[Eternal Student (OP)]

Hi.

    Hopefully the poll speaks for itself.   The forum looked quiet this evening and this might provide some discussion or activity.
    There are some answers for this already.   You don't need to go and look for them in textbooks or on the internet but you can if you had a few minutes to spare.  As always, use your own discretion and try to select a sensible and safe website.

Best Wishes.

[Halc]

There is a well known model in Quantum mechanics called "the particle in a box" (also "a particle in an infinite square well").   Considering that model, does the particle exert pressure on the walls of the box?

I had to look up exactly what these words meant. "Box" and "Infinite square well" sort of imply a large container, where in fact they're talking about a very small one in a deep potential well from which escape (by tunneling say) isn't possible.

Without reading any bit about pressure, I'd say yes, it would since it could occupy a lower energy state if it had more room for a longer wavelength. It can exist only in certain quantized energy states in there, and a wider box allows a lower energy one. So it applies pressure the same way that water would since water could occupy a lower energy state with more leg room.

[Eternal Student (OP)]

Hi.

   That's a fair answer.  I need to lay down some opposition just to keep the poll interesting.   (Just to be clear, it doesn't matter what I think, we just need arguments on both sides).

One important feature of the model is that the wave function of the particle looks like this:

    Diagram A is NOT the quantum model,  it is the classical Newtonian understanding of what should be happening.  In that model, the particle is sometimes at the walls and can exchange momentum there.
    Diagrams B through to F illustrate the wave function representation which is part of the quantum mechanical model of a particle in a box.   One of the important points about this model is that the wave function MUST always be zero at the walls of the box.   The modulus squared of that wave function  (or in simpler English "the size" of it) tells you the probability of finding the particle at that location.   So the particle is NEVER at the walls of the box.   So it cannot impart momentum to those walls.

Best Wishes.

[Eternal Student (OP)]

Hi.

We might need another counter-argument.

    Just considering the 1-dimensional situation for simplicity (like the diamgrams in the previous post):   If you measure a force on a wall at one moment of time,   then you know precisely where the particle was and the momentum that it had at that time.   (It was bouncing off the wall).   This would violate the uncertainty principle.  You cannot know the precise location and momentum of the quantum particle simultaneously.   The only sensible escape is that it cannot happen - you cannot measure a force on a wall at any moment in time.

Best Wishes.

[Halc]

So the particle is NEVER at the walls of the box.   So it cannot impart momentum to those walls.

Does it say that somewhere? Because this seems a non sequitur to me.
First of all, the particle cannot be found at the walls. Wrong to say it is somewhere. But I'm talking about the conclusion that it cannot impart pressure (not momentum) to the walls just because it never gets measured right there.

[Eternal Student (OP)]

Hi.

Does it say that somewhere?

   Where does it state that the particle does or can exert force on the walls?   Ideas like obtaining a force are macroscopic interpretations.    Quantum Mechanics for the particle in the box consists of just a handfull of postulates.  They can be written in a few different ways, the wording is certainly flexible,  but here is one version:

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/qm.html
    (It's the second boxed section that lists 6 postulates.    The last box just expands on what is meant by items 1. and 5. and provides precisely the result we needed to determine the probability of finding the particle somewhere).

   If you wish to assert that QM does directly permit particles to transmit a force to something else (like the walls) then you need to construct or exhibit a suitable Hermitian operator that represents that force  (or provide some reference to where that is done).

Wrong to say it is somewhere.

    That is actually fair enough,  it is only true that it won't be found at the walls when you measure its position.  However what constitutes a measurement?    If you did observe an impulse at one wall for one brief moment of time,  doesn't that constitute a measurement of the position of the particle?  See the discussion earlier about the uncertainty principle.   This simple QM model does not include multiple particles:   It is not as if the particle could have ejected a photon from a remote location to impact on the wall.

Final Note:   I'm not stating my own views, just deliberately presenting arguments for the other side.

Best Wishes.

[Eternal Student (OP)]

Hi.

   I've had one response to the poll so far and that was me.

*   It is open to guests,  you do not need to have created an account.

*   This is in the Just Chat section and not serious.   It's also anonymous (the poll is anonymous but written replies will include your username).

Best Wishes.

[Halc]

Where does it state that the particle does or can exert force on the walls?

That's what is under discussion I thought. I'm not asserting anything.

I'm not stating my own views, just deliberately presenting arguments for the other side.

In that spirt, yes.

I just said that the zero probablilty of finding the particle right at the wall does not necessitate that no pressure is applied to the walls. It's not even true in classical physics where some atom gets close to the wall but never actually touches it, but nevertheless transfers momentum to it when it accelerates away.

