# Scaling - Inverse relationship between uncertainty and mass

Scaling - Inverse relationship between uncertainty and mass

I’m trying to express Heisenberg's Uncertainty Principle in a simplified formula that is not boundary unlimited and still capture what I believe is an inverse relationship between uncertainty and mass - the "scaling hypothesis".

I started with this mathematical definition of uncertainty: h: m Delta x Delta v >h which I got from a lecture by Prof. Wolfson. As usual, m = mass; delta x = the uncertainty in position, delta v = the uncertainty in velocity not momentum but I don't think that matters here; and h = Planck’s constant). I don't know what the equation form of this definition is but later in his lecture, Prof. Wolfson says: "If we look at the mass and divide both sides of the equation by m [mass], we get: [Uncertainty =] Delta x Delta v greater than h over the mass. He goes on to say: "That means for massive objects like a tennis ball or me or a car, the product of the uncertainty in position and uncertain velocity is miniscule."

I took this to be mathematical support for what I'm calling the "scaling hypothesis". A friend, while not disagreeing with the claim that uncertainty decreases as mass increases, says the math does not support it. Why? He says the equation contains a boundary variable which I'm guessing is the >h factor. If in fact he is right, I would like to restate the equation to remove the boundary and retain the uncertainty/mass relationship.

Prof. Wolfson does not express any concern about the inclusion of >h in the equation leaving me to believe he thinks the math supports his conclusion. He goes on to say that Planck’s constant is a tiny tiny number so the uncertainty principle has a negligible effect on macroscopic objects. Even so, I think my friend is on solid ground in pointing out that >h does not set an upper boundary only a lower one; hence, the math does not support the claim.

Moreover, since I'm trying to relate increases in mass to decreases in uncertainty in a simple formula, I'm not sure what role Planck’s constant has. I know it must be a non-zero value, if it weren't, uncertainty would not be an issue. But I don't know how Planck’s constant bears on mass and uncertainty; except, perhaps, that h provides a minimal value for mass. In any case, by trimming the formula to: (Delta x Delta v over m), I get rid of the >h boundary issue without undermining the physics since >h has a miniscule effect on uncertainty.

BTW, What is the equation form of the definition h: m Delta x Delta v >h?

Dave1939

bhobba
Mentor
I think a precise statement of the uncertainty principle is needed to discuss this stuff meaningfully.

It has been detailed in a number of threads on this forum but here it is again.

The exact statement is suppose you have a large number of similarly prepared systems. Divide it into two lots. In the first lot you measure position. You can get an answer you want for each measurement as accurately you like. Then in the second lot you measure momentum, and again you get an answer as accurately as you like. But when you compare the standard deviations of the results (ie their statistical spread) they are as per the Heisenberg uncertainty relations.

Note:
There is no limit to the accuracy of a position or momentum measurement.
It a statistical statement about similarly prepared systems
Its a theorem proved from the commutation relations of position and momentum operators.

You will find a definitive discussion of it in page 225 of Ballentine - QM - A Modern Development.

This means the answer to the original question is easy - if you know the momentum exactly later measurements of position will be completely unknown.

Thanks
Bill

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bhobba
Mentor
I started with this mathematical definition of uncertainty: h: m Delta x Delta v >h which I got from a lecture by Prof.

That's not the definition of uncertainty - its a loose statement of the uncertainty principle. The Deltas are standard deviations from statistical theory with what it means you can get from any book on statistics. In QM also note that mV where V is the velocity operator is not always momentum which is a bit trickier than classical mechanics - it usually means the conjugate momentum from poisson bracket theory. For a free particle its correct - but otherwise it can get complicated. You will find a discussion of this as well as a derivation of the formulas in chapter 3 of Ballentine where its true foundation - symmetry - is used. It also explains where h comes from - its basically a units conversion factor. You can use units where its one if you like - and in theoretical calculations that's often done for convenience.

Thanks
Bill

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Ok, Thanks. What about the relationship claim (scaling)? Is there an inverse relationship between uncertainty and mass? If so, how would you express this mathematically?

Dave,I think all your friend is saying is that the principle gives minimum uncertainty, which does depend on mass, but you certainly can be much more uncertain, with no upper limit, even if the item is huge.

