Heisenberg uncertainty principle

[lennybogzy]

I got in an argument with my dad over the weekend. Is the Heisenberg uncertainty principle a result of technological limitation or a basic premise in quantum mechanics?

Basically is the information fundamentally "unknowable" or simply unmeasurable?

 

[DrChinese]

I got in an argument with my dad over the weekend. Is the Heisenberg uncertainty principle a result of technological limitation or a basic premise in quantum mechanics?

Basically is the information fundamentally "unknowable" or simply unmeasurable?

Definitely a basic premise of quantum mechanics. There are several ways to see this is so experimentally:

a) Only non-commuting observables of a particle are limited. Commuting observables - such as spin and position - are not.

b) If you make precise independent observations on separated entangled particles, then they will still provide statistics in accordance with the HUP.

 

[lennybogzy]

yeah entanglement is a good example. Seems to me there's a deep philosophical point here...

 

[glengarry]

I got in an argument with my dad over the weekend. Is the Heisenberg uncertainty principle a result of technological limitation or a basic premise in quantum mechanics?

Basically is the information fundamentally "unknowable" or simply unmeasurable?

Just think of the idea of non-commuting operators in terms of mutually exclusive conceptual paradigms. That is, when we are speaking of an object's location, the idea of velocity simply doesn't "fit in," because you necessarily need two time points in order to do a velocity calculation. On the other hand, since a moving object implies a continuum of locations (i.e., a one-dimensional line), then the idea of the singular, zero-dimensional location no longer has any meaning. It's all just the http://en.wikipedia.org/wiki/Zeno%27s_paradoxes" , but in a highly rigorous format.

 

[lennybogzy]

Well said. My fascination lies in the very fact that these non-commuting operators are "mutually exclusive conceptual paradigms". That goes against everything we feel is right and leads to some intriguing implications for the very nature of reality.

Granted human 'feeling' has proven time and time again to be of no authority

 

[glengarry]

Well said. My fascination lies in the very fact that these non-commuting operators are "mutually exclusive conceptual paradigms". That goes against everything we feel is right and leads to some intriguing implications for the very nature of reality.

Granted human 'feeling' has proven time and time again to be of no authority

Well, from where I stand, this whole mess (including the QM vs GR schism), is just a result of the fact that continuity, by its very nature, is perfectly "ungraspable." That is, I believe that the universe is truly a continuum, and that we are each functions of the "universal equation," and furthermore that, deep down inside, we each believe this to be the case.

But whenever we sit down with our pens and notebooks, it is inevitably the case that our reasonings take a decidedly "quantum" turn. What happens, therefore, is that there becomes an unbridgeable gap between what we feel we are, and what we can rigorously say about this feeling.

 

[lennybogzy]

But whenever we sit down with our pens and notebooks, it is inevitably the case that our reasonings take a decidedly "quantum" turn. What happens, therefore, is that there becomes an unbridgeable gap between what we feel we are, and what we can rigorously say about this feeling.

I say bridging that gap is our destiny as a species. The trick is avoiding the word 'say'.

 

[IcedEcliptic]

I say bridging that gap is our destiny as a species.

Of course, QM destroys determinism, and therefore destiny. ;)

 

[glengarry]

I say bridging that gap is our destiny as a species. The trick is avoiding the word 'say'.

I agree with you entirely. And you can always help me out by joining in on this thread. It started out in this forum, then got moved to philosophy. There will be quite a lot of reading for you to catch up on, but I think it'll be well worth it in the end!

(Don't let the opening post fool you... the discussion takes very many interesting turns!)

 

[glengarry]

Of course, QM destroys determinism, and therefore destiny. ;)

No, the essential lesson of QM is the destruction of the externally imposed form of destiny. That makes us free to pursue our own, self-determined destinies. That is, the opposite of external causation is not mere blind randomness, but rather, freely-willed, intentional causation.

 

[IcedEcliptic]

No, the essential lesson of QM is the destruction of the externally imposed form of destiny. That makes us free to pursue our own, self-determined destinies. That is, the opposite of external causation is not mere blind randomness, but rather, freely-willed, intentional causation.

How is that a destiny? Destiny is the opposite of free-will.