Ideas like obtaining a force are macroscopic interpretations.

Agree, but you get macroscopic effects from many quantum effects. So you might measure a pressure of sorts, but it's probably wrong to suggest that a super-sensitive meter would register spike accelerations of the wall on a periodic basis like you would with a red ball bouncing back and forth.

I based my pressure on the naive suggestion that the tight walls necessarily put the particle in a higher minimal energy state. It has potential energy, a bit like a compressed spring.

If you did observe an impulse at one wall for one brief moment of time,  doesn't that constitute a measurement of the position of the particle?

I imagine it would, but I'm not suggesting a periodic impulse any more than a nitrogen tank under high pressure exhibits back&forth motion as the particles collide with the container. The pressure would be balanced, exerted on both sides, as would the pressure from said compressed spring. Again, my naive guess. I'm hardly an authority here.

[Eternal Student (OP)]

Hi.

   Classical results should be recoverable from QM when you take large numbers of particles and work only with averages or expected values.  How does your situation match up with the temperature dependency that we do observe for a gas?

This formula:
   Energy (of state n) =  E_n   =   
with   n = energy level,   L = length of one side   (assuming the box has all sides the same length for 2-D or 3-D),   can be derived from the QM model.    One important feature is that n is a strictly positive integer  n =1, 2, 3, ....   we cannot have n=0 or else the wave function becomes 0 and the particle has disappeared.     So there is a minimum energy or ground state.    The best you can hope for is that lowering temperature tends to put the particle in a lower energy state but eventually it hits the minimum energy state.  This is unavoidable.
     However, that energy would always vary ~ 1/L²  so that, by your argument, the particle can always obtain a lower energy by expanding the space available to it.   It will always exert some pressure on the walls of the box no matter how cold you make it.    How do you reconcile that with what we know about ideal gases?

Best Wishes.

[alancalverd]

Worth considering Einstein's equivalence of energy density with pressure.

This aligns with Halc's explanation. If the box were wider, the energy density inside it would be lower, so there is pressure on the walls. However Planck stuck with classical mechanics to the extent that his walls were perfectly rigid and elastic, so no work is done on reflection and the particle retains all its energy.

That also explains the nodes at the walls. The particle bounces off in an infinitesimal time (because no energy is transferred) so the time it resides in the vicinity of the wall, and thus the probability of finding it there, tends to zero. 

[alancalverd]

How do you reconcile that with what we know about ideal gases?

We don't know anything about ideal gases, because there aren't any! The classical ideal gas consists of particles with mass but no radius, so it is infinitely compressible.

[Eternal Student (OP)]

Hi.

Worth considering Einstein's equivalence of energy density with pressure.

    I'm not aware of a statement of such an equivalence principle.    However, there is almost an equivalence in General Relativity, if that's what you meant.   
     The stress-energy tensor of GR includes components due to the pressure so that pressure could influence curvature in a way broadly similar to the distribution of energy through space.


     Figure 1:   The stress-energy tensor is shown with pressure components in green, energy density in red and related energy flux in yellow.

   However, it is not clear that you could obtain an identical curvature tensor from the Einstein Field Equation of GR by replacing all energy density with pressure, even allowing for a change of co-ordinates to be applied to the resulting curvature tensor.   More generally, I am not aware of any straight forward relationship like E=mc² (expressing an equivalence of mass and energy) that expresses an equivalence of pressure and energy density.

    If you have a better reference for this "pressure and energy equivalence" of which you speak, please let me know.  It would be interesting to read.

 - - I'm keeping the post short and just ending this here ---

Best Wishes.

[Bored chemist]

Interestingly, if you consider this

 

You run into a different conundrum.
In the first excited state- labelled C- your particle has zero chance of being in the middle of the box.
But it has a 50% chance of being one either side.
So, how does it get from left to right without ever  being in the middle?
You can pretend it doesn't matter- it's actually really on one side of the box and, unless it's kicked into a higher state, it remains there.
 But it's worse than that.
The observable universe can be treated as a huge box with very high walls.
Because the box is big, the energy level separation is tiny.
So any particle you see will actually be in a very high excited state.
It will be passing through a massive number of the "zeroes" of probability density every second.
How does any particle travel past the bits where its probability is zero?

[Bored chemist]

We don't know anything about ideal gases, because there aren't any!

So... that's one thing we know about them.
I think we know other stuff about them too.
They are colourless. (If they interacted with photons they would interact with those emitted by the walls of the container (and one-another) and that's forbidden by their definition.)

Arguably, we know everything about them for the same reason that Tolkien knew everything about Hobbits.

