both to 'collapse' into whatever that position is. It will no longer have any probability of existing at the other location.

So the question is: how has that been verified experimentally?
 
  • #27
f95toli
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So the question is: how has that been verified experimentally?

Couldn't tunneling of states (as opposed to particles), which after all is more general, by viewed as evidence of this?
There are a few well-known systems (a Josephson junction would be one example) where the system is initially trapped in an insulated potential well but can tunnel from this (nearly) dissipationless state to a dissipative state where the system is subject to very rapid decoherence (and is therefore "measured"). This system has e.g. been used as a qubit.

As far as I know no one has ever observed a situation where the systems BOTH stays in the non-dissipative state AND stays in the well.
 
  • #28
alxm
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Have you considered that the barrier effectively performs a "measurement" on the particle represented by the original wave function, causing it to collapse into 2 separable wave functions (packets) that are not entangled?

No, it does not perform a 'measurement'. There is no need to bring in the Copenhagen interpretation into this problem. Go read up on scattering theory.

Whether you call it a "bimodal wave function" or 2 "unimodal wave functions" is just semantics. You can certainly call it two "wave packets", because the definition of a wave packet is one localized probability density function.

You cannot treat it as such. If you have one localized wave function (wave packet) at one point in time, which then hits a barrier and disperses, the two (or however many) dispersal nodes you get will be interdependent. They do not form a linear superposition of states, as the case would be for seperate, independent wave packets.

A wave packet can be split into 2 wave packets by a barrier. That is established by the tunneling page I linked and associated simulation rendering which makes this quite apparent.

Yes, but you don't seem to have done the math. The fact that a wave function has two separate nodes does not make it two separate wave functions representing two independent states of two independent particles.

By simple logic, it follows directly from these 2 facts that "A particle may be split into 2 particles by interaction with a barrier." And hence, these two resulting split wave packets will not be bound to each other.

Show it mathematically then.
 
  • #29
jambaugh
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Thank you conway for trying to stay on track with my original question. However you are not asking the same question I was asking anymore.

...
My question: is there any specific evidence or experiment (as opposed to simply quoting theory) supporting one case over the other?

You can't separate these issues. Via CI the wave function is a theoretical construct used to describe the behavior of the physical particle. It is the theory which dictates how the wave function behaves and then it is the wave function which tells us what the theory predicts about the physical behavior of the particle.

In short we don't measure wave functions they are not observable. In CI they are not physical. In other interpretations they are given different ontological status but are none-the-less still not observable.

The theory dictates case 2 w.r.t. the wave function and the probabilistic behavior of the particle is as far as experiments have been able to determine consistent with the theory. But the theory cannot and indeed explicitly prohibits (under CI) any ontological descriptions of the physical particle except in terms of what is/was/will be/might be... measured.

Quantum theory is not based on an ontological model like classical theory but rather a distinctly phenomenological model.
 
  • #30
alxm
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So the question is: how has that been verified experimentally?

Through the experimental verifications of quantum mechanics.
 
  • #31
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Couldn't tunneling of states (as opposed to particles), which after all is more general, by viewed as evidence of this?
There are a few well-known systems (a Josephson junction would be one example) where the system is initially trapped in an insulated potential well but can tunnel from this (nearly) dissipationless state to a dissipative state where the system is subject to very rapid decoherence (and is therefore "measured"). This system has e.g. been used as a qubit.

As far as I know no one has ever observed a situation where the systems BOTH stays in the non-dissipative state AND stays in the well.

I'm sorry, but I'm not able to comment on your example.
 
  • #33
f95toli
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I'm sorry, but I'm not able to comment on your example.

That is a shame, because it it is a good example:smile:

But the basic idea is quite simple.
Imagine a particle trapped in a well, to the left there is an infinite wall; to the right a barrier with some finite height (meaning there is a non-zero tunneling probability). Hence, to start with we have a "particle in a box" situation, the wavefunction is localized to the well,
Now, in addition to this we assume that if the particle leaves the well by tunneling to the right it goes into a state with a lot of dissipation; i.e. it becomes a "classical particle" and is instantly "measured".

Now, this turns out to be a good description of several REAL systems and has been studied experimentally for over 25 years (there are whole books about this, see e.g. Takagi's "Macroscopic Quantum Tunneling" which is slightly odd in places, but otherwise quite good).