 

[DrChinese]

Just think of the idea of non-commuting operators in terms of mutually exclusive conceptual paradigms. That is, when we are speaking of an object's location, the idea of velocity simply doesn't "fit in," because you necessarily need two time points in order to do a velocity calculation.

As mentioned previously, this is wrong for 2 reasons. It has nothing to do with problems of needing 2 time points.

 

[JDługosz]

I got in an argument with my dad over the weekend. Is the Heisenberg uncertainty principle a result of technological limitation or a basic premise in quantum mechanics?

Basically is the information fundamentally "unknowable" or simply unmeasurable?

It is very common to confuse the Uncertainty principle with "The Measurement Problem." It doesn't help when "measurement problems" are introduced as analogy or seems to be introduced to show why uncertainty spreads out to cover all kinds of effects.

Uncertainty involves pairs of observables. You might come up with fancy ways to measure a position far better than you thought you could before (e.g. "near field" microscopes, quantum nondemolition measurement bases), but doing so will cause the complementary observable to not exist with any degree of precision at all.

 

[IcedEcliptic]

It is very common to confuse the Uncertainty principle with "The Measurement Problem." It doesn't help when "measurement problems" are introduced as analogy or seems to be introduced to show why uncertainty spreads out to cover all kinds of effects.

Uncertainty involves pairs of observables. You might come up with fancy ways to measure a position far better than you thought you could before (e.g. "near field" microscopes, quantum nondemolition measurement bases), but doing so will cause the complementary observable to not exist with any degree of precision at all.

I blame schools for thinking that all kids need to know of QM is some vague "Can't know momentum, position, velocity at the same time" nonsense. They don't explain HUP, so the analogy is no analogy; they teach the measurement problem as if it WERE the HUP! Criminal!

 

[SpectraCat]

Uncertainty involves pairs of observables. You might come up with fancy ways to measure a position far better than you thought you could before (e.g. "near field" microscopes, quantum nondemolition measurement bases), but doing so will cause the complementary observable to not exist with any degree of precision at all.

No this is incorrect ... the HUP has nothing to do with measurements per se, and it certainly has nothing to do with the precision of a given measurement technique. The HUP constrains the probabiity distributions from which two non-commuting observables are sampled. Therefore, it constrains the distributions of a series of repeated measurements of those observables on identical systems (somewhat hard to accomplish in practice). So, no matter how good or bad the precision of your measurement, the minimum width of the distribution of a given observable according to the HUP is unchanged.

 

[lennybogzy]

Ok... is the measurement problem a result of HUP?

Can noncommuting observables be described as "mutually exclusive conceptual paradigms"?

 

[SpectraCat]

Ok... is the measurement problem a result of HUP?

Can noncommuting observables be described as "mutually exclusive conceptual paradigms"?

For your first question, no. The measurement problem arises when we make a measurement of an observable from a superposition state. The measurement postulate of QM states that the result of such a measurement will always be an eigenstate of the observable. The measurement problem points out some technical difficulties involved with actually realizing such a result in a real experiment. For the most part, these have been resolved by the theory of decoherence, but there are still some lingering issues I believe. Search "measurement problem" in this forum and you will get plenty of reading material. Or you could just google it ...

For your second question, that looks like double speak to me .. I really have no idea what you are asking. Non-commuting observables have a precise mathematical definition ... the four-word phrase you used has no clear definition whatsoever as far as I can tell. Please explain more clearly what you are trying to ask.

 

[lennybogzy]

i'm trying to convert a "precise mathematical definition" as honestly as possible into a conceptual framework that my mind will accept

 

[SpectraCat]

i'm trying to convert a "precise mathematical definition" as honestly as possible into a conceptual framework that my mind will accept

In that case it is best to keep it specific at first, so that you can understand how things work. The idea of non-commuting observables is fundamental to quantum mechanics. In classical mechanics, if you have some particle moving in space, at any point you can define it's position and momentum to arbitrary precision, and furthermore, if you know the values, you can accurately predict the future trajectory of that particle. Because of this, we say classical mechanics is deterministic.

In QM, this is just not possible at a fundamental level, because position and momentum are non-commuting observables, and you cannot simultaneously define them to arbitrary precision. Rather, the HUP tells us that the products of the uncertainties in those variables can never be less than one-half times Planck's constant divided by 2 time pi. This means that the future trajectories of quantum particles cannot be accurately predicted from measurements of their position and momentum at any given time. We can only know the probabilities of finding them at some later point with a particular position and momentum. For this reason, QM is called a "probabilistic theory".

