[Eternal Student (OP)]

Hi.

Some interesting posts which I have spent nearly a day thinking about.

You have to remember that Planck invented quantum mechanics by derivation from a classical particle in a box model, not the other way around....

     A minor but noteworthy issue is that Planck did not invent the modern version of Quantum Mechanics.   It was never a single-handed development.  Many of the ideas from Planck and a few others like Bohr were motivational but did not become axioms of modern quantum mechanics.
       This phase (pre 1925 and involving many of Planck's ideas and Bohr's quantised model of the Hydrogen atom) is known as the old quantum theory.  Never complete or self-consistent, the old quantum theory was rather a set of heuristic corrections to classical mechanics...
       Building on de Broglie's approach, modern quantum mechanics was born in 1925, when the German physicists Werner Heisenberg, Max Born, and Pascual Jordan developed matrix mechanics and the Austrian physicist Erwin Schrödinger invented wave mechanics.

Extracts taken from the History of the development of Quantum Mechanics,   https://en.wikipedia.org/wiki/Quantum_mechanics#History

    It is perfectly reasonable to start from the postulates of modern QM (let's say as it existed post 1930) and work outwards from there.  None-the-less it's interesting to consider it the other way around and your posts were well considered, thank you.

Best Wishes.

[paul cotter]

Hi ES, this discussion is WAY over my head as merely being a nuts'n'bolts, matter of fact retired engineer. However as this is just "chat", I thought I would stick my head above the parapet  I feel there is a degree of incongruity in the question in that you are asking does a quantum phenomenon produce a macroscopic classical effect( pressure ). You yourself have alluded to a multiplicity of quantum "events" being necessary to produce a macro effect.  I now await a hail of spoiled fruit and rotten eggs etc, coming my direction!!

[alancalverd]

does a quantum phenomenon produce a macroscopic classical effect

It must, albeit a very small one. One test of a quantum hypothesis is whether you can derive a macroscopic observable simply by integrating lots of quantum events. A photon has no knowledge of the existence of others, so radiation pressure cannot suddenly come into being when n > 1!

[Eternal Student (OP)]

Hi.

There's nothing wrong with what you've said @paul cotter .   I'm grateful for anyone spending some time here and it is just chat.

The poll has nearly reached the end of its time, so I might as well say a bit more about it.

*   I don't think there is a correct answer, arguments exist on both sides.   However, the most conventional answer or the one which is generally accepted is "yes", the particle would exert pressure on the walls.

   The general reasoning is of some small interest and easily understood.  So I'll take 2 minutes and present that here.

The energy that a particle in a box can have is given by: 
E =   

[Equation 1]

with L = length of the box,  m = mass of the particle,  h = plancks constant   and  n = 1,2,3,4,......   
   The derivation of that can be found in most places discussing Quantum Mechanics.   The formula holds for a 1-dimensional, 2-D or 3-D box without adjustment provided that in the higher dimensions we insist the box is a square or cube - it has the same length along all sides.
    Now you just assume that the walls of the box were not fixed in place but could be moved a small amount.   We'll have the length increase by  δL >0 .    By [Equation 1] this causes the energy of the particle in the box to drop,  we have a change in the energy  δE.
     We'll have a closed system, so that energy can only pass from the particle in the box to/from the walls.  Assuming the law of conservation of energy, then any energy lost by the particle must have been work done on the walls.
The work done on the walls should be  F.δL  =  (Force on the wall)  x (Distance moved).
   So we have   FδL =  -δE       =>     F =   -δE / δL      =>    F =     taking the limit δL -> 0.
From [Equation 1] we then see  F = k / L³       for k = =  a constant.
For a 3-D box we would then have  Pressure = P = F / (Area of a wall) =    k / L⁵.

    Anyway, that's the basic argument to show that a pressure is exerted on the walls.   Arguments for and against can be made just by taking the assumptions of this method apart piece by piece.

