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- 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?

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?