That's not the definition of uncertainty - its a loose statement of the uncertainty principle. The Deltas are standard deviations from statistical theory with what it means you can get from any book on statistics. In QM also note that mV where V is the velocity operator is not always momentum which is a bit trickier than classical mechanics - it usually means the conjugate momentum from poisson bracket theory. For a free particle its correct - but otherwise it can get complicated. You will find a discussion of this as well as a derivation of the formulas in chapter 3 of Ballentine where its true foundation - symmetry - is used. It also explains where h comes from - its basically a units conversion factor. You can use units where its one if you like - and in theoretical calculations that's often done for convenience.

Thanks
Bill

Bill and others in this thread:

Putting the issue of mathematical support aside for a moment. Solely based on what you know about uncertainty, can one say: There is an inverse relationship between uncertainty and mass or uncertainty and frequency/wavelength?

As to support for the latter, would the de Broglie's equation (lambda = h/mv provide better support for scaling overall or at least in terms of wavelength? As I understand it, wave packets can be constructed in ways that demonstrate how uncertainty varies with wavelength and frequency. Apparently, wave packets can be made that are highly localized because they have low frequency and long wavelengths; as such, measurement can give the precise location of a photon. Or, we can make wave packets that are less localized with high frequency and short wavelengths which reduces the probability of finding a photon. And most importantly as related to my question, I read somewhere that wave packets can be made with varying localization and with in-between frequencies and wavelengths. I think the Heisenberg quantum microscope thought experiment is a way of thinking about this.

As to: h: m Delta x Delta v >h, is it a good enough approximation of uncertainty that when expressed as an equation* and both sides are divided by m, to say:There is an inverse relationship between uncertainty and mass or uncertainty.

*Prof. Wolfson did not give the equation form of his definition of uncertainty: h: m Delta x Delta v >h. What would that be? Please let me know.

bhobba
Mentor
As to support for the latter, would the de Broglie's equation (lambda = h/mv provide better support for scaling overall or at least in terms of wavelength?

Forget about De-Broglies hypothesis - its wrong and was consigned to the dustbin of history when Dirac came up with his Transformation theory in 1926.

I have no idea what Wolfson says or its context but the Wikipedia article is generally sound:
http://en.wikipedia.org/wiki/Uncertainty_principle

It is not sound however regarding Heisenberg's microscope which was correctly criticised by Bohr:
http://www.informationphilosopher.com/solutions/experiments/heisenberg_microscope/
'Heisenberg said (erroneously as it turns out) that the act of observing the electron "disturbs" it and causes the resulting uncertainty. Niels Bohr embarrassed Heisenberg with his frank criticism and public discussion of Heisenberg's error in Bohr's 1927 Como Lecture'

Thanks
Bill

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bhobba
Mentor
Is there an inverse relationship between uncertainty and mass?

No. Its exactly as the uncertainty principle says - it's got nothing DIRECTLY to do with mass.

In QM there is never any uncertainty in mass - whenever you measure it its always the same.

In that regard don't get confused by relativistic mass which is an incorrect concept:
http://math.ucr.edu/home/baez/physics/Relativity/SR/mass.html

Thanks
Bill

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Here is an excerpt from a physics site that is representative of what I'm reading and what everybody, but you and apparently others here, take to be a relationship between mass and uncertainty. I made several attempts to express this in terms of mass and wavelengths without any acknowledgement on your part of the basic idea. This is very disconcerting to a newbie in that it tends toward obfuscation rather than illumination. Clearly there is a relationship between size and the degree to which quantum effects are noticed or felt in the macroworld. There are even articles in Nature about physicists who are trying to find the classical-quantum boundary. Anyway, it is just as clear that my attempts to state this have fallen short in some, yet to be fully explained, way.

If saying: "There is an inverse relationship between uncertainty and mass or uncertainty and frequency/wavelength" is wrong, how should one express the experimentally proven fact that quantum effects are neither noticed nor felt by us or even by microbes? Is: "The degree of measurement uncertainty (re any paired attributes like momentum and location) is proportionate to an object's mass or its frequency/wavelength" any better?

While the subtleties of the uncertainty principle are often lost on nonphysicists, it turns out the idea is frequently misunderstood by experts, too. But a recent experiment shed new light on the maxim and led to a novel formula describing how the uncertainty principle really works.

The uncertainty principle only applies in the quantum mechanical realm of the very small, on scales of subatomic particles. Its logic is perplexing to the human mind, which is acclimated to the macroscopic world, where measurements are only limited by the quality of our instruments.

But in the microscopic world, there truly is a limit to how much information we can ever glean about an object.