The reason why I thought this might be a good example is that the tunneling is a very well defined process; in some of these systems it is quite literally a SINGLE state (described by a single-particle wavefunction) that tunnels (as opposed to ensembles of particles and other messy situations) which in turn means that it is an ideal "toy system"; both theoretically and experimentally (and the experimental date agree very well with theory).
 
  • #34
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That is a shame, because it it is a good example:smile:

I'm sure it is and I appreciate your elaboration on it. I hope you don't think the idea of tunneling itself was in question, by the way. These discussions have a way of going off in a hundred different directions. And they often end with someone weighing in with the important revelation that "quantum mechanics is the most accurate theory ever devised by man".

I still think the OP had a reasonable question in terms of what kind of experimental verification could be done for the oft-cited thought experiment in question. I'm reluctant to get any further into the correspondence because it seems to have taken a vaguely unpleasant turn.
 
  • #35
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You can't separate these issues. Via CI the wave function is a theoretical construct used to describe the behavior of the physical particle. In short we don't measure wave functions they are not observable. In CI they are not physical. In other interpretations they are given different ontological status but are none-the-less still not observable.

In this case, the question has a physical meaning and so the question can be answered via direct measurements without resorting to wave packet theory. ie, take a closed system containing a barrier and a free particle. Fire the free particle at the barrier with no holes, then put a detect on both sides of the barrier. If a particle is detected on both sides of the barrier, then this proves that a particle can split itself into non-entangled reflected and transmitted particles via quantum tunneling, which effectively answers the original question without resorting to wave theory
 
  • #36
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In this case, the question has a physical meaning and so the question can be answered via direct measurements without resorting to wave packet theory. ie, take a closed system containing a barrier and a free particle. Fire the free particle at the barrier with no holes, then put a detect on both sides of the barrier. If a particle is detected on both sides of the barrier, then this proves that a particle can split itself into non-entangled reflected and transmitted particles via quantum tunneling, which effectively answers the original question without resorting to wave theory

This has been done already. The double-slit experiment is not much different from the passage of a particle through a barrier. The details vary, but you are still dealing with the same wave interference effect. I think it is well established that particles do not "split" in the double slit setup, so there is no reason to believe that they would "split" in the barrier setup.
 
  • #37
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This has been done already. The double-slit experiment is not much different from the passage of a particle through a barrier. The details vary, but you are still dealing with the same wave interference effect. I think it is well established that particles do not "split" in the double slit setup, so there is no reason to believe that they would "split" in the barrier setup.

No. Not even close to the same thing...and there is no interference effect. The double slit experiment tests wave interference and is done at a scale that assumes tunneling is negligible...whereas this would be done on a much smaller scale to test tunneling and there are no interference effects...have you been following this discussion at all?
 
  • #38
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Against my better judgement, I keep trying to figure out what the heck we're talking about here. I get the impression that junglebeast makes a qualitative distinction between a wave function that is physically spread out versus one that is actually detached from itself. I have my doubts that this is really a significant distinction: the same mathematics creates these cases so they ought to follow the same physics. I also sympathize with those posters who have offered other physical systems as evidence of something or other...the probelm is, just what point is being made? Feynmann once said that all of the mysteries of quantum mechanics are contained in the double slit experiment, but I think people have taken that too literally.

When junglebeast talks about a particle splitting in two at the barrier, I don't know what to make of that. But if he's just asking for experimental proof that the two branches of his wave function cannot be simultaneously detected, I think he hasn't been answered yet. The Josephson junction example might in fact be pertinent, even more so than the double slit, but can we first decide whether the experiment can be done more or less directly?
 
  • #39
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Against my better judgement, I keep trying to figure out what the heck we're talking about here. I get the impression that junglebeast makes a qualitative distinction between a wave function that is physically spread out versus one that is actually detached from itself. I have my doubts that this is really a significant distinction: the same mathematics creates these cases so they ought to follow the same physics.

Because I think there is a far more simple representation for the mathematics of QM, which also has much more sensible meaningful interpretation than the attempted "Copenhagen interpretations," and one of the only places its predictions differs from the overly complicated existing theory is in this specific case. So if this specific case has no experimental evidence to support the current model, then I would be more inclined to believe the more simple model, which does comply with all the other QM experiments I know of.
 
