Momentum of a stationary particle/wave?

[Philip Land]

We are all familiar with Heisenbergs uncertainty principle. When we determine the position of a particle or wave, the uncertainty of momentum reach infinity.

So let's say I have a machine that measures the position very very precisely. Then the uncertainty of this non-moving particles momentum reach infinity, the momentum could have big magnitude. But how can a particle that don't move have momentum? Or does the particle oscillate with a de broglie wavelength within the discrete finite volume we have measured it to be within (because we can not measure exactly)?

Or does the uncertainty principle don't apply once measurement is done? It just tells uncertainty if I "would" make a measurement?

Feel free to go complex on me.

 

[A. Neumaier]

The uncertainty relation only applies to a fixed moment in time. The position of your particle will be very uncertain the very next moment.

 

[PeterDonis]

lets say I have a machine that measures the position very very precisely. Then the uncertainty of this non-moving particles momentum reach infinity

Who said the particle wasn't moving? You measured its position, not its momentum. You can't say it isn't moving just because you measured its position very precisely one time.

 

[Philip Land]

Who said the particle wasn't moving? You measured its position, not its momentum. You can't say it isn't moving just because you measured its position very precisely one time.

Let me rephrase my question.

Let's say we trap it in a well with infinite potential such that the particle can't escape, the let the size of this well be small. Then I could take the limit until the particle no longer can move in space.

I can easily solve Schrödinger for this case, $\psi(x) = Constant * sin(k_n x)$ for even n, where $k=n \pi / 2a$. where a is half the length of a simple square well.

Decreasing $a$ makes the frequency very high.

As writing this I realize that the only thing that will happen if you trap a particle is that the energy / unit length (in this 1 dim case) will increase and we concentrate the energy which cannot escape.

But dosen't this also mean that now the position is always given at any time, hence we can never know the momentum of this trapped particle? Which is strange because it's fairly straightforward to calculate the energy-levels for each quantum number n.

 

[Nugatory]

Which is strange because it's fairly straightforward to calculate the energy-levels for each quantum number n.

Yes, but the state isn't anyone of those energy eigenstates. It's a superposition of many of them.

 

[PeterDonis]

Then I could take the limit until the particle no longer can move in space.

No, you can't, because an "infinite square well" with $a = 0$ is no longer a square well; it's an infinite potential everywhere, which has no solution.

 

[A. Neumaier]

Then I could take the limit until the particle no longer can move in space.

For any size of the cavity, the particle would oscillate very fast inside the tiny cavity, so that its momentum is unpredictable.

 

[Nugatory]

Original poster might also want to consider the negative delta function potential, which is as close as we can get to a particle that "can only be one place". This is not, of course, the zero-width limit of the infinitely deep square well, for which no solution exists (although solutions do exist for arbitrarily small non-zero widths).

 

[vanhees71]

For any size of the cavity, the particle would oscillate very fast inside the tiny cavity, so that its momentum is unpredictable.

There's no way to define a momentum observable for this case anyway. You can define the Hamiltonian, but not the momentum observable. We have discussed this at length once in this forum. The infinite-square-well potential is not an easy problem despite it's treated by many textbooks as if it were. They never carefully explain the important difference between Hermitean and self-adjoint operators!

For a handwaving treatment, see

https://www.physicsforums.com/threa...uous-in-an-infinite-well.916171/#post-5773830

 

[Vanadium 50]

No, you can't, because an "infinite square well" with a=0a = 0 is no longer a square well; it's an infinite potential everywhere, which has no solution.

Or maybe it's a delta function (which has one bound state).

This is an example of where one needs to be very careful about explaining not just that one is taking a limit, but exactly how they are taking the limit.

Classically, if I take ball bouncing between two walls and bring the walls closer together, the ball speeds up. In the limit that the walls get very close, the position of the ball is well determined, but its momentum is not.

 

[vanhees71]

On top, it's a completely unphysical situation. I've never had a clue what these examples are supposed to teach me. It's a nice mathematical exercise, but no more. Even finite potential steps or wells are not very illuminating for me, although you can study easily some basic notions of scattering theory in 1D motion with them. From the particle-point of view, they don't help much. Note that such potentials imply $\delta$-distribution valued forces for the analogous classical problem.

 

[Philip Land]

This was indeed very tricky. For everyone interested in reading about this particular problem in detail, here's a pdf that goes in depth on self adjoint operators.

 

[A. Neumaier]

There's no way to define a momentum observable for this case anyway.

