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Is it true that put any particle (non-charged; no spin; only translational and potential energy considered) on a force field $U$, it's probability distribution can be calculated through Schrodinger Equation?

I am curious because if the answer is yes, some easy computer simulation will give interest quantum effects.

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  • $\begingroup$ As far as we know, yes. Quantum mechanics is the fundamental theory according to which everything behaves. $\endgroup$
    – Javier
    Jul 10, 2017 at 23:57
  • $\begingroup$ The interesting quantum effects often involve many particles and are rarely trivial to simulate: the example of complicated molecules comes to mind. $\endgroup$ Jul 11, 2017 at 0:12
  • $\begingroup$ To expand on zero's comment, even a helium atom needs a (not easy) numerical simulation, as no analytical solution is possible. Look up the three body problem in closed solution form for a classical example. $\endgroup$
    – user154420
    Jul 11, 2017 at 0:20
  • $\begingroup$ It depends on precision. QM holds to an excellent precision in most cases, but it is non-relativistic so we know that in some situations it must fail and be replaced by QFT. For some potential effects see Does QFT prevent preparation of an entangled particle pair as in EPR experiment? $\endgroup$
    – Conifold
    Jul 11, 2017 at 2:14

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Quantum mechanics, as well as quantum field theory or classical mechanics, are nothing but frameworks that are used to predict results given an experiment; as such, the question should not be whether or not some framework hold in some cases, but rather if it gives useful and correct predictions.

In classical physics, where classical means small velocities and small energies, one can actually measure classical observables (position and momentum, energy, temperature and the like) without affecting them very much (namely without having the problem that measuring position will increase the uncertainty on momentum): as such, one asks questions like where the particle is, what the velocity is, what is the total angular momentum.

Is it true that put any particle (non-charged; no spin; only translational and potential energy considered) on a force field $U$, its probability distribution can be calculated through Schroedinger Equation?

Yes, you can always calculate things given some initial frameworks, the actual question is what you can do with it: probability distributions are used to calculate scattering amplitudes in high energy experiments where (for some reasons) one cannot calculate the trajectory of the particles but only derive some indirect measurements (decay rates and similar). If the particle is "classical", you had better just look at the actual position and state where the particle is.

I am curious because if the answer is yes, some easy computer simulation will give interest quantum effects.

How does this have anything to do with quantum mechanics?

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A stronger statement is that all probabilities ultimately originate from quantum fluctuations. In principle, this doesn't have to be true for QM to be always valid. The stronger statement means that in all cases where classical probabilities are used, these probabilities are in principle given as the squared norms of quantum mechanical amplitudes. This is investigated in this article, the authors reach the conclusion that there are no known counterexamples.

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  • $\begingroup$ Thanks for your insightful answer! Could you please elaborate on how all probabilities are originated from QM fluctuations? Could you please provide some more references? I am so excited to hear about it. $\endgroup$
    – High GPA
    Jul 11, 2017 at 20:30
  • $\begingroup$ It's very risky to state that "all probabilities ultimately originate from quantum fluctuations" unless you exactly specify what it means. In quantum mechanics, for example, it needs not necessarily be true as probabilities more or less originate from the fact that the wave function is a superposition of some certain eigenstates of some certain observables. $\endgroup$
    – gented
    Jul 11, 2017 at 22:07
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In principle yes. Classical mechanics is really a convenient convention. A gnat with a mass $m~=~10^{-8}kg$ moving at $v~=~.1m/s$ has by the de Broglie rule $p\lambda~=~h$, momentum $p~=~mv$ and $h = 6.7\times 10^{-33}js$ has a wavelength $\lambda~=~6.7\times 10^{-24}m$. That is really small. A gnat is made of many atoms with their own wavelength and these wave all cancel each other to to give this tiny wave.

W. Zurek showed that the moon Hyperion around Saturn rotates in a chaotic manner. Quantum mechanics acts to introduce fluctuations that have a measurable impact on the mechanics. In this case because of the “butterfly effect” of chaos theory quantum effects are amplified.

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    $\begingroup$ I don't think we can attribute the chaotic orbit of Hyperion to quantum effects. Gravitational perturbations from other Solar System objects would have a far, far, far greater effect than quantum fluctuations in the position/momentum of the planet. $\endgroup$ Jul 11, 2017 at 0:48
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    $\begingroup$ Here is the W. Zurek paper. In my extremely brief perusal, Zurek argues quantum mechanics predicts an (absurd) spreading of Hyperion's wavefunction on the order of 20 yr unless he considers decoherence. As far as I can tell in my quick glance he doesn't seem to be arguing there is any actual (macroscale) measurable deviation from classical mechanics. $\endgroup$
    – GPhys
    Jul 11, 2017 at 1:40
  • $\begingroup$ Hey, enough already! Besides, Zurek does argue for a change outside of classicality. Anyway, enough of this. $\endgroup$ Jul 11, 2017 at 17:51

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