Quantization & Indetermination

In summary: I don't understand what you're getting at.The reasoning is; - The position of a particle multiplied by its momentum is the action of the particle - The action of the particle is quantized, that's it, is an integer number multiplied by the constant of Planck - So being, the product of position by momentum must be an integer number multiplied by the constant of Planck - A theory that incorporates the fact that action is quantized, and at the same time the contradictory assumption that the position and momentum can adopt any value, will predict a minimum error of about a quantum of action in the measures - For example, the product of position by momentum could result in any value, say
  • #1
Rigel XIX
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Hello,

Is the quantization of the action the origin of the indetermination?

The reasoning goes like this:

- Just talking abut one example, a single particle with position and momentum - keep it simple -
- The measure of the position could have any value, in principle
- The same for the measure of the momentum
- However, the simultaneous measure of both of them cannot, as their product can only be an integer number multiplied by the constant of Planck
- This is so because, as far as the measures are simultaneous, they are in fact a measure of the action of the particle (it is enough to multiply them to obtain the action)
- Neglecting the fact that the measures are actually linked, results is a error in the measure of about a the magnitude of the constant of Planck, even in theory, even in a mental experiment

Is this what the Principle of Heisenberg says for position & momentum?

Thanks beforehand for any response, and please, keep it B-Level, if possible

Thanks!
 
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  • #2
Rigel XIX said:
- However, the simultaneous measure of both of them cannot, as their product can only be an integer number multiplied by the constant of Planck
The product can be anything.
The product of the uncertainty has a minimum, but that is a different statement. This product is also not quantized.
Rigel XIX said:
- Neglecting the fact that the measures are actually linked, results is a error in the measure of about a the magnitude of the constant of Planck, even in theory, even in a mental experiment
I don't understand that point.
 
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  • #3
The reasoning is;

- The position of a particle multiplied by its momentum is the action of the particle
- The action of the particle is quantized, that's it, is an integer number multiplied by the constant of Planck
- So being, the product of position by momentum must be an integer number multiplied by the constant of Planck
- A theory that incorporates the fact that action is quantized, and at the same time the contradictory assumption that the position and momentum can adopt any value, will predict a minimum error of about a quantum of action in the measures
- For example, the product of position by momentum could result in any value, say 27,4*h (J*s) But the theory knows that it cannot correspond to anything real, as the action of the particle is quantized, and therefore predicts a mistake in the measure, simply because it could be 27*h or 28*h, but not 27,4*h

And so, the question is if the indetermination in position and momentum is a consequence of the quantification of the action

Thanks !
 
  • #4
You misunderstand what an action is. Multiplying position and momentum doesn’t give you anything interesting. In particular, it depends on the completely arbitrary and meaningless definition of the coordinate system.
 
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  • #5
mfb said:
You misunderstand what an action is. Multiplying position and momentum doesn’t give you anything interesting. In particular, it depends on the completely arbitrary and meaningless definition of the coordinate system.
... just to emphasise this point. If you take your origin to be where the particle was measured to be, then the product of position and momentum is zero.
 
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  • #6
mfb said:
You misunderstand what an action is. Multiplying position and momentum doesn’t give you anything interesting. In particular, it depends on the completely arbitrary and meaningless definition of the coordinate system.
Hm, I'd say that (orbital) angular momentum, ##\vec{r} \times \vec{p}##, is pretty interesting ;-).

Also in statistical physics (which is much more self-consistent within QT than classical physics to begin with) the natural unit of phase-space volumes is ##h^{3N}=(2 \pi \hbar)^{3N}##, since the number of microstates of an ideal gas in a phase-space cell is ##g \mathrm{d}^{3N} x \mathrm{d}^{3N} p/h^{3N}##. Here ##N## is the number of particles and ##g=2s+1## the spin-degeneracy factor.
 
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  • #7
It is a B-level thread, and I’m sure OP was not talking about the cross product of position and momentum, or phase space in statistical mechanics.
 
  • #8
You misunderstand what an action is. Multiplying position and momentum doesn’t give you anything interesting. In particular, it depends on the completely arbitrary and meaningless definition of the coordinate system.
... just to emphasise this point. If you take your origin to be where the particle was measured to be, then the product of position and momentum is zero.