Only if the cavity has hard walls. Momentum is perfectly well defined in a cavity with steep wall where position can take any value but absolutely large values have an extremely low probability. The prime example is a harmonic oscillator with very large stiffness and hence very large frequency.

 

[A. Neumaier]

it's a completely unphysical situation. I've never had a clue what these examples are supposed to teach me.

It is an excellent approximation for a particle in a tiny cavity viewed from far away. The details of the cavity potental no longer matter, and the (renormalized) delta-potential says all there is to say. It is also the simplest (and hence useful) toy model where to study renormalization.

 

[vanhees71]

Only if the cavity has hard walls. Momentum is perfectly well defined in a cavity with steep wall where position can take any value but absolutely large values have an extremely low probability. The prime example is a harmonic oscillator with very large stiffness and hence very large frequency.

Sure. It's much more valuable to study the harmonic oscillator in great detail than to deal with mathematically complicated artificial problems. Of course, you could as well argue that the harmonic-oscillator potential is also always an approximation...

 

[hilbert2]

To me, the original question seems to be based on a belief that the property $\left<p\right>=0$ for a stationary state of 1D quantum system would imply that $\left<p^2 \right>$ and the variance of momentum should also be zero, which isn't true.

And I don't think the state that a quantum system collapses into upon a position measurement is very likely to be a stationary state. To make it become a stationary state you should measure the total energy observable instead.

 

[vanhees71]

Despite the fact that I don't believe in the collapse religion, I don't understand your statement. Usually stationary states (eigenstates of the Hamiltonian) do not imply well localized particles (see the stationary states of the hydrogen atom as an example).

 

[A. Neumaier]

To make it become a stationary state you should measure the total energy observable instead.

But there is no experiment that would allow you to do that...

 

[hilbert2]

Despite the fact that I don't believe in the collapse religion, I don't understand your statement. Usually stationary states (eigenstates of the Hamiltonian) do not imply well localized particles (see the stationary states of the hydrogen atom as an example).

Ok, I think I misinterpreted the "stationary" in the threat title to mean "solution of time-independent SE", while it probably means "something with very small variance of position".

 

[vanhees71]

That's how I also understand "stationary". That's why I was puzzled by your claim that then the particle's position should be pretty well determined. Of course, the OP #1 is a common misunderstanding of the meaning of the uncertainty relation. This is due to Heisenberg, who didn't understand his own uncertainty relation right in the beginning. He submitted a paper during Bohr's absence from Copenhagen, and the damage was done. Bohr immediately corrected the misunderstanding by Heisenberg, but it was too late, and the wrong interpretation stuck. Errors are particularly persistent by diffusing rapidly through the textbook literature. That's a kind of Murphy's law of science communication.

The correct interpretation according to Born's Rule and the minimal statistical interpretation is that the uncertainty relation is a general property of all quantum states of a particle, i.e., due to its preparation. If you prepare the particle well localized, i.e., with small $\Delta x$, then necessarily the momentum of the particle is pretty undetermined, i.e., $\Delta p$ is large (and vice versa). This has nothing to do with how accurately you can measure position or momentum, which you can do in principle with arbitrary precision (of course the technical difficulty rises with the demand of higher precision, but that's not a principle obstacle for measuring either position or momentum precisely).

Now take a free particle. Its energy $E=\vec{p}^2/(2m)$. So for a stationary state the momentum should be completely determined (which is impossible since the corresponding plane-wave solution of the Schrödinger equation is not a proper state, because it's not square integrable) and thus the position is completely undetermined. Physicswise this means to get an approximately stationary state for a free particle the standard deviation of momentum should be very small, implying through Heisenberg's uncertainty relation that the standard deviation of position must be very wide.

This is also true, if there are bound states, i.e., true stationary states like the electron in a hydrogen atom in its ground state. It's not very well localized too.

 

[Philip Land]

But there is no experiment that would allow you to do that...

That's how I also understand "stationary"

I have one essential question. The uncertainty in momentum and space, is that the "size" of the interval we can find it within OR number of potential states in the given interval? Because I assume the former in my original question below:

If we restrict the space that a particle can be in, this case by a theoretical infinite square well. The uncertainty in position is no longer big, since the particle has to be in the well. And the smaller we make the well, the more accurately we know its position without actually measuring it.

This mean that uncertainty in momentum gets bigger as we shrink the well.

Hence the momentum ($\alpha$ Energy) could be 'extremely' large. How do we explain a particle that is in rest (since it must me in the very small well) having almost infinite momentum? Where does all that momentum comes from? Vacuum fluctuation we have 'forced' the presence of?