  • #40
I've briefly googled ftl tunneling and it appears possible but doesn't tunneling end up with a lower energy state after crossing the barrier? So if the barrier is c wouldn't the energy of the particle be greater after crossing c or does the wavelength of the photon lower itself enough to satisfy the tunneling rules and yet still be ftl because the energy of the photon is greater than usual for the wavelength it dropped to?

Frank
 
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  • #41
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what in god's name is going on in here. has anyone in here actually taken a qm class?

Thank you conway for trying to stay on track with my original question. However you are not asking the same question I was asking anymore.

This is a rendering of the wave function being partially reflected and transmitted through a barrier:
EffetTunnel.gif


An initial wave function, call it W, is split into 2 separate wave functions call them Wa and Wb, one of which is transmitted and the other reflected.

Case A: Wa and Wb can be treated as separate wave functions; it is possible for Wa to collapse and it does NOT cause collapse of Wb.

Case B: When Wa collapses, Wb simultaneously collapses because they are really still part of the same wave function, regardless of their spatial separation.

The wording on wikipedia's page seems to indicate support for Case A. So far I have only seen support for case B in the replies to this thread.

My question: is there any specific evidence or experiment (as opposed to simply quoting theory) supporting one case over the other?


the answer is resoundingly B. you've asked it twice already. it is always B. in fact there are actually 3 "parts" to the wavefunction because there's the nonzero probability of the particle being in the barrier!

jambaugh has told you already twice. the point of collapse is not that the wavefunction goes to zero everywhere. the point is that it peaks around your measurement. note that i say peaks and not dirac deltas since the probability of a particle being a point [itex]x_0[/itex] in space is zero:

[tex]\int_{x_0}^{x_0}\psi^2 dx = 0[/tex]

so the wavefunction peaks/collapses/updates and then spreads out again after sometime ( unless it's in a coherent state, which it's not in this case ).

you need to look at what a wavefunction really is: a solution to a pde whose square is a probability distribution function. stop playing these metaphysics games.
 
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  • #42
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what in god's name is going on in here. has anyone in here actually taken a qm class?

the answer is resoundingly B. you've asked it twice already. it is always B. in fact there are actually 3 "parts" to the wavefunction because there's the nonzero probability of the particle being in the barrier!

jambaugh has told you already twice. the point of collapse .....
...you need to look at what a wavefunction really is: a solution to a pde whose square is a probability distribution function. stop playing these metaphysics games.

Again, that wasn't his question. We know the "correct" answer is B. His question: what is the direct experimental evidence?
 
  • #43
jambaugh
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Again, that wasn't his question. We know the "correct" answer is B. His question: what is the direct experimental evidence?

The fact that you observe only one particle after the fact! But again you don't observe wave-functions they are the mathematical description not something experimentally observed. The issue is what is the correct formal mathematical description and the answer is: we are here talking about a single system with its single wave-function.

Now to understand it better we should consider how the logic parses through the quantum description. The two pieces of wave-function in this example are formally added together:

[tex]\psi(x) = \psi_L(x) + \psi_R(x)[/tex]
This superposition is simply the resolution of a single vector in terms of two components, like
v = xi + yj. Underlying this vector sum is a tensor sum of spaces spanned by the components. The logical analogue of the tensor sum in QM is the logical or. In this case we are resolving the superposition of components into "the particle reflected or it tunneled". Note the coherent sum is not the 'or' operation but rather reflects the fact that the two component descriptions are both "off a bit". Superposition is a property of our description not of the system itself. It reflects our "poor" choice of basis in describing a system.

Compare all this to the tensor product of two system descriptions. This is what we're thinking when we talk of "two wave-functions" instead of "two pieces of a single wave-function". The tensor sum of two modes corresponds somewhat to the compositional "and".
[tex] \Psi = \psi_1\otimes \psi_2[/tex]
reflects a logical statement system 1 was observed to correspond to mode [itex]\psi_1[/itex] and system 2 was observed to correspond to mode [itex]\psi_2[/itex].

When we take the product of spaces we are composing separate systems to treat them as a single composite system. (It is in this case were we may invoke entanglement.) We again get a single "wave-function" but it will be a function of two independent position parameters corresponding to classical configurations of two component systems e.g. two separate electrons (or more properly to positions of two separate system-detection events).
This is not the correct interpretation of the case being discussed here.

In vernacular language we distinguish these two cases by the correspondence "two wave-functions" = tensor product of two wave functions= composite system, and "two components of a single wave function" = superposition = coherent sum of two modes for a single system.
 

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