This is the point. Let's look at the issue the other way around;

Reference systems are a human device, while action - and the fact that is quantized - looks to be something real, because there is a constant of Planck that we didn't invent. Whatever action could be, it is quantized and the value of the quantum of action is the constant of Planck
Assuming that position and momentum could adopt any value when measured simultaneously, is quite the same that to say that we can adopt any reference system at will
This seemingly does deny the quantization of the action, because the product of position by momentum is expressed in Jules*second. That's an action, it should be quantified
Go and tell the Universe that our reference system denies the quantization of action, or that it says that is cero, and imagine the possible answer
Action is quantized, whatever system denies this reality, exists in our imagination, but doesn't correspond to anything real. All our assumptions are not going to change the value of the constant of Planck, or the fact that it exists
If you imagine a reference system such that action is equal to cero in it, I would say your have done something possible in our imagination - similar to think in terms or space or time separately - but not representing anything real - as action equal to cero is similar to say non-existence -
 
  • #9
Rigel XIX said:
This is the point. Let's look at the issue the other way around;

Reference systems are a human device, while action - and the fact that is quantized - looks to be something real, because there is a constant of Planck that we didn't invent. Whatever action could be, it is quantized and the value of the quantum of action is the constant of Planck
Assuming that position and momentum could adopt any value when measured simultaneously, is quite the same that to say that we can adopt any reference system at will
This seemingly does deny the quantization of the action, because the product of position by momentum is expressed in Jules*second. That's an action, it should be quantified
Go and tell the Universe that our reference system denies the quantization of action, or that it says that is cero, and imagine the possible answer
Action is quantized, whatever system denies this reality, exists in our imagination, but doesn't correspond to anything real. All our assumptions are not going to change the value of the constant of Planck, or the fact that it exists
If you imagine a reference system such that action is equal to cero in it, I would say your have done something possible in our imagination - similar to think in terms or space or time separately - but not representing anything real - as action equal to cero is similar to say non-existence -
Where are you getting all this from?
 
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  • #10
Rigel XIX said:
Whatever action could be, it is quantized and the value of the quantum of action is the constant of Planck

This is a common pop science misconception, but it's still a misconception. The fact that Planck's constant has a particular value does not mean that all actions must be some multiple of that value.
 
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  • #11
Rigel XIX said:
while action - and the fact that is quantized

What makes you think that - or rather it is quantized but probably not in any way a beginner would understand.

We know, from the path integral formalism of Feynman, how to derive the classical action - but exactly how is it quantized? In fact the path integral approach is the analogue of the action in QM. Feynman was not actually the first to figure it out - his hero Dirac was:
http://www.ifi.unicamp.br/~cabrera/teaching/aula 15 2010s1.pdf

See equation 11.

But of course Feynman took it a lot further.

This is not something discussed in beginner texts/popularization's so I would be curious where you got it from? My guess is from a popularization that doesn't explain things correctly - just a guess. They often give meaningless throw away lines like in QM things come in a quantum of action. Its not true.

Another misconception often promulgated in such sources is that energy is quantizied - it isn't - the counter example is a free particle - it can have any energy at all.

Thanks
Bill
 
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  • #12
I think the culprit are the many books (even university-level textbooks) who think you have to present outdated models first. That's why many books contain an introductory chapter working out the socalled "old quantum theory", which was a set of predecessor models of "modern quantum theory". The strongest motivation for the physicists at the time to work out "modern QT" was the inconsistencies and ad-hoc assumptions of "old QT". Particularly in atomic physics the Bohr-Sommerfeld model, which contains the ad-hoc assumption that at least for quasiperiodic systems the action should be quantized in integeger multiples of ##h=2 \pi \hbar##. Already for the helium but also for all other atoms, to get the spectra right, you needed more ad-hoc assumptions, and this dissatisfied of course all physicists involved in this research, including Bohr and Sommerfeld themselves. Particularly Sommerfeld's approach in teaching physics (the most successful approach ever, perhaps only comparable to Feynman's with the clear difference that Sommerfeld produced more Nobel prize winners than anybody else in history of physics) by letting young people jump right away into current reseach proved very successful. Heisenberg was among the people who discovered "modern QT", which is the most successful theory today. That the Bohr-Sommerfeld approach works quite well for the hydrogen energy levels is due to the huge symmetry of the Kepler problem and in this sense a special case.
 
