4 ms·

QM is a linear theory (its equations are linear). Chaos arises from non-linear interactions. Until we have a non-linear version of of the Shrodinger/Dirace/Kl

by dhimes 4y ago

QM is a linear theory (its equations are linear). Chaos arises from non-linear interactions. Until we have a non-linear version of of the Shrodinger/Dirace/Klein-Gordon/Whatever-the-cool-kids-are-using-these-days equations, we're going to have a problem reconciling them.

d_tr 4y ago
Newton's second law is also linear but you can study chaotic systems just fine. You can study coupled weird oscillators in QM too. You just take the classical problem and derive the Hamiltonian operator. I do not understand what the problem is here (I mean with the SE being linear) and I saw this repeated a couple times in the other thread too.
- dhimes 4y ago
  The linearity of NII depends on the linearity of F and a. QM has certain postulates that have to be overcome first, such as any state can be written as a linear combination of eigenstates, etc. You can see them here: http://vergil.chemistry.gatech.edu/notes/quantrev/node20.html http://vergil.chemistry.gatech.edu/notes/quantrev/node20.htm...
catchclose8919 4y ago
...or you just take the wave function as the real-reality, and in this reality a "chaotic system" is one with even macroscopic position probabilities "smeared out" too uniformly have any one be "classically-real". If classic reality is the illusion, and our intuition is fundamentally limited and incapable of grasping the actual reality, then even many-worlds is a bit of clutch... we imagine an infinity of "parallel" classical universes simply because it's the best we can do: one wave function universe where all the variants are "real" at the same time only that being real is a complex-numbered-probability so some are more real than other with different orientations and magnituded too. ...we could just give up hoping that our ape-brains can fully "grasp" reality, and just focus on getting the math to work: if some math systems make chaos and qm both work, then move on with that regardless of our primitive intuitions and preconceptions/superstitions of that the workld ought to be!
jostylr 4y ago
The evolution of the system in Bohmian mechanics includes the motion of the particles which is a non-linear and highly non-trivial equation which allows for chaos to appear. Classical mechanics emerges from the theory under suitable conditions. An outline of the reasoning can be found in https://arxiv.org/abs/quant-ph/0112005 https://arxiv.org/abs/quant-ph/0112005 There is a strong suspicion in the Bohmian community that the sensitivity to initial conditions in Bohmian mechanics is an important piece of the sociological explanation for the embrace of the collapse theory of QM a hundred years ago, long before a full appreciation of such sensitivity was widely appreciated in non-linear dynamic systems.
credit_guy 4y ago
Is that so? As a rule, when you take a limit new things can emerge. For example, pi is the limit of a sequence of rational numbers. For example 3, 3.1, 3.14, 3.141, 3.1415, etc. All these rational numbers are "perfectly predictable", while the digits of pi are not. One could say pi is a "chaotic number", and is the limit of non-chaotic numbers. Nobody would find this paradoxical. Take turbulence, the textbook example of chaos. I can approximate any solution of the Navier-Stokes equation with a sequence of periodic functions (with longer and longer period). Each of these is non-chaotic, but the limit is chaotic. Quantum mechanics converges to classical mechanics when h->0. So, the solutions of Schroedinger's equation (which are all non-chaotic) converge to something that exhibit chaos. Where's the paradox?
- hkksksjsjsjd 4y ago
  h is not zero. According to Sabine, after the timescale of ~20 years, the difference between h=0 and the small non-zero value in nature should exhibit in observations of chaotic motion of Hyperion. In this instance, observation apparently agrees with h=0, which is a paradox.

by dhimes 4y ago

d_tr 4y ago

Newton's second law is also linear but you can study chaotic systems just fine. You can study coupled weird oscillators in QM too. You just take the classical problem and derive the Hamiltonian operator. I do not understand what the problem is here (I mean with the SE being linear) and I saw this repeated a couple times in the other thread too.

dhimes 4y ago
The linearity of NII depends on the linearity of F and a. QM has certain postulates that have to be overcome first, such as any state can be written as a linear combination of eigenstates, etc. You can see them here: http://vergil.chemistry.gatech.edu/notes/quantrev/node20.html http://vergil.chemistry.gatech.edu/notes/quantrev/node20.htm...

catchclose8919 4y ago

...or you just take the wave function as the real-reality, and in this reality a "chaotic system" is one with even macroscopic position probabilities "smeared out" too uniformly have any one be "classically-real". If classic reality is the illusion, and our intuition is fundamentally limited and incapable of grasping the actual reality, then even many-worlds is a bit of clutch... we imagine an infinity of "parallel" classical universes simply because it's the best we can do: one wave function universe where all the variants are "real" at the same time only that being real is a complex-numbered-probability so some are more real than other with different orientations and magnituded too. ...we could just give up hoping that our ape-brains can fully "grasp" reality, and just focus on getting the math to work: if some math systems make chaos and qm both work, then move on with that regardless of our primitive intuitions and preconceptions/superstitions of that the workld ought to be!

jostylr 4y ago

The evolution of the system in Bohmian mechanics includes the motion of the particles which is a non-linear and highly non-trivial equation which allows for chaos to appear. Classical mechanics emerges from the theory under suitable conditions. An outline of the reasoning can be found in https://arxiv.org/abs/quant-ph/0112005 https://arxiv.org/abs/quant-ph/0112005 There is a strong suspicion in the Bohmian community that the sensitivity to initial conditions in Bohmian mechanics is an important piece of the sociological explanation for the embrace of the collapse theory of QM a hundred years ago, long before a full appreciation of such sensitivity was widely appreciated in non-linear dynamic systems.

credit_guy 4y ago

Is that so? As a rule, when you take a limit new things can emerge. For example, pi is the limit of a sequence of rational numbers. For example 3, 3.1, 3.14, 3.141, 3.1415, etc. All these rational numbers are "perfectly predictable", while the digits of pi are not. One could say pi is a "chaotic number", and is the limit of non-chaotic numbers. Nobody would find this paradoxical. Take turbulence, the textbook example of chaos. I can approximate any solution of the Navier-Stokes equation with a sequence of periodic functions (with longer and longer period). Each of these is non-chaotic, but the limit is chaotic. Quantum mechanics converges to classical mechanics when h->0. So, the solutions of Schroedinger's equation (which are all non-chaotic) converge to something that exhibit chaos. Where's the paradox?

hkksksjsjsjd 4y ago
h is not zero. According to Sabine, after the timescale of ~20 years, the difference between h=0 and the small non-zero value in nature should exhibit in observations of chaotic motion of Hyperion. In this instance, observation apparently agrees with h=0, which is a paradox.