4 ms·
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
by 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 agoAs 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 agoYes, 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.