How Sacrosanct is Conservation of Momentum in QM?

[sethoflagos]

Quote

Tunneling limits the minimum size of devices used in microelectronics because electrons tunnel readily through insulating layers and transistors that are thinner than about 1 nm.

I've seen statements such as this kicking around for decades and always been troubled by a niggling doubt. I imagine that the phenomenon can be explained by normal electron scattering and tunnelling without invoking any deviation to conservation of momentum. But is that all there is to it?

Is the momentum vector of a particle subject to variability due to, say, random quantum fluctuation in local field strength? 

This idea seems somewhat belied by the near point image my eyes can make of a distant star, but even so...

My primary contextual interest is in the absolute deterministic nature of gas molecule trajectories between collisions - ie how 'straight' are their paths through free space. 

[exchemist]

39 minutes ago, sethoflagos said:

I've seen statements such as this kicking around for decades and always been troubled by a niggling doubt. I imagine that the phenomenon can be explained by normal electron scattering and tunnelling without invoking any deviation to conservation of momentum. But is that all there is to it?

Is the momentum vector of a particle subject to variability due to, say, random quantum fluctuation in local field strength? 

This idea seems somewhat belied by the near point image my eyes can make of a distant star, but even so...

My primary contextual interest is in the absolute deterministic nature of gas molecule trajectories between collisions - ie how 'straight' are their paths through free space. 

I suppose the expectation value of a measurement will be in accordance with conservation, which would seem to leave a bit of Heisenbergian wiggle room on individual measurements.  But I'm not a physicist. 

[Genady]

In QM, as well as in QFT, momentum is conserved. Lagrangians are translationally invariant. Momentum conservation follows by the Noether's theorem.

[sethoflagos]

1 hour ago, Genady said:

In QM, as well as in QFT, momentum is conserved. Lagrangians are translationally invariant. Momentum conservation follows by the Noether's theorem.

Presumably this is consistent with a classical picture of momentum exchange in the interaction of a gas molecule and a photon.

How about interaction with a random quantum fluctuation? Can the gas molecule 'leak' momentum into displacement of such 'virtual' particles?

 

1 hour ago, exchemist said:

I suppose the expectation value of a measurement will be in accordance with conservation, which would seem to leave a bit of Heisenbergian wiggle room on individual measurements.  But I'm not a physicist. 

I'm really asking about phenomena distinct from the measurement problem. I can buy some fundamental uncertainty in the actual path taken. More, I'm trying to narrow it down from some kind of random walk to a spectrum of possible paths all of which are straight lines.

[exchemist]

16 minutes ago, sethoflagos said:

Presumably this is consistent with a classical picture of momentum exchange in the interaction of a gas molecule and a photon.

How about interaction with a random quantum fluctuation? Can the gas molecule 'leak' momentum into displacement of such 'virtual' particles?

 

I'm really asking about phenomena distinct from the measurement problem. I can buy some fundamental uncertainty in the actual path taken. More, I'm trying to narrow it down from some kind of random walk to a spectrum of possible paths all of which are straight lines.

It’s not a measurement problem though. It’s an uncertainty built into reality. I only mention “measurement” as that is something that resolves uncertainty into a defined value in the course of an interaction. As I understand it, momentum and energy are conserved on average, but there is a fuzzy halo around individual values for a given QM entity at a given instant.
 

 

 

 

[sethoflagos]

29 minutes ago, exchemist said:

As I understand it, momentum and energy are conserved on average, but there is a fuzzy halo around individual values for a given QM entity at a given instant.

That's actually quite a big deal.

[exchemist]

6 minutes ago, sethoflagos said:

That's actually quite a big deal.

Yes. That's why I would welcome comment on my understanding of this point by a real physicist. @Genady's previous contribution did not seem to me to tackle it head on.

But for instance, vacuum fluctuations imply a temporary violation of conservation of energy, I think, which averages out to zero. 

 

[Genady]

Momentum is conserved on every line and at every vortex of every Feynman diagram. 

The path integral calculation sums up all available paths, not only straight ones.

1 hour ago, exchemist said:

vacuum fluctuations imply a temporary violation of conservation of energy, I think, which averages out to zero

There is an inconsistency in this claim: energy is non-negative, so if it becomes positive even temporarily, its average cannot be zero.

[swansont]

4 hours ago, sethoflagos said:

I've seen statements such as this kicking around for decades and always been troubled by a niggling doubt. I imagine that the phenomenon can be explained by normal electron scattering and tunnelling without invoking any deviation to conservation of momentum. But is that all there is to it?

What is the connection to conservation of momentum? The issue is tunneling.

10 minutes ago, Genady said:

There is an inconsistency in this claim: energy is non-negative, so if it becomes positive even temporarily, its average cannot be zero.

The contribution can average to zero; the value being slightly larger or smaller than the average.

[exchemist]

8 minutes ago, Genady said:

Momentum is conserved on every line and at every vortex of every Feynman diagram. 

The path integral calculation sums up all available paths, not only straight ones.

There is an inconsistency in this claim: energy is non-negative, so if it becomes positive even temporarily, its average cannot be zero.

Hmm, fair point about energy. I'm afraid I don't know QFT, so I am not sure of the connection between the path integral formalism and the uncertainty principle. AS I understand the concept of expectation values of a property it is the average result one would get from a series of measurements on a series of identical systems. Some individual members of the series would be below and some above. So if the expectation value corresponds to the value predicted by conservation, some of the results might not. Is this a wrong picture?

[Genady]

6 minutes ago, swansont said:

The contribution can average to zero; the value being slightly larger or smaller than the average.

Please explain: if the average is zero, then a slightly smaller value would be negative, wouldn't it?