For example, if you make a measurement to find out exactly where an electron is, you will only be able to get a hazy idea of how fast it's moving. Or you might choose to determine an electron's momentum fairly precisely, but then you will have only a vague idea of its location.

Frustrated in Texas

Once again, there is an inverse relation between mass & minimum uncertainty. Not uncertainty per se- that depends on the guy doing the measurment.

bhobba
Mentor
If saying: "There is an inverse relationship between uncertainty and mass or uncertainty and frequency/wavelength" is wrong, how should one express the experimentally proven fact that quantum effects are neither noticed nor felt by us or even by microbes? Is: "The degree of measurement uncertainty (re any paired attributes like momentum and location) is proportionate to an object's mass or its frequency/wavelength" any better?

You need to study decoherence:
https://www.amazon.com/dp/3540357734/?tag=pfamazon01-20

The reason the world is classical is because it's being observed all the time by the environment.

Its very difficult to remove this, even a few stray photons from the CBMR is enough to decohere a dust particle and give it a definite position.

However it can be done and then quantum effects appear:
http://physicsworld.com/cws/article/news/2010/mar/18/quantum-effect-spotted-in-a-visible-object

There is no uncertainty relationship involving mass - mass as an observable is a scalar, hence commutes with everything so the uncertainty theorem does not apply:
http://en.wikipedia.org/wiki/Uncertainty_principle

Thanks
Bill

bhobba
Mentor
Once again, there is an inverse relation between mass & minimum uncertainty. Not uncertainty per se- that depends on the guy doing the measurment.

Do you have a link supporting that assertion?

Mass is a scalar so commutes with everything. It always gives the same value when measured.

Thanks
Bill

I (and the OP) are just talking about the standard ΔpΔx>h, and the fact that p is proportional to mass, so that greater mass allows more exact measurement of position & velocity. Nothing new, and nothing about an uncertainty in mass.

bhobba
Mentor
I (and the OP) are just talking about the standard ΔpΔx>h, and the fact that p is proportional to mass,

Where do you get the idea p is proportional to mass? In QM p is the generalised momentum.

Thanks
Bill

Where do you get the idea p is proportional to mass? In QM p is the generalised momentum.
We are talking about measuring a simple cartesian position and velocity, so p here should be ordinary momentum (the potential energy has no dependance on velocity)

bhobba
Mentor
We are talking about measuring a simple cartesian position and velocity, so p here should be ordinary momentum (the potential energy has no dependance on velocity)

In QM that is not momentum except for a free particle with actual mass. It doesn't even make sense for particles of zero mass.

There is no inverse relationship between uncertainty and mass except in a roundabout way.

Thanks
Bill

In QM that is not momentum except for a free particle with actual mass
Of course we are talking about objects that have mass. But you are making an important point: the inverse relation we are discussing is meaningful only for a small class of measurements, and is not fundamental to QM or to understanding what massive particles are "like". Thank you for clarifying this.

You need to study decoherence:
https://www.amazon.com/dp/3540357734/?tag=pfamazon01-20

The reason the world is classical is because it's being observed all the time by the environment.

Its very difficult to remove this, even a few stray photons from the CBMR is enough to decohere a dust particle and give it a definite position.

Given decoherence and the Consistent Histories Interpretation of QM, it seems that there is a negligible but not zero probability of an object's wave function collapsing into a state other than its antecedent state. In saying this, I am relying on these claims: 1) decoherence has been going on in the quantum foam prior to the Big Bang and continues today; 2) with each collapse, the probability of a wave function collapsing into a state other than an antecedent state decreases over time with each subsequent collapse.

I read somewhere about an opposing view that claims a wave function collapse is independent of its antecedent collapses. With each collapse the probabilities remain unchanged. Do you have a position on this?

bhobba
Mentor
Given decoherence and the Consistent Histories Interpretation of QM, it seems that there is a negligible but not zero probability of an object's wave function collapsing into a state other than its antecedent state.

Antcedent state? In interpretations that have collapse (not all do - Consistent histories is one that doesn't - it doesn't even have observations - in that interpretation QM is the stochastic theory of histories) what it collapses into follows from the Born Rule and physical continuity to be the eigenvector associated with the outcome of the observable.

I read somewhere about an opposing view that claims a wave function collapse is independent of its antecedent collapses. With each collapse the probabilities remain unchanged. Do you have a position on this?

I suspect you are thinking of the Quantum Zeno Effect:
http://en.wikipedia.org/wiki/Quantum_Zeno_effect

Its a simple consequence of the continuity assumption.