I do know that high uncertainty in momentum does not equal high momentum, but potentially high momentum. So the case above is possible. And sorry for my abuse of language, just trying to make this as intuitive as possible.

 

[Nugatory]

How do we explain a particle that is in rest (since it must me in the very small well) having almost infinite momentum?

As several posters have already pointed out, the particle is NOT at rest. You can imagine it bouncing back and forth between the walls of the well as a way forming an intuitive picture of how this might be.

Be aware, however, that this classical image is misleading in several ways. First, it suggests that the particle has a position within the box and a momentum even when neither of these quantities have been measured. Second, it suggests that reducing the width of the well to zero would give you an immobile particle with a fixed position, and that doesn't work - set the width to zero and there is no solution.

Where does all that momentum comes from?

The total momentum of the system comprised of the particle and the measuring device is conserved.

 

[vanhees71]

I have one essential question. The uncertainty in momentum and space, is that the "size" of the interval we can find it within OR number of potential states in the given interval? Because I assume the former in my original question below:

If we restrict the space that a particle can be in, this case by a theoretical infinite square well. The uncertainty in position is no longer big, since the particle has to be in the well. And the smaller we make the well, the more accurately we know its position without actually measuring it.

This mean that uncertainty in momentum gets bigger as we shrink the well.

Hence the momentum ($\alpha$ Energy) could be 'extremely' large. How do we explain a particle that is in rest (since it must me in the very small well) having almost infinite momentum? Where does all that momentum comes from? Vacuum fluctuation we have 'forced' the presence of?

I do know that high uncertainty in momentum does not equal high momentum, but potentially high momentum. So the case above is possible. And sorry for my abuse of language, just trying to make this as intuitive as possible.

The standard deviation of a quantity $A$ is defined by
$$\Delta A^2=\langle A^2 \rangle-\langle A \rangle^2.$$
This is a socalled cumulant of the distribution function. If you know all cumulants or moments, you can reconstruct the distribution function. If you have only the mean value and the standard deviation, you have incomplete knowledge about the distribution function. Then information theory tells you that the best you can do with the given information is to use a Gaussian distribution with this mean value and standard deviation. This then tells you that the probability to find a value for $A$ in an interval of twice the width of the standard deviation is around 68%. Have a look at the first figure in

https://en.wikipedia.org/wiki/Standard_deviation

As I already stressed, the infinite-square well is a very bad example to argue about localizability and the position-momentum uncertainty since you cannot define a proper momentum observable in this case.

The most simple case, you should carefully study, are Gaussian wave packets, where you can easily tune the position uncertainty (uncertainty=standard deviation; it's just the jargon of the Copenhagen gang trying to confuse the issue) and then calculate the standard deviation ("uncertainty") of momentum. Then you can also analytically solve the time evolution via the time-dependent Schrödinger equation. I'm sure that with this exercise, you'll get a pretty good understanding what's going on concerning the position-momentum uncertainty relation.

 

[A. Neumaier]

How do we explain a particle that is in rest (since it must me in the very small well) having almost infinite momentum?

Did you ever solve a problem with a square well? In the most general state the expectation of the position and the velocity oscillates with a superposition of very high frequencies, corresponding to the energy levels obtained by diagonalizing the Hamiltonian. Thus the particle is never at rest though it is confined to a very small region, and the expected momentum (mass times velocity) is almost always huge and oscillating in sign.

 

[bob012345]

There's no way to define a momentum observable for this case anyway. You can define the Hamiltonian, but not the momentum observable. We have discussed this at length once in this forum. The infinite-square-well potential is not an easy problem despite it's treated by many textbooks as if it were. They never carefully explain the important difference between Hermitean and self-adjoint operators!

For a handwaving treatment, see

https://www.physicsforums.com/threa...uous-in-an-infinite-well.916171/#post-5773830

Can you explain the physical implications of the difference between Hermitian and self-adjoint operators. The terms seem to be used interchangeably. There may be different mathematical nuances in linear algebra but don't they both refer to operators with real observables? Thanks.

 

[Ian J Miller]

As others have pointed out, if you merely measure exactly the position of the particle, you know absolutely nothing about the momentum, therefore you cannot say it is stationary. A stationary wave is slightly different because it refers to the state. As in classical stationary waves, it is mathematically equivalent to two equal and opposite waves going in opposite directions. If you look at the wave as usually given for a particle in a potential well, you get a crest but no trough. Net momentum may be zero, but the particle has speed and kinetic energy.