The above description is somewhat oversimplified, but hopefully it gives you a qualitative picture of what it means for variables to be non-commuting, and with the other posts in this thread, some idea of the significance of the HUP.

 

[Fredrik]

Ok... is the measurement problem a result of HUP?

No, it isn't. The uncertainty theorem is just a theorem in QM that tells us something about how far from the average we should expect the results to be when we measure the same two observables on a large number of identical systems prepared in the same state.

The "measurement problem" isn't even a consequence of QM. It's a consequence of assuming that both of the following statements are true (in addition to all the axioms of the standard version of QM):

a) A state vector is a mathematical representation of all the properties of a system.
b) A measurement has only one classical result.

So if the "measurement problem" bothers you, just stop insisting that both of the above must be true. Drop a) and you have the "ensemble interpretation". Drop b) and you have some sort of "many-worlds interpretation". Another option is to try to replace the standard version of QM with a theory that's equivalent to QM in the sense that it makes the same predictions, but is still a different theory because it's defined by different axioms. Such a theory may or may not have a "measurement problem".

Can noncommuting observables be described as "mutually exclusive conceptual paradigms"?

I don't know what that would mean. "Observables" should be defined as equivalence classes of measuring devices. So an observable is something in the real world, with an operational definition. In the theory, observables are represented mathematically by bounded self-adjoint operators. When two such operators don't commute, the corresponding observables are said to be incompatible. The best way to think of non-commuting self-adjoint operators, or the corresponding incompatible observables, is as two (equivalence classes of) measuring devices that would interfere with each other if we tried to use both of them at the same time.

For example, take a Stern-Gerlach apparatus oriented to measure the spin of a silver atom along the z axis, and an identical Stern-Gerlach apparatus oriented to measure the spin along the x axis. Suppose you took the magnets from both and put them in the same place. The result wouldn't be a measuring device that measures S_x and S_z at the same time. The magnetic fields would add up to a new field, in a new direction, so if this new apparatus can measure anything at all, it would define some other observable (probably spin along the line x=z, y=0, in the direction of increasing x and z...but I haven't really thought that through).

 

[JDługosz]

No this is incorrect ... the HUP has nothing to do with measurements per se, and it certainly has nothing to do with the precision of a given measurement technique. The HUP constrains the probabiity distributions from which two non-commuting observables are sampled. Therefore, it constrains the distributions of a series of repeated measurements of those observables on identical systems (somewhat hard to accomplish in practice). So, no matter how good or bad the precision of your measurement, the minimum width of the distribution of a given observable according to the HUP is unchanged.

So was my explanation simply not rigorous (necessary in a casual explanation) or was it missing something important? I don't see how what you said is different from what I was getting at.

"constrains the probabiity distributions from which two non-commuting observables are sampled." sampled != measurements ?

 

[JDługosz]

Ok... is the measurement problem a result of HUP?

To clarify your question after reading SpectraCat's reply, the "problem" we are talking about is that you can't measure something precisely without trashing it. If you measure its position by bouncing a high-energy probe off of it (high energy gives high resolving power), the object bounces away. In classical physics, you can then put the object back exactly where it was, so no problem. In QM, the HUP means that (in this example) if you measure the position with high precision, you don't know what the original momentum was, so after you disturbed it you can't track it back to its original state. To measure something like position, the HUP indeed means that you can't win. The problem is that the conjugate variable (momentum) has an effect on the original (position).

However, it's not generally true for all kinds of measurements. If you measure exactly the kind of state it's already in, you don't mess it up. That requires knowledge of QM and designing the situation to do something useful with that kind of measurement. But, that is exactly what went into the design of sensors for certain gravitational wave experiments.

You can make repeated measurements of a state x that is carefully chosen so its conjugate x' is not something that affects x, then you can make repeated precise measurements of x, without HUP trashing your subject of study. Exactly what x is isn't something easily understood since it doesn't map to classical terms.

 

[JDługosz]

i'm trying to convert a "precise mathematical definition" as honestly as possible into a conceptual framework that my mind will accept

Try https://www.amazon.com/gp/product/0674741137/?tag=pfamazon01-20, Quantum Mechanics and Experience by David Z Albert.