Examples:   (i)   You may have noticed that E as given by [Equation 1] does depend on n,  the quantum number of the particle.   There is, a priori, no reason why the particle had to start and end with that same quantum numbers for a change in length of the box.   Take n = 1 and L = 1m  initially.    Adjust to n=2 and L = 2m  finally.   From [Equation 1]  we will still have initial Energy = final Energy, no work had to be done on the walls.
    You could try and argue that the box had a length = 1.5m  and all fractions between 1m and 2m during that change and there was no suitable integer n that could maintain a constant energy for the particle in the box.   However, you could have had the particle exert force on the walls for half of the expansion and then retard the walls to recover work from it for the other half of the expansion.   While a "negative pressure" is not something we would like in ordinary Newtonian mechanics, it is exactly what we need in theories like General relativity.  We cannot explain the current expansion of space without assuming the existence of a component of the cosmological fluid that does exert negative pressure.  So you can dismiss complaints about "negative pressure" and just recognise that if QM permits "tunneling" and other strange effects then negative pressure cannot be taken out of the running.
     I said "a priori" and we should qualify that a litle:  A state does not evolve in time arbitrarily.  Unless a measurement is made, the time dependant Schrodinger equation is taken as the only thing which governs the evolution of the state.   However, to apply it rigorously, the potential V which the particle experiences will need to be time dependant.   As the walls move, the potential barrier is retreating.  The full evolution of the state of the particle is going to be complicated and no-one  (well no-one I've found from an hours search on the internet) seems to have done the calculation. It could very well be that exactly how you allow the wall to move affects the final state of the particle in the box, i.e. it's not enough to know that you end up with a Length L = 2m,  you need to know how you got there.

---  This post is already too long.  I'm ending.   There are several arguments for and against and I'm sure I don't know half of them ----

Best Wishes.

[Bored chemist]

In reality, a particle in a box exerts a pressure on the walls.
Why would using a QM description of it stop that?

[Eternal Student (OP)]

Hi.

In reality, a particle in a box exerts a pressure on the walls.
Why would using a QM description of it stop that?

   The model is just a model of one thing.  It is not a model of everything that exists or can be exhibited in the universe.

    A Newtonian model of a marble rolling down a hill is a perfectly fine model.  In reality, the marble was red.  The Newtonian description of a sphere rolling down an inclined plane just isn't going to tell you that

    For the "particle in a box" model,  the walls were not included as a quantum mechanical object.   One reasonable argument is that the QM model cannot say much about the walls until you stop treating them as Newtonian objects and instead consider a much more complicated model.   For example, replace the walls with a million particles each of which has its own description as a QM object and is included in the wave function of the system.

Best Wishes.

[paul cotter]

Hi again ES. I didn't express myself very well: what I meant to say is that your original question involves the use of classical mechanics and quantum theory in an interaction-can one do this in a rigorous manner? Previously I have been castigated( not really, only joking ) for mixing these two methodologies together. 

[Eternal Student (OP)]

Hi.

what I meant to say is that your original question involves the use of classical mechanics and quantum theory in an interaction-can one do this in a rigorous manner?

   Well yes and no.

No  -->  There are lots of replies (especially from me) already stating that at some point a jump from Quantum mechanics to Newtonian mechanics is required.   This is dangerous and reduces the validity.   There has been one very recent reply suggesting that perhaps you do need to start modelling the wall as a quantum mechanical object in its own right   etc.

Yes --->  Almost exactly this question, much as it appeared in the poll, can be asked in undergraduate lectures about Physics.   Usually it's just a discussion but sometimes it's homework, or even a multiple choice exam question.
   See:  https://www.physicsforums.com/threads/force-exerted-by-a-particle-in-a-box-on-the-boundary.935415/
for one example from 2017.
    What I mean is that the sense of the question and the validity of the answer is sufficiently well accepted that it could even be a multiple choice question like the one the poor lady was facing in 2017  - there is no room for any discussion in a multiple choice question.
     More generally, a whole bucket of results in thermodynamics and statistical mechanics are based on starting with a Quantum Mechanical model and quickly passing over to a macroscopic effect or property that can be measured.   This is usually done by allowing an expected value of a measurement of a QM system (i.e. an average as if the measurement was done many times) to be regarded as a macroscopic property that would be present in classical physics.
     Overall QM is just too complicated to model everything with it and "bridges" to macroscopic and classical physics are required all the time.   In a forum like this, we at least have some time and space to discuss the limitations of the model and the bridges that were taken to reach a macroscopic effect.

Best Wishes.

[alancalverd]

The energy that a particle in a box can have is given by:

And there is the weakness in the argument - inadequate specification. 

