/sci/ - Science & Math

>>10898843
>what are limiting-case approximations?

>>>/pol/

Replace all observables with the expectation values of those observables.

bump

>>10898843
[math]\hbar\rightarrow 0[/math]

>>10898843
Look up "decoherence"

Basically a lot of the weirdness about quantum mechanics is based on interference effects. A boring but good example is shooting electrons through two slits. What happens is not simply the sum of either going through one slit or the other.

The idea of decoherence is that interactions with the large number of degrees of freedom in the environment wash out these interference effects, and it becomes simple classical probability.

There are still problems of interpretation, but this explains a lot on how quantum systems become classical as the number of particles increase.

>>10898843
https://youtu.be/-YqL0TQnvFI?t=25m

>>10898843
>How do the quantum phenomena emerge into classic physics phenomena?
they emerge in the classical limit.
The time derivative of a quantum operator [math]\Omega[/math], according to ehrenfest's theorem, is
[eqn]
\left\langle \frac{d}{d t} \Omega \right\rangle = i \langle[H, \Omega]\rangle + \left\langle\frac{\partial \Omega}{\partial t}\right\rangle
[/eqn] where the angle brackets have their usual meaning as the expectation value of an operator given a state [math]|\Psi\rangle[/math], i.e. [math]\langle \Omega \rangle = \langle \Psi | \Omega | \Psi \rangle[/math]
consider the time derivative of the position and momentum operators, and dispensing with the angle brackets since I don't want to write \langle and \rangle again, just understand that everything is an expectation value of a quantum state.

If we have the hamiltonian [math] H = \frac{P^2}{2m} + V(X) [/math], then
[eqn]
\frac{d P}{d t} = i [H, P] + frac{\partial P}{\partial t} = i[V(X),P]
[/eqn] the partial derivative vanishes since the momentum operator has no explicit time dependence.
Now it is easy to show by writing [math]f(X)[/math] as a taylor series and using the canonical commutation relation that
[eqn][f(X),P] = i f'(X)[/eqn], thus we have that
[eqn]
\frac{d P}{d t} = -V(X)
[/eqn] i.e. Newton's second law. Note that this is not exactly the same as Newton's second law in classical mechanics, since this relates expectation values of the operators. However, in the classical limit, the probability distribution of the operators approaches a delta function, and thus we have
[eqn]
\frac{d p}{d t} = -V(x)
[/eqn] where the lower case denotes that they are classical variables rather than quatum operators.

>>10898843
>How do the quantum phenomena emerge into classic physics phenomena?
they emerge in the classical limit.
The time derivative of a quantum operator [math]\Omega[/math], according to ehrenfest's theorem, is
[eqn]
\left\langle \frac{d}{d t} \Omega \right\rangle = i \langle[H, \Omega]\rangle + \left\langle\frac{\partial \Omega}{\partial t}\right\rangle
[/eqn]where the angle brackets have their usual meaning as the expectation value of an operator given a state [math]|\Psi\rangle[/math], i.e. [math]\langle \Omega \rangle = \langle \Psi | \Omega | \Psi \rangle[/math]
consider the time derivative of the position and momentum operators, and dispensing with the angle brackets since I don't want to write \langle and \rangle again, just understand that everything is an expectation value of a quantum state.

If we have the hamiltonian [math] H = \frac{P^2}{2m} + V(X) [/math], then
[eqn]\frac{d P}{d t} = i [H, P] + \frac{\partial P}{\partial t} = i[V(X),P]
[/eqn]
the partial derivative vanishes since the momentum operator has no explicit time dependence.
Now it is easy to show by writing [math]f(X)[/math] as a taylor series and using the canonical commutation relation that
[eqn][f(X),P] = i f'(X)[/eqn], thus we have that
[eqn]
\frac{d P}{d t} = -V(X)
[/eqn]i.e. Newton's second law. Note that this is not exactly the same as Newton's second law in classical mechanics, since this relates expectation values of the operators. However, in the classical limit, the probability distribution of the operators approaches a delta function, and thus we have
[eqn]
\frac{d p}{d t} = -V(x)
[/eqn]where the lower case denotes that they are classical variables rather than quantum operators.

>>10899128
>However, in the classical limit, the probability distribution of the operators approaches a delta function

This is exactly the problem of the classical limit that you are assuming here. Why are classical objects localized and have a well defined momentum rather than being in some energy eigenstate?

Ehrenfest's theorem does look like classical mechanics already, but it holds for very quantum systems too. It isn't a statement about the limit

>>10899159
>Why are classical objects localized and have a well defined momentum rather than being in some energy eigenstate?
because of the uncertainty principle. [math]\Delta x \Delta p \approx \hbar \implies \Delta x \Delta v \approx \hbar / m [/math]. m is relavitively large for a classical object, so correspondingly [math] \Delta x \Delta v [/math] is small.

>>10899179
An energy eigenstate is usually taken as the 'natural' state to talk about physics at the atomic or molecular level. This is not a wave packet like you want to consider.

I agree one aspect of the classical limit is actions get large with respect to hbar, but it is not the only thing.

Why when the number of particles increases do things start to look more wave packet like? (Hint, I've already posted what physicists think is the answer)

It's called the classical approximation

>>10898843
What do you mean by "how"? That's how they already work. You must do an extremely contrived experiment to see anything non-classic.

>>10898843
Here, learn a thing that might be true.

quantamagazine.org/quantum-darwinism-an-idea-to-explain-objective-reality-passes-first-tests-20190722/

>>10899128
>Why when the number of particles increases do things start to look more wave packet like?
Because many convolutions will eventually converge to a gaussian distribution?