But what that has to do with your original post has me beat.

Thanks
Bill

bhobba
Mentor
Frustrated in Texas

The reason you are getting frustrated is you are reading popularisations or semi popularisations that in attempting to explain complex issues in lay language don't give the full facts. Based on this you then assume things the theory doesn't say - the theory does not have an inverse relation between mass and uncertainty. You said 'I’m trying to express Heisenberg's Uncertainty Principle in a simplified formula that is not boundary unlimited and still capture what I believe is an inverse relationship between uncertainty and mass - the "scaling hypothesis"'.

If you want to understand QM properly then here is the book I would recommend:
https://www.amazon.com/dp/0465036678/?tag=pfamazon01-20

Thanks
Bill

Once again, there is an inverse relation between mass & minimum uncertainty. Not uncertainty per se- that depends on the guy doing the measurment.

Do you have a link supporting that assertion?

Mass is a scalar so commutes with everything. It always gives the same value when measured.

Thanks
Bill

The article says that these guys measured photons & got very close to the minimal uncertainty- impressive. What are you trying to show?

bhobba
bhobba
Mentor

I cant see anything about mass there - so exactly what are you posting it in support of?

After thinking about what you wrote are you simply, for a free massive particle, moving mass to the other side of the inequality so from delta p delta x >= hbar/2 you get from p = mv for a free particle delta v delta x >= hbar/2m.

Thanks
Bill

The reason you are getting frustrated is you are reading popularisations or semi popularisations that in attempting to explain complex issues in lay language don't give the full facts. Based on this you then assume things the theory doesn't say - the theory does not have an inverse relation between mass and uncertainty. You said 'I’m trying to express Heisenberg's Uncertainty Principle in a simplified formula that is not boundary unlimited and still capture what I believe is an inverse relationship between uncertainty and mass - the "scaling hypothesis"'.

If you want to understand QM properly then here is the book I would recommend:
https://www.amazon.com/dp/0465036678/?tag=pfamazon01-20

Thanks
Bill

Bill, Thanks for your patience in answering my questions and especially for all the reading recommendations. I'm mostly interested in the philosophical implications of the weird aspects of quantum mechanics so I'm currently reading The Quantum Moment by Crease (philosopher) and Goldhaber (physicist). However, it is clear that I need a more reliable foundation from which to consider quantum weirdness so I will check out Quantum-Mechanics-The-Theoretical-Minimum. Thanks again, Dave

The article says that these guys measured photons & got very close to the minimal uncertainty- impressive. What are you trying to show?

Scaling. That quantum effects are not noticed or felt in the macroscopic world and therefore the uncertainty associated with the microscopic world is of no consequence accept when measuring objects at near absolute zero.

bhobba
Mentor
Hi Dave

Thanks for persevering with the crazy world of quantum theory. It is unfortunate that a lot of the things written for the lay reader are wrong, even some stuff written at the technical level is wrong.

I recently was made aware of a paper when a lot of these misconceptions have been collected together in one place:
http://arxiv.org/pdf/quant-ph/0609163.pdf

BTW Sussinds book avoids all those problems - but at a cost - it does involve some math.

Thanks
Bill

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bhobba
Mentor
Scaling. That quantum effects are not noticed or felt in the macroscopic world and therefore the uncertainty associated with the microscopic world is of no consequence accept when measuring objects at near absolute zero.

Broadly yes. But its not from 'scaling' - which I don't understand your meaning of. It's because objects in the everyday world are constantly observed and decohered by the environment.

Susskind explains a bit about decoherence in his book - return here when you have gone through it and we can have a longer chat about it.

The following may also help:
https://www.amazon.com/dp/0465067867/?tag=pfamazon01-20

Thanks
Bill

I cant see anything about mass there - so exactly what are you posting it in support of?

After thinking about what you wrote are you simply, for a free massive particle, moving mass to the other side of the inequality so from delta p delta x >= hbar/2 you get from p = mv for a free particle delta v delta x >= hbar/2m.