These are the only possible states for an ideal particle making elastic collisions with the sides of a rigid box. As soon as you introduce the possibility of movement by δL you have broken the boundary conditions.

That's where the quantum model needs to be replaced by the classical continuum if you want to derive radiation pressure of a macroscopic system by differentiating energy density with respect to length. As long as the results are consistent, no problem - and they are.

[Eternal Student (OP)]

Hi.

*  Poll closes in a few hours.
*  Open to guests.
*  Members can alter their selection.   I wouldn't bother - but I think I did tick that option when I set it up.

    There is no great reward but equally no penalty for voting.   Thank you to everyone who has spent any time here.

Best Wishes.

[Bored chemist]

QM model cannot say much about the walls

Nope,
But observation can.
A particle in a box is observed to exert a  pressure on the walls.
If QM says it doesn't then it's not reality which is wrong.

[alancalverd]

For the "particle in a box" model,  the walls were not included as a quantum mechanical object. 

Neither was the particle! The idealised classical model led to the concept of quantised energy levels. Cart/horse inversion strikes again?

Once you have established the notion of the probability wave function, you can consider the possibility of a nonzero value inside the boundary of a real wall, and the concept of tunneling makes sense, but it's not a good starting point.

[Eternal Student (OP)]

Hi.

A particle in a box is observed to exert a  pressure on the walls.

    Yes, I would have thought so   BUT  I haven't tried it.
Trying to get one small particle, say an electron, into a box is tricky.   It's even harder if you want to make sure it isn't just absorbed into the walls and I'll guess you need very delicate equipment to measure a tiny force on the walls.

Neither was the particle! The idealised classical model led to the concept of quantised energy levels. Cart/horse inversion strikes again?

    Yes.  Cart/horse inversion.  Historically, certain things lead toward the development of modern Quantum mechanics but modern QM is a self-contained and hopefully consistent system now.   It does not need to maintain any of those pre-1925 assumptions, many of those assumptions were dropped and NOT carried forward into the postulates of modern QM.   It is perfectly reasonable to start from the postulates of modern QM and work outwards from there.   The modern "particle in a box" model is something that can be derived using modern QM.   This is only 6 postulates (depending on how you write them down but you can do it sensibly in 6 short postulates),  none of which mention anything like rigid walls, elastic collisions or the QM object acting like an idealised particle  (see your sentence which is quoted next).

These are the only possible states for an ideal particle making elastic collisions with the sides of a rigid box.

     I've noticed a few posts you've made that seem to be based on some belief that historical and discontinued theories and models must still be adhered to.    It is certainly interesting to consider the history and perhaps that is the "bridge" that you would choose to take.  By the term "bridge" I mean your preferred method to link the modern QM "particle in a box" model to much more macroscopic and classical phenomena but it is not the only bridge or option available and it does not make your choice of bridge more correct or absolute.   
     Modern QM is under no obligation to maintain all the assumptions of some theory which preceded it.

Best Wishes.

[alancalverd]

I refer the hon gent to the previous post # 30 by my learned friend BC. If modern QM can't predict radiation pressure, it is wrong.

Planck's model was just that - a model from which you can derive useful hypotheses such as the quantisation of electron energy levels in an atom, and the black body spectrum, which stand up to experimental investigation. It is not a "discontinued theory" as it doesn't actually imply that these phenomena are due to particles rattling about in boxes, nor does it make any assumptions: it is merely a mathematical analysis of an ideal particle in an ideal box.

And it probably wouldn't apply to an electron in a real box. One of the problems we have with  designing x-ray tubes is that electrons, being charged,  tend to stick to the sides instead of making nice elastic collisions. But the x-ray spectrum is pretty much as Planck predicts.

A lot of quantum mechanics actually derives from the impossibility of classical assumptions, in particular the Bohr atom with orbiting electrons, which is inherently unstable. Is it still taught in schools, I wonder? I did complain when it appeared in the Warwick University Coat of Arms, granted in 1967, about 50 years after it had been abandoned as ludicrous.

[Eternal Student (OP)]

Hi.

Well the poll has closed so it's time to do something with the data.   
However, we're just going to leave everyone on the cliff edge - because that's where we have the greatest potential.   
 

  ba..da...da...  (on the drums)

Best Wishes.