Atomic Spins Evade Heisenberg Uncertainty Principle
scientificamerican.com
scientificamerican.com
The authors of the original Nature article say that "spins obey non-Heisenberg uncertainty relations", which is true if "Heisenberg uncertainty relation" is (narrowly) defined as the uncertainty relation for position and momentum, Δx⋅Δp ≥ ħ/2. But since spins don't have position and momentum, they are not expected to observe this uncertainty relation anyway. The more general uncertainty relation for any two observables A and B reads ΔA⋅ΔB ≥ |⟨[A,B]⟩|/2 (where [A,B]=AB-BA is the commutator, and ⟨⟩ is the expectation value) and for spins which have three components Sx, Sy, Sz the commutator is [Sx,Sy]=iħSz etc, therefore the uncertainty relation for two spin components reads ΔSx⋅ΔSy ≥ ħ/2⋅|⟨Sz⟩|, hence it does not seem so surprising that it can be made rather small if the expectation value of the third spin component can be made small.
Edit: typo
There's a good article on him almost failing his PhD exam because he had difficulty with a related concept (https://www.aps.org/publications/apsnews/199801/heisenberg.c...). It's amusing because he was later awarded the Nobel prize for "the creation of quantum mechanics".
The actual uncertainty comes from the fact that the two quantities are Fourier transforms of each other... and just by that relationship, inherently, if one gets very localized (== very high frequency bump in its space), its Fourier dual gets spread out very far through space. (You can sort-of analogize this if you know about audio ... a sharp spike in temporal space, when transformed into frequency space, becomes a very big spread of values, because all those frequencies are relatively blunt and they have to somehow fit together to make this sharp thing, which requires an enormous number of them. Or if you go the other way, a sharp spike in frequency space means one frequency, which transforms into an infinitely-long sine wave in temporal space. So think about that kind of thing, except instead these are waveforms where the y value is kind-of the probability of getting that particular x-value as a result if you perform a measurement.)
> Historically, the uncertainty principle has been confused[5][6] with a somewhat similar effect in physics, called the observer effect, which notes that measurements of certain systems cannot be made without affecting the systems, that is, without changing something in a system. Heisenberg offered such an observer effect at the quantum level (see below) as a physical "explanation" of quantum uncertainty.[7] It has since become clear, however, that the uncertainty principle is inherent in the properties of all wave-like systems,[8] and that it arises in quantum mechanics simply due to the matter wave nature of all quantum objects. Thus, the uncertainty principle actually states a fundamental property of quantum systems, and is not a statement about the observational success of current technology.[9] It must be emphasized that measurement does not mean only a process in which a physicist-observer takes part, but rather any interaction between classical and quantum objects regardless of any observer.[10][note 1]
> Bohr was compelled to modify his understanding of the uncertainty principle after another thought experiment by Einstein. In 1935, Einstein, Podolsky and Rosen (see EPR paradox) published an analysis of widely separated entangled particles. Measuring one particle, Einstein realized, would alter the probability distribution of the other, yet here the other particle could not possibly be disturbed. This example led Bohr to revise his understanding of the principle, concluding that the uncertainty was not caused by a direct interaction.
Disagree, it's a perfectly reasonable explanation of what's happening.
'Measuring' the system is really 'operating' on it. A quantum operation puts the system into one of the operators eigenstates, and also yields the eigenvalue as the measurement itself.
Two non-commuting operators, A and B, will have different eigenstates. Applying operator A on the system puts the system into some eigenstate a, which will be some linear combination of the eigenstates b, of operator B.
So yes, operating on the system with operator A will put the system into a superposition of b states. If the system was in a well-defined b eigenstate previously, it's not anymore. It was 'disturbed' by the 'measurement' done by operator A.
Position and momentum operators are examples of non-commuting operators. So are the 'spin' operators that measure angular momentum along different axes.
It's not false, but it's not entirely true. The uncertainty principle doesn't state that you can't _measure_ p and q with arbitrary precision. It states that you cannot _imagine_ them with arbitrary precision. It's like the double slit experiment: when you measure which slit is traversed, interference patterns disappear; but, when you don't measure, you cannot even _think_ that the photon is passing through one slit instead of the other.
Uncertainty Principle refers to the arbitrary precision of two operators. You can measure each operator individually to arbitrary precision. In theory, that is.
If the two operators commute you can get theoretical arbitrary precision on both measurements together. Since their commutator is zero.
If they don't commute, the product of standard deviations of each measurement operator is restricted to no less than half the expectation value of their commutator. (photon-torpedo's comment at the top of this thread gives a better summary).
The UP is not only: "You can't know position and momentum together." It rather: "Position and measurement _cannot be defined_ together."
In classical mechanics, you can imagine a particle being at a certain position with a certain momentum. In quantum mechanics, you cannot.
That the product of the standard deviations of position and momentum are at least half of h-bar.
It says nothing else about what that "means", the mechanism by which it comes about, or whether it is a fundamentally "true" description of reality.
Per Wikipedia (not the best source, but accurate here):
https://en.wikipedia.org/wiki/Uncertainty_principle
"Historically, the uncertainty principle has been confused[5][6] with a somewhat similar effect in physics, called the observer effect, which notes that measurements of certain systems cannot be made without affecting the systems, that is, without changing something in a system. Heisenberg offered such an observer effect at the quantum level (see below) as a physical "explanation" of quantum uncertainty.[7] It has since become clear, however, that the uncertainty principle is inherent in the properties of all wave-like systems,[8] and that it arises in quantum mechanics simply due to the matter wave nature of all quantum objects.
...
Bohr was compelled to modify his understanding of the uncertainty principle after another thought experiment by Einstein. In 1935, Einstein, Podolsky and Rosen (see EPR paradox) published an analysis of widely separated entangled particles. Measuring one particle, Einstein realized, would alter the probability distribution of the other, yet here the other particle could not possibly be disturbed. This example led Bohr to revise his understanding of the principle, concluding that the uncertainty was not caused by a direct interaction."
You're casually ignoring how it got to eigenstate a in the first place. The particle is in an eigenstate a because it was acted on by the A operator.
If it was in an eigenstate b prior to this, or any other state that isn't a single eigenstate of A, the action of operator A disturbed that original state.
The HUP itself is a direct consequence of measuring the uncertainty of two noncommuting operators, and its derivation is a fairly straightforward, though a bit annoying, mathematical result.
And that's about as far as I understand it. The whole thing seems intended to make my primitive brain hurt. But any experiment which starts off by cooling things down to "a few microkelvin" definitely has my appreciation.
If you can prove that, your Nobel prize awaits. :)
The truth is that nobody knows why there is quantum uncertainty. The common belief is that the uncertainty is "real", but non-local hidden variable theories have not been proven to be impossible.
(In a recent survey of physicists, 47% favoured explicitly real uncertainty - https://arxiv.org/pdf/1612.00676.pdf)
It's interesting, but squeezing is a common technique, and this headline misrepresents the importance of this experiment.
> ...measure the spin precession rate much more accurately than previously thought possible.
since no one thought different.
Good for QM because it gives you a sink to dump uncertainty, good for hiring because it helps you not get sued, good for avoiding miscommunications in general, ......