 

[vanhees71]

Can you explain the physical implications of the difference between Hermitian and self-adjoint operators. The terms seem to be used interchangeably. There may be different mathematical nuances in linear algebra but don't they both refer to operators with real observables? Thanks.

The only reason, why this very sloppy use of math by many physicists works out is "due to the extremely good nature of separable Hilbert space, which is nearly as simple as a finite-dimensional unitary vector space" (according to my functional-analysis professor). On the other hand, it's not difficult to find examples, where get nonsensical results:

https://arxiv.org/abs/quant-ph/9907069

 

[A. Neumaier]

Despite the fact that I don't believe in the collapse religion

Why then do you teach the collapse (without giving it this name) in your lecture notes on statistical mechanics where you define the foundations? In the version of 05th March 2019, you state after (2.14) that

After a simultaneous measurement the system is in a corresponding simultaneous eigenstate.

Thus you state explicitly that the system is no longer in the state given by the unitary dynamics, but in the state projected to the eigenspace corresponding to the measurement. This is the collapse, even in a version known to be invalid; see, e.g., Section 7 of the quantum mechanics volume 3 by Landau and Lifschitz, 4th paragraph after (7.8).

You make use of the collapse in the paragraph preceding (2.1.6):

This preparation of a system is possible by performing a precise simultaneous measurement of a complete complete set of observables.

You also refer to it in the second paragraph of Section 2.2:

we know from the above mentioned basics that the system is in an eigenstate of the corresponding self-adjoint operator

 

[vanhees71]

As I said, these notes are very old, and I've not the time to reformulate everything in terms of the minimal interpretation right now. However, even within the minimal interpretation, if you've prepared a complete set of compatible observbles, you are in the corresponding pure state. Of course, that's not possible for position and momentum, because there is no such pure state, because plane waves (for momentum eigenstates expressed in the position representation) or Dirac $\delta$ distributions (for position in the position representation) are no proper states.

 

[bob012345]

As several posters have already pointed out, the particle is NOT at rest. You can imagine it bouncing back and forth between the walls of the well as a way forming an intuitive picture of how this might be.

Be aware, however, that this classical image is misleading in several ways. First, it suggests that the particle has a position within the box and a momentum even when neither of these quantities have been measured. Second, it suggests that reducing the width of the well to zero would give you an immobile particle with a fixed position, and that doesn't work - set the width to zero and there is no solution.
The total momentum of the system comprised of the particle and the measuring device is conserved.

If the particle has no momentum until it's measured then the question I've always had is what does that mean and must it involve a sentient being doing the measurement? I think that's been debated for a very long time...so I don't expect a definitive answer.

 

[Nugatory]

If the particle has no momentum until it's measured then the question I've always had is what does that mean ?

It means that it has no momentum the same way that I have no lap when I'm standing up and no fist when my hand is open with fingers outstretched.

and must it involve a sentient being doing the measurement?

No. An inanimate device whose output no one ever looks at is quite sufficient.

 

[bob012345]

It means that it has no momentum the same way that I have no lap when I'm standing up and no fist when my hand is open with fingers outstretched.No. An inanimate device whose output no one ever looks at is quite sufficient.

If an inanimate device is sufficient, then that device could conceivably be any other particle since any interaction might be considered a "measurement". Then, everything would be in a definite state between "measurements" which in the real world continuously happen.

 

[DarMM]

It means that it has no momentum the same way that I have no lap when I'm standing up and no fist when my hand is open with fingers outstretched.

What exactly do you mean by this? Do you mean that fundamentally quantum objects don't have an energy/momentum/position unless forced to adopt one by interaction with a classical object?

 

[Nugatory]

?...since any interaction might be considered a "measurement"...

Not "any" interaction, but "any thermodynamically irreversible" interaction. Yes, and that's why Schrödinger's cat isn't in a superposition of dead and alive before we open the box - the cat is continuously interacting irreversibly with its environment, and this is generally the case for all the macroscopic objects around us. The phenomenon is called "decoherence", and it explains why the macroscopic world generally behaves classically. The particle in a box is a different situation altogether; it is interacting only with the square-well potential, reversibly.

You can google for "quantum decoherence", although the math might be a bit daunting. David Lindley's book "Where does the weirdness go" is a layman-friendly overview worth reading if you don't want to take on the math.

 

[Nugatory]

What exactly do you mean by this? Do you mean that fundamentally quantum objects don't have an energy/momentum/position unless forced to adopt one by interaction with a classical object?

Not quite... but the context for the quote goes back to #22 of the thread.