Thanks
Bill

I'm not sure if this is applicable but recall that Prof. Wolfson of Middlebury College started with what I now take to be an approximate mathematical definition of uncertainty: h: m Delta x Delta v >h (m = mass; delta x = the uncertainty in position, delta v = the uncertainty in velocity and h = Planck’s constant). He converts this loose definition into an equation without stating what that equation is and then, after emphasizing the mass component of the equation and that the product of Delta x Delta v "cannot come out smaller than approximately h", he divides both sides by mass. In his own words: "Also, look at the mass in there. I could divide both sides of that equation by the mass, and I'd have Delta x Delta v greater than h over the mass. That means for massive objects like a tennis ball or me or a car, the product of the uncertainty in position and uncertain velocity is miniscule. That's why the uncertainty principle doesn't affect macroscopic-sized objects..." As to h, he says: "Whether its exactly h or h/ 2π ... depends exactly on the details of your wave packet, but it's approximately a number on the order of h, and h is a very small number."

Since uncertainty is a component of the numerator and mass is the denominator of Prof. Wolfson's equation, I mistakenly inferred an inverse relationship between the two. Are you now saying that there may be something to this and whatever that is it is better expressed by the math in your latest reply?

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Broadly yes. But its not from 'scaling' - which I don't understand your meaning of. It's because objects in the everyday world are constantly observed and decohered by the environment.

Susskind explains a bit about decoherence in his book - return here when you have gone through it and we can have a longer chat about it.

The following may also help:
https://www.amazon.com/dp/0465067867/?tag=pfamazon01-20

Thanks
Bill

The Kindle version of this book is only $10.44 so I bought it for my library. bhobba Mentor That means for massive objects like a tennis ball or me or a car, the product of the uncertainty in position and uncertain velocity is miniscule. That's why the uncertainty principle doesn't affect macroscopic-sized objects..." That's a reasonable argument at the beginner level. But as you get more advanced you realize it has issues such as only applying to free particles - most of the objects we see around us are not free - in fact its the interaction with the environment that is responsible for the classical world. Added Later: The other reason its not really adequate in explaining the classical world is it's a lower bound on uncertainty and a multiple of delta v and delta x - it doest explain why delta v and delta x are both small - you could have a large uncertainty in velocity and a negligible uncertainty in position - but this is not what we see. Thanks Bill Last edited: bhobba Mentor The Kindle version of this book is only$10.44 so I bought it for my library.

That reminded me - I have been meaning to get a copy as well and just got the Kindle edition.

I read a lot of more advanced texts on QM but enjoy those at a more elementary level.

Thanks
Bill

Broadly yes. But its not from 'scaling' - which I don't understand your meaning of. It's because objects in the everyday world are constantly observed and decohered by the environment.

Bill, I think you are owed an explanation of what I mean by scaling. What follows is a few quotes and a paraphrasing of Brian Greene’s comments on page 97 of The Fabric of the Cosmos.

The uncertainty as to momentum and location of an electron and that of a car are vastly different. The Heisenberg principle not only declares the uncertainty of knowing anything about an electron’s momentum when its position is known, “it also specifies - with complete certainty - the minimum amount of uncertainty in any situation”. I take this to mean that at the macroscopic scale we can know both momentum and location with negligible uncertainty. Hence, uncertainty is a function of scale or as you say environment; things like the electron are clearly subject to the uncertainty principle and things like a car, not so much - scaling.

As to the scaling connection to the inverse relationship between mass and uncertainty, the mass of an electron is magnitudes of difference from that of a car. To illustrate this, Green says: “In day-to-day life we routinely speak about things like a car passing a particular stop sign (position) while traveling at 90 miles per hour (velocity).” He goes on to say, I paraphrase, such talk has no precise meaning in QM as we cannot simultaneously measure a definite speed and a definite position. Yet, we get away with this technically incorrect statement because on macroscopic scales the amount of uncertainty is tiny and generally unnoticed or unfelt. He says the position of the speeding car as it passes the stop sign is known within a centimeter and the uncertainty in speed is just shy of a billionth of a billionth of a billionth of a billionth of a mile per hour.

To consider what happens on microscopic scales, replace the massive car with an almost massless electron having a known position within a billionth of a meter (almost certain), then the uncertainty in its speed would be a whopping 100,000 miles per hour. Scaling changes the amount of uncertainty we get when measuring things; on microscopic scales it is very apparent but this is not the case on macroscopic scales. In Greene's own words: "Uncertainty is always present, but it becomes significant only on microscopic scales.

This is probably another example of a physicist writing for a popular audience who sacrifices exactness for simplicity but surely not, I hope, to the extent of saying something that is wrong. Probably, like Prof Wolfson, it is a reasonable approximation of uncertainty that is good enough for philosophers but not physics students.

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