>>10898843
Either it doesn't have a property, or you need to assume its property.

>>10900032
I wasn't talking about the central limit theorem. There are all sorts of basis states in quantum mechanics corresponding to eigenstates of different observables, and most look nothing like a minimum uncertainty wave packet, even a non-gaussian one.

It's probably wasn't worth bringing up this topic on this board, but look up decoherence if you're interested.

>>10898843
<X>

>>10900048
>I wasn't talking about the central limit theorem
What were you talking about?

>There are all sorts of basis states in quantum mechanics corresponding to eigenstates of different observables
I'm familiar with elementary QM, thanks for the refresher though. How again is this relevant to the conversation?

>nothing like a minimum uncertainty wave packet, even a non-gaussian one
I wasn't aware that non-gaussian minimum uncertainty states exist. In fact they don't, but maybe I'm misreading your post.

I could talk about decoherence all day and I'm glad you're interested in it, but it's really not necessary to bring up when explaining the classical limit. Even if it was even more relevant, be careful bringing up a topic on which no scientific consensus exists.

>>10899128
Can you actually do Latex on 4chan ? Nice
Also OP, maybe you can find some answers on statistical physics.

>>10899128
Nice, thanks.

>>10900118
No need to get so defensive

I was talking about the problem of preferred basis. If you have a coherent superposition of states it can be expanded in any number of different bases, and if ultimately if we want it to become a classical incoherent superposition of states the question is which one is selected?

I completely disagree that decoherence has nothing to do with the classical limit, unless you are arguing about semantics. It is why we don't find macroscopic systems in coherent superpositions.

>>10898868
*and approximate higher order moments as vanishing

>>10900137
>If you have a coherent superposition of states it can be expanded in any number of different bases
Not any number, it can be expanded as eigenstates of any quantum operator that commutes with the hamiltonian.

>if we want it to become a classical incoherent superposition of states the question is which one is selected?
So as long as you're consistent, it doesn't matter. This is the first thing to know about basis vectors.

You also need to clarify if you're using the physicist's definition of a coherent state (eigenstate of the lowering operator) or the laymen's definition. Which is why I don't like the term "decoherence," it uses a different definition of coherence than physicists use. I would read up on some introductory QM and linear algebra before getting into the popsci interpretations.

>>10900122

>>10900411
I'm not talking about popsci. This has been established physics for decades. People trying to build quantum computers have to account for decoherence for anything to work.

Coherence has (almost) nothing to do with coherent states. It is a term that comes from optics with a long history.

It sounds like you know a little bit of physics, but please, it is in your own interest to lose that ego a bit. If you want to know more about the preferred basis problem and why it really is a problem try searching for it. I probably won't check back in this thread.

>>10900713
>>10900411

Actually I tried searching the term myself just now and didn't great results. Try looking at the Wikipedia article for "einselection" instead if you want to know what I meant

>>10900770
Meh, that's brainlet shit all over again. Appearance of collapse is provided by the structure of entanglement, which is completely independent of any basis. Obviously. It works the same in any basis, nothing preferred there. And those environment shenanigans server no purpose there. Collapse is the business between two interacting bodies, environment has no business there at all. What is environment for universe? Classical behavior is provided by the size of macroscopic objects, they try to use environment to provide coherence, but it's done by size, any macroscopic object is big enough to do the same.

>>10901069
Environment also refers to the microscopic degrees of freedom that we are ignoring (tracing out) in a macroscopic system.

In principle I could have a macroscopic body that is an eigenstate of z angular momentum. This is a state that is absolutely perfectly axially symmetric, and we don't observe this in the classical world. This state would of course obey everything you could throw at it in terms of Ehrenfest's theorem. And since its angular momentum is so much bigger than hbar, it formally is the "classical limit"

The fact that we do not observe such states, is more or less what people mean by the preferred basis problem, and it is part of the quantum to classical transition.

Quantum Darwinism explains why the classical world appears. But it still doesn't explain why some events end up not being explained by classical mechanics, such as truly stochastic events on a macroscopic scale.
Why do some outcomes happen over others in stochastic events? Quantum Darwinism doesn't explain it completely. All it concludes is that quantum events will act more or less in synchrony with the classical laws due to decoherence, but not always.

Why does quantum foam and spin foam end up taking this or that path over all others?

https://youtube.com/watch?v=mlztKOlqiEQ

>>10901139
I agree decoherence doesn't explain everything philosophically. I'm more interested in the practical questions related to the quantum to classical transition. I hate the name "quantum darwinism" by the way

https://youtube.com/watch?v=PKwq7b2i-vc

>>10901127
Really clear explanation, thanks! I'm convinced you have a point now. Sorry for being such an asshole earlier.

>>10899066
Add to this non commutative operators that force time's arrow

>>10898843
quantum physics is basically saying, we can't explain this sht, lets just write down our observations and figure out the probability of their reocurrance.

>>10901155
>I hate the name "quantum darwinism"
Why? Are you religious? Would you prefer it to be called Quantum God's Will? The name doesn't matter.

>>10902537
It does though. Quantum Darwinism sounds like the kind of name of some crackpot pseudoscience. First time I heard it I assumed it was for brainlets until I saw the name of the scientist who coined it

>>10903802
>Quantum Darwinism sounds like the kind of name of some crackpot pseudoscience.
Why, though? This association may well be singular to you. The name succinctly implies what the concept is. You could call it Quantum Selection, but that conveys slightly less information.