Does our inability to decide (by any meaningful procedure) which state an electron emits in prove that the nature is essentially non-computable?
Also what is the actual trick with the Stern-Gerlach experiment?
I mean, can you prepare a system such that it won't be possible to measure spin or down regardless of how precise your measuring device is?
>>7975673
>system such that it won't be possible to measure spin or down r
yes. take a spin up in the z direction, put it into a stern-gerlach measuring the x-direction, now, in whatever branch out of the apparatus, you know nothing now about the spin in the z-direction.
>>7975701
Erm, that's not quite what I meant. Also, I don't know QM that well, is it a kinda of what's described in the OP-pic?
>>7975673
You can always measure up or down. Predicting the outcome of said measurement is another thing altogether.
>>7975916
Do you know what the OP pic is about? Like, put simply, what's the issue described there?
>>7976028
tell us the source of your picture
>>7977478
If I knew.
>>7975586
>computable
To what resolution?
>>7977629
Read through what computable means
>>7980828
So in the stern galach experiment, the particle is emitted with a spin that is a superposition of different states. If you know about vectors, this state can be though of as a vector with a basis. This basis can be chosen to whatever you want, it doesn't change the state of the particle.
When you measure the particle, you are measuring it in a direction in the basis, so - for example -measuring its z spin, you force its state to be in the z direction with a value of [math]\pm\frac{\hbar}{2}[/math]. Forcing it to be in this state destroys any information about it in the x or y directions. If you measure again in the z direction, it will have the same value. If you measure in the x or y direction, it will have [math]\pm\frac{\hbar}{2}[/math] in those directions, and this destroys the information in the z direction.
Nature is non-deterministic, and it seems intrinsically probabilistic. This doesn't mean that it's non-computable though - look at statistical mechanics.
>>7980888
I mean, can you prepare the system so that the beam splitting be completely unrecognizable?
>>7981194
Unrecognizable how? Too small?
>>7981263
Yeah
>>7981282
You can prepare it so that the magnetic field you're splitting with makes the difference too small to see, but the effect is still going on. Can you make what you're asking clearer?
>>7980888
maths are not thing to do with nature though
>>7981894
...
>>7981894
)))
>>7981318
Is there any physical limit on this "too small"?
The uncertainty principle would limit it.
>>7982623
How exactly? I'd appreciate a couple of formulas. Like if the density matrix is exactly
[math] \displaystyle \begin{bmatrix}
\frac{1}{2} & 0 \\
0 & \frac{1}{2}
\end{bmatrix} [/math]
and we measure exactly along the z-axis?
What apparatus can distinguish between exactly equal eigenvalues 1 and 1?
>>7982646
If you're measuring the spin along any axis, the eigenvalues are always ±hbar/2. This is the physical quantum nature of the system. Along any axis you measure the spin, it will always have one of these values.
Now, our ability to resolve these values may be limited by equipment, or ultimately, the uncertainty principle. If we can measure the spin then it will always have one of those two values.
>>7982646
I think you're misunderstanding the density matrix. The eigenvalues of it don't mean anything for this problem.
The density matrix given just means that the system is in a mixed state with equal probability to have spin up, or spin down along the z axis. So when you measure the spin z it will be either ±hbar/2 with a probability of ½ for each.
It's the eigenvalues of the operator that are the values you measure.
The spin z operator matrix is given by:
Sz=hbar/2(1, 0/0, -1)
>>7982658
What means "resolve these values"?
>>7982900
Resolving the values means you can see the difference between them.
>>7982935
What does it mean "you can see"?
>>7983625
Wait-wait before you start trolling me. Still, what precisely did you mean by "you can see the difference between them."?
>>7983705
When it's resolvable, you can see the difference between the spins.
When it's not resolvable, you can't see the difference.
>>7983733
I understand that what is "resolvable" and "to see" precisely? Or you mean beam splitting may be, say, beyond Plank scale? I am trying to think in concrete numbers.
>>7983828
It wouldn't mean anything if the beam was said to split smaller than Planck scale.
>>7983880
So what do you MEAN? Literally "seeing" is bullshit.
>>7983917
I mean any method of measuring.
>>7983932
So, any measurement device wouldn't determine whether the spin is up or down even though the eigenvalues are distinct? Is that what you mean?
Can there actually be a real physical system whose operator would have indisinguishable eigenvalues?
>>7987465
What operator?
You can have different states that have the same eigenvalue under a particular operator - they're called degenerate.
>>7987901
And how to cope with degeneracy?
>>7988115
Generally you don't use either state, but the sum of both of them.
A simple example would be a particle in a box, where two of the side lengths are the same.
Another example would be the three different p-orbitals in a hydrogen atom. In these examples there are multiple mutually orthogonal states that all have the same energy.
I assumed here when you talk about the operator of a system you mean the Hamiltonian. there are lots of operators associated to a given quantum system, but the Hamiltonian defines it in some sense.
>>7988164
So what do we actually measure if the eigenvalues are degenerate?
>>7988192
you measure the eigenvalue and you do not know what final state you have
>>7990804
Is what exactly happens in the experiments or is it a thought construction?