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  • #13
vanhees71 said:
I think the culprit are the many books (even university-level textbooks) who think you have to present outdated models first. That's why many books contain an introductory chapter working out the socalled "old quantum theory",

Dah o:)o:)o:)o:)o:)o:)o:)

Forgot that.

Of course.

Thanks
Bill
 
  • #14
so I would be curious where you got it from?

From nowhere, just my reasoning, assuming the fact that action is quantized. If it is not, the reasoning is wrong

or rather it is quantized but probably not in any way a beginner would understand.

Quite true, in particular, I don't understand equation 11

I will ask this; If action is not quantified, why is so that it cannot be lower than the constant of Planck?

Thanks !
 
  • #15
Rigel XIX said:
If action is not quantified, why is so that it cannot be lower than the constant of Planck?

Why do you think it can't?
 
  • #16
As discussed above already: You can choose to have the particle centered around zero, then the product of expected position and momentum is zero. You can even make both factors zero at the same time.
 
  • #17
mfb said:
As discussed above already: You can choose to have the particle centered around zero, then the product of expected position and momentum is zero. You can even make both factors zero at the same time.

Say that we do it for a photon. It was my understanding that the total action of a photon is always equal to the constant of Planck

Is it so that the action expressed as E*t (Jules*second) is equal to the constant of Planck and the action expressed as position by momentum (Jules*second) is cero? I take the answer is that those products are not actions?

Thanks!
 
  • #18
Rigel XIX said:
It was my understanding that the total action of a photon is always equal to the constant of Planck

Where are you getting this from? Please give specific references. You appear to have a fundamental misunderstanding about what Planck's constant means, but we can't help much in fixing it unless we know where you got it from.
 
  • #19
Rigel XIX said:
Say that we do it for a photon.
A photon doesn't even have a well-defined position. I was talking about massive particles, massless particles are a bit more difficult.
Rigel XIX said:
It was my understanding that the total action of a photon is always equal to the constant of Planck
You keep repeating this misconception, but it stays a misconception. Maybe you are thinking about the photon spin?
Rigel XIX said:
Is it so that the action expressed as E*t (Jules*second) is equal to the constant of Planck and the action expressed as position by momentum (Jules*second) is cero?
I don't understand what you are asking.
 
  • #20
Thanks, mfb, for your explanations
 

1. What is quantization?

Quantization is the process of representing a continuous signal or value with a discrete set of values. It is commonly used in physics and engineering to describe the behavior of particles and energy.

2. How does quantization relate to indetermination?

Quantization and indetermination are closely related because they both describe the limitations of our ability to measure and understand the behavior of particles and energy. Indetermination, or Heisenberg's uncertainty principle, states that it is impossible to know both the position and momentum of a particle simultaneously. Quantization helps us understand this limitation by showing that particles can only exist at certain discrete energy levels.

3. What is the significance of quantization in quantum mechanics?

Quantization is a fundamental concept in quantum mechanics. It allows us to understand the behavior of particles and energy at the microscopic level, where classical physics breaks down. By quantizing energy levels, we can explain phenomena such as the discrete energy levels of atoms and the behavior of subatomic particles.

4. How does quantization affect our understanding of the universe?

Quantization is a crucial component of our understanding of the universe. It has led to breakthroughs in physics and technology, such as the development of transistors and lasers. It also plays a key role in our understanding of the fundamental forces and particles that make up our universe.

5. Can quantization be observed in everyday life?

While quantization is a fundamental concept in physics, it is not directly observable in everyday life. However, its effects can be seen in many technologies that rely on quantum mechanics, such as computer processors and GPS systems. Additionally, the discreteness of energy levels can be observed in phenomena such as the emission of light from atoms and the behavior of electrons in a conductor.

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