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Time-dependent Schrödinger equation for the time evolution of a quantum system is a linear equation. Essentially: d/dt Ψ = H Ψ There is no second, or higher,

by sampo 6y ago

Time-dependent Schrödinger equation for the time evolution of a quantum system is a linear equation. Essentially: d/dt Ψ = H Ψ There is no second, or higher, powers of Ψ. There is even no constant term. It's linear. https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation#Time-dependent_equation https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation#Time... Now, how does nonlinear macroscopic world appear, if the fundamental time evolution at quantum scale is linear? So obviously Schrödinger equation alone is not enough to describe the time evolution of the universe. One way is to introduce wavefunction collapse, which is a nonlinear process. But there is no physical theory (well, there are propositions) of how and when collapse happens. It just happens somehow, sometime. This problem is at the core of why quantum mechanics is an incomplete theory.

prof-dr-ir 6y ago
I do not think that you need to introduce wave-function collapse here, because there is no need for the universe to be fundamentally non-linear - it suffices for the Poincare recurrence time to be extremely large. Like phase transitions in statistical mechanics, could it no be that chaos is just an emergent effect that arises from the quantum dynamics of infinitely many particles? (The appearance of non-linearities in that limit happens to also be discussed in the MIT paper of the quanta article.)
btilly 6y ago
Your comment presupposes that the Everett interpretation cannot be true. But it offers an explanation of the appearance of collapse without having a collapse. Namely if a quantum mechanical observer observes a quantum mechanical system in a superposition of states, we get a superposition of quantum mechanical observers who can no longer meaningfully interact, each of which observed a different state of the quantum mechanical system. This is exactly what the Schrödinger equation predicts MUST happen. As a side note, this is the most popular interpretation of quantum mechanics among cosmologists. It turns out that if you're using quantum mechanics to explain things like the birth of galaxies, taking seriously what quantum mechanics says for human sized quantum mechanical systems becomes very easy.
pontus 6y ago
While the Schrodinger equation is linear, that doesn't mean that the time evolution of various derived quantities are also linear. For example, the expectation value of position follows a nonlinear equation of motion even though the schrodinger equation is linear. This is known as Ehrenfest's theorem: https://en.m.wikipedia.org/wiki/Ehrenfest_theorem https://en.m.wikipedia.org/wiki/Ehrenfest_theorem In other words, nonlinear time evolution is natural in quantum mechanics for quantities other than the wave function and does not require collapse.
- prof-dr-ir 6y ago
  As your link shows, the time evolution of the expectation value of position does in fact not generally obey a closed-form non-linear differential equation; instead one needs the expectation value of V'(x) which is a different quantity altogether. But the easiest way to compute that quantity is of course to solve the Schrodinger equation... Edit: removed an accusation because I misread the original comment.
  - pontus 6y ago
    Yes, that's right: you need <V'(x)> rather than V'(<x>). It's still the case that <x> does not follow a linear DE. My point was that while the SE is linear, that does not mean that everything derived from it is also linear. The original comment was asking where all the nonlinearities in the world could come from since the SE is linear. It was suggested that either QM is incomplete because it is linear or that we need wave function collapse to introduce nonlinearities. I think my counterexample shows that both of those suggestions are incorrect.
Lichtso 6y ago
I am quite sure chaos can be an emergent property. E.g. think of Conways game of life: One could build a contraption which amplifies a very small event into a gigantic one (like a Geiger-Müller tube) and spawn a few gliders (as particles). Now, a very small change in the initial configuration changes the outcome drastically, even though all the rules of the simulation are still perfectly deterministic and linear.
tagrun 6y ago
H can depend on Ψ, which is why Bose-Einstein condensates referred in the new article are nonlinear.

by sampo 6y ago

prof-dr-ir 6y ago

I do not think that you need to introduce wave-function collapse here, because there is no need for the universe to be fundamentally non-linear - it suffices for the Poincare recurrence time to be extremely large. Like phase transitions in statistical mechanics, could it no be that chaos is just an emergent effect that arises from the quantum dynamics of infinitely many particles? (The appearance of non-linearities in that limit happens to also be discussed in the MIT paper of the quanta article.)

btilly 6y ago

Your comment presupposes that the Everett interpretation cannot be true. But it offers an explanation of the appearance of collapse without having a collapse. Namely if a quantum mechanical observer observes a quantum mechanical system in a superposition of states, we get a superposition of quantum mechanical observers who can no longer meaningfully interact, each of which observed a different state of the quantum mechanical system. This is exactly what the Schrödinger equation predicts MUST happen. As a side note, this is the most popular interpretation of quantum mechanics among cosmologists. It turns out that if you're using quantum mechanics to explain things like the birth of galaxies, taking seriously what quantum mechanics says for human sized quantum mechanical systems becomes very easy.

pontus 6y ago

While the Schrodinger equation is linear, that doesn't mean that the time evolution of various derived quantities are also linear. For example, the expectation value of position follows a nonlinear equation of motion even though the schrodinger equation is linear. This is known as Ehrenfest's theorem: https://en.m.wikipedia.org/wiki/Ehrenfest_theorem https://en.m.wikipedia.org/wiki/Ehrenfest_theorem In other words, nonlinear time evolution is natural in quantum mechanics for quantities other than the wave function and does not require collapse.

prof-dr-ir 6y ago
As your link shows, the time evolution of the expectation value of position does in fact not generally obey a closed-form non-linear differential equation; instead one needs the expectation value of V'(x) which is a different quantity altogether. But the easiest way to compute that quantity is of course to solve the Schrodinger equation... Edit: removed an accusation because I misread the original comment.
- pontus 6y ago
  Yes, that's right: you need <V'(x)> rather than V'(<x>). It's still the case that <x> does not follow a linear DE. My point was that while the SE is linear, that does not mean that everything derived from it is also linear. The original comment was asking where all the nonlinearities in the world could come from since the SE is linear. It was suggested that either QM is incomplete because it is linear or that we need wave function collapse to introduce nonlinearities. I think my counterexample shows that both of those suggestions are incorrect.

Lichtso 6y ago

I am quite sure chaos can be an emergent property. E.g. think of Conways game of life: One could build a contraption which amplifies a very small event into a gigantic one (like a Geiger-Müller tube) and spawn a few gliders (as particles). Now, a very small change in the initial configuration changes the outcome drastically, even though all the rules of the simulation are still perfectly deterministic and linear.

tagrun 6y ago

H can depend on Ψ, which is why Bose-Einstein condensates referred in the new article are nonlinear.