5 minutes ago, exchemist said:

Hmm, fair point about energy. I'm afraid I don't know QFT, so I am not sure of the connection between the path integral formalism and the uncertainty principle. AS I understand the concept of expectation values of a property it is the average result one would get from a series of measurements on a series of identical systems. Some individual members of the series would be below and some above. So if the expectation value corresponds to the value predicted by conservation, some of the results might not. Is this a wrong picture?

Energy is conserved in every possible outcome, not only on average. In no quantum or particle experiment conservation of energy was ever violated.

[swansont]

15 minutes ago, Genady said:

Please explain: if the average is zero, then a slightly smaller value would be negative, wouldn't it?

The linewidth of a de-excitation transition is related to the lifetime, owing to the uncertainty relation. The nominal value of the transition might be e.g. 1 eV, but the value of any particular photon might be slightly higher or lower than 1 eV. Not a negative amount, since the linewidth is on the order of MHz to GHz, but a negative contribution because the energy is smaller than the average

[Genady]

20 minutes ago, swansont said:

The linewidth of a de-excitation transition is related to the lifetime, owing to the uncertainty relation. The nominal value of the transition might be e.g. 1 eV, but the value of any particular photon might be slightly higher or lower than 1 eV. Not negative, since the linewidth is on the order of MHz to GHz. 

Right. The fluctuations are above and below a positive average, but not below zero average.

Edited January 27 by Genady

[swansont]

4 hours ago, Genady said:

Right. The fluctuations are above and below a positive average, but not below zero average.

What’s an example of a fluctuation about zero?

[Genady]

4 minutes ago, swansont said:

What’s an example of a fluctuation about zero?

We were discussing this example:

 

6 hours ago, exchemist said:

for instance, vacuum fluctuations imply a temporary violation of conservation of energy, I think, which averages out to zero.

 

[swansont]

2 hours ago, Genady said:

We were discussing this example:

Vacuum fluctuations are not an example of variations about zero energy, though. The energy of the vacuum - zero-point energy - is not zero. 

[Genady]

2 minutes ago, swansont said:

Vacuum fluctuations are not an example of variations about zero energy, though. The energy of the vacuum - zero-point energy - is not zero. 

Exactly. This was my message:

 

7 hours ago, Genady said:

its average cannot be zero.

 

[sethoflagos]

15 minutes ago, swansont said:

Vacuum fluctuations are not an example of variations about zero energy, though. The energy of the vacuum - zero-point energy - is not zero.

Exactly. Can we now develop @exchemist's point and address whether the random quantum fluctuations of the vacuum state can perturb the momentum of a gas molecule?

Edited January 27 by sethoflagos
typo

[swansont]

2 hours ago, Genady said:

Exactly. This was my message:

 

 

exchemist didn’t say it was. They said the average perturbation was zero.

2 hours ago, sethoflagos said:

Exactly. Can we now develop @exchemist's point and address whether the random quantum fluctuations of the vacuum state can perturb the momentum of a gas molecule?

I thought Genady addressed this. 

[exchemist]

9 hours ago, sethoflagos said:

Exactly. Can we now develop @exchemist's point and address whether the random quantum fluctuations of the vacuum state can perturb the momentum of a gas molecule?

I've had a thought about what may be the source of my confusion.

It only makes sense to apply conservation laws in the context of an interaction. An interaction "collapses the wave function", or at any rate the property in question takes on a specific value at that instant. So then there is no uncertainty principle issue, as the value of the property has become definite.

(But I am conscious of struggling with half remembered stuff from half a century ago, so I may well be speaking ex ano.)   

[sethoflagos]

10 hours ago, swansont said:

I thought Genady addressed this. 

Not explicitly.

 

[MigL]

Going by  purely Heisenberg considerations, if at a certain scale translational symmetry ceases to have meaning, can a classical theorem like Noether still say anything about momentum conservation ?
If position and momentum are 'fuzzy', or smeared out, translational symmetry and momentum conservation must necessarily be also.

Feynman diagrams  essentially put a box around an interaction within which all possible series of interaction can happen. The largest series of interactions, with the most nodes are least likely to be realizable as they describe virtual interactions, which are 'off shelf' and don't need to satisfy energy-momentum considerations.

I would think that means our current models do not necessarily conserve energy or momentum at small enough scales.
The conservation condition is approached, or emerges, when off shelf virtual particle effects become negligible, and you 'zoom back' out of the Feynman box.
And it is already exceedingly difficult to keep the few electrons trapped in narrow, shallow potential wells, at current feature size of modern semiconductors in order to define logic levels, but Moore's Law has had a good run, going from a couple of thousand transistors 50 years ago, to over 100 billion in Apple's latest ARM implementation.

 

[sethoflagos]

Am I right in reading 'off shelf' as 'off shell' ie not a solution of the mass/energy equivalence eqn?

[MigL]

Sorry.
I guess I picked the wrong choice from the auto-correct suggestions.

[swansont]

On 1/27/2024 at 10:27 AM, Genady said:

Momentum is conserved on every line and at every vortex of every Feynman diagram. 

The path integral calculation sums up all available paths, not only straight ones.

Quote

Energy is conserved in every possible outcome, not only on average. In no quantum or particle experiment conservation of energy was ever violated.

 

4 hours ago, sethoflagos said:

Not explicitly.

 

The above seems explicit to me. From your view, what issue isn’t being addressed?

Momentum is conserved in interactions.

The notion in QM that light slows down in a medium but takes a straight path is because the virtual excitations take time but don’t result in a real absorption because momentum has to be conserved - there’s nothing there to recoil. Only the straight path is permitted.