What is the Meaning of the Schrödinger Equation?

[iaM wh]

TL;DR Summary

Meaning of the Schrödinger equation

I would like to discuss the Schrödinger equation in order to get some insight.

The equation, as I understand it, is essentially an expression of the conservation of energy. What it says is that ∆Total Energy= ∆ Kinetic Energy + ∆ Potential Energy.

In Schrödinger's day, there were various mathematical expressions for these energies. For example, in Newtonian mechanics kinetic energy is expressed as p²/2m, and that expression is explicitly in the Schrödinger equation. But, for a wave, Energy was (h-bar)f. Schrödinger plugged these numbers into his equation. Potential energy is difficult, because it depends upon the system one is working with. And potential energy does not depend only on the individual potential energies of the particles that make up a system. It also depends upon the system as a whole, and the spacio-temporal relationships of the particles that make up a system. That is, potentisl energy of a system seems to be irreducible. You can't extract it by knowing everything about the particles that make up the system. You also have to know how those particles relate to one another.

Now it seems that Schrödinger, being unable to come up with a general expression for systemic potential energy, and seeing that in the equation there appeared a scalar (E) in the exact place in which it would have been expected there be an operator (E-hat) simply put a hat on the E scalar and turned it into an operator. What this did, in effect, was give us an explicit explicit mathematical expression (generalized, and including both particles and waves) for total energy. Since kinetic energy had already been expressed mathematically, and total energy - kinetic energy = potential energy, we got access to total potential energy. Indeed, when solving the Schrödinger equation, one of the keys is to plug in potential energy, as it pertains to the individual constituents that make up a system. But there is usually no easy way to plug in the systemic potential energy, due to the spacio-temporal relationships between the particles. The mathematical expression pertaining to this value of systemic potential energy is really what Schrödinger gave us with his equation.

Now, since QM is based on this equation, and since this equation contains systemic potential energy (which it seems, based on the equation, is non-local; that is, it seems that the equation allows systemic potential energy to vary, across the whole system, as the spacio-temporal relationships that make up the particles/waves of the system change) is it any wonder that QM seems to be non-local in nature. I mean, is it reasonable to expect that a theory based on the Schrödinger equation could possibly be interpreted as local?

 

[Swamp Thing]

I mean, is it reasonable to expect that a theory based on the Schrödinger equation could possibly be interpreted as local?

Sometimes this idea is introduced with a simple thought experiment: If a source emits a single photon with equal amplitudes in all directions, and if a detector catches it, say one second later, then the rest of the universe knows instantly that the photon is gone, and that no detector has any chance of catching that photon any more. This includes all parts of the universe that are light years away from that detector.

I'd like to add a question here that the original post suggests to me: Did Einstein (or anyone else) raise any objections about spooky action at a distance before EPR, based on just the idea of a global wavefunction collapsing into a local detection event that then precludes further detections at distant locations?

 

[iaM wh]

Yes, the wavefunction collapsing instantaneously, and therefore transmitting information throughout the universe instantaneously was an idea that Einstein never liked, even before EPR. It blatantly violates his theory of relativity, and the causal link is broken.

I'm just wondering from your example why you say that the rest of the universe "knows" instantly that it can't detect the photon, even places that are light years away. Is this something that has been verified by experiment? Or is it a conclusion drawn from the theory?

 

[Swamp Thing]

why you say that the rest of the universe "knows" instantly that it can't detect the photon, even places that are light years away. Is this something that has been verified by experiment? Or is it a conclusion drawn from the theory?

I think it's both. But I'm not very sure.

Edit: Of course, an experiment probably wouldn't be on a light year scale, but just big enough to prove faster-than-causal correlations. (In a sense, it is a correlation -- an anti-correlation, actually)

 

[Michael Price]

Wavefunctions are allowed to collapse instantaneously, because the Schrödinger equation is a non-relativistic equation. So there is no problem with FTL collpase, apart rom the observer-dependence issues. Schrödinger knew of what we now call the Klein-Gordon equation, but he wanted to get a 1st order equation.

 

[Swamp Thing]

What did EPR add to the pot that wasn't there before?

 

[Michael Price]

What did EPR add to the pot that wasn't there before?

Nothing. Entanglement is the unavoidable consequence of systems with more than one particle in them.

 

[iaM wh]

Wavefunctions are allowed to collapse instantaneously, because the Schrödinger equation is a non-relativistic equation. So there is no problem with FTL collpase, apart rom the observer-dependence issues. Schrödinger knew of what we now call the Klein-Gordon equation, but he wanted to get a 1st order equation.

how is FTL collapse not a problem? If I am in a place with some probability of observing a particle and you, space-like separated from me, actually observe that particle and instantaneously collapse the wave function, you change the statistics where I am.

 

[Michael Price]

how is FTL collapse not a problem? If I am in a place with some probability of observing a particle and you, space-like separated from me, actually observe that particle and instantaneously collapse the wave function, you change the statistics where I am.

And in non-relativistic physics there is no problem with that. If you see the particle at X you won't see it at Y.

 

[Nugatory]

What did EPR add to the pot that wasn't there before?

EPR proposed that quantum theory (as it was understood in 1935!) was incomplete in the sense that there could be a more complete specification of the state of the system including hidden variables (that is, internal degrees of freedom of which we are not aware) which would allow us to predict the measurement outcome at one detector without considering the setting of the other detector.

 

[HomogenousCow]

The non-relativistic Hamiltonian is explicitly non-local, in that it contains non-local terms (fixed potential, say). To my knowledge this is not related to the EPR-type non-locality, which remains even with explicitly local Hamiltonians.

 

[A. Neumaier]

Born's rule together with the SE is explicitly nonlocal in that they predict that a free particle prepared here in a state where the momentum is small still has a nonzero probability to be found the very next moment light-years away.