[alancalverd]

 ;)You get a smiley for that!

[alancalverd]

  I'm not aware of a statement of such an equivalence principle.

I've only come across its ascription to Einstein in a few lectures and can't find it on line, but I'm sure it's in at least one of his publications. Anyway the story is

Dimensions of energy: ML²T^-2

Dimensions of volume: L³

Dimensions of pressure = force/area  = MLT^-2/L² = ML^-1T^-2  = energy/volume

It's an extremely useful equivalence in astrophysics and mechanical engineering!

PS apologies - I just ticked the wrong box in the poll! I refer the hon gent to the answer I gave yesterday.

[Eternal Student (OP)]

Hi and thanks for recent replies.

You run into a different conundrum.......how does it get from left to right without ever  being in the middle?

    A very good set of questions or discussion points.   "Quantum weirdness" is a short answer that might be given.   We could have a longer discussion but there's already plenty of articles on the internet.  In this particular situation it's not too difficult to try and match up some Newtonian explanation but it may be best not to even try.   Anyway, that's the attitude I need for the next paragraph.

    One of the points which I still think is important is that there is a jump from the Quantum Mechanical model to the notion that a pressure would exist on the walls.   At the moment those who have submitted a response to the poll have all said "yes" there is a pressure rather than "no" with one possible explanation being simply that the QM model does not take you that far.   No-one has exhibited an operator that represents pressure and acts on the wave function.  I need to make it clear again that without a suitable operator, "Pressure" or Force on the wall has not been established as an observable under the postulates of Quantum Mechanics.   Instead all we have is a jump from the QM model to much more macroscopic properties and very Newtonian mechanics.
    (Note that I am still trying to present arguments on the opposing side, no-one else is doing that at the moment).

The observable universe can be treated as a huge box with very high walls.

   Maybe....  The potential that a particle is being exposed to is not likely to be 0 (or some constant reference point) everywhere inside that box and I can see little reason why a particle is being kept away from the edges of the universe by some large potential (if indeed there are edges to the universe).   It's a possible model but with enough assumptions and simplifications that any results can't be taken too literally.
    While we're discussing the assumptions of a model,   the entire "particle in a box" model is idealised and almost certainly unrealistic.   We don't think there are any potentials that could produce vertical walls (change from zero to non-zero potential over 0 distance) and we don't think there are potentials that could reach an infinite value either.  That is not a criticism of your attempt to model a particle in the universe.   All uses of the "particle in a box" model should be viewed as just an approximation.

It's an extremely useful equivalence in astrophysics and mechanical engineering!

   Oddly enough dimensional analysis often does pre-empt or herald some deeper connection between two quantities - but we seem to have a small way to go yet.

Best Wishes.

[alancalverd]

    One of the points which I still think is important is that there is a jump from the Quantum Mechanical model to the notion that a pressure would exist on the walls.   At the moment those who have submitted a response to the poll have all said "yes" there is a pressure rather than "no" with one possible explanation being simply that the QM model does not take you that far.   No-one has exhibited an operator that represents pressure and acts on the wave function.  I need to make it clear again that without a suitable operator, "Pressure" or Force on the wall has not been established as an observable under the postulates of Quantum Mechanics.   Instead all we have is a jump from the QM model to much more macroscopic properties and very Newtonian mechanics.

Something of a cart/horse inversion, I think.

You have to remember that Planck invented quantum mechanics by derivation from a classical particle in a box model, not the other way around, by stating that the only way that the particle can conserve energy is by having nodes at the perfectly elastic and immovable walls. This then defines the permissible wave functions within the box (must have an integer number of antinodes) and the force exerted on the walls by the particle. 

Since the particle is also a molecule of a perfect gas, you could derive radiation pressure from the work done to compress the gas, again a classical macroscopic process, and to nobody's surprise it delivers the  experimental result for photon (the next best thing to a perfect gas) radiation pressure.

There's surely nothing weird about it. The quantum number of a particle in a one-dimensional box is the number of antinodes in its probability distribution.

[alancalverd]

Diagram A is NOT the quantum model,

Oh yes it is! Everything else derives from it because it only imposes one boundary condition (nodes at the walls) and permits any number of solutions that meet that criterion, without suggesting how they could be achieved.

[alancalverd]

So, how does it get from left to right without ever  being in the middle?

That is indeed the difference between continuum and quantum physics. If the particle were charged, it couldn't actually oscillate because it would lose energy in doing so. The connection between continuum and quantum is only that the continuum model of a particle in a box correctly predicts the existence of nodes and antinodes of probability density, and the quantisation of energy levels.