# What is the Meaning of the Schrödinger Equation?

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• iaM wh
In summary, the Schrödinger equation is an expression of the conservation of energy and provides a mathematical expression for total energy by incorporating both kinetic energy and potential energy. However, potential energy is difficult to determine as it depends on the system as a whole and its spacio-temporal relationships. Schrödinger introduced an operator to account for systemic potential energy, making the equation non-local in nature. This idea was questioned by Einstein and has been both theorized and experimentally supported.
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?

iaM wh said:
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?

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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?

iaM wh said:
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)

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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.

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

Swamp Thing said:
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.

Michael Price said:
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.

iaM wh said:
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.

Swamp Thing said:
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.

Swamp Thing
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.

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.

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dextercioby

## 1. What is the Schrödinger Equation?

The Schrödinger Equation is a fundamental equation in quantum mechanics that describes the behavior of a quantum system over time. It is named after Austrian physicist Erwin Schrödinger and is used to determine the probability of finding a particle in a particular state.

## 2. What is the meaning of the Schrödinger Equation?

The Schrödinger Equation represents the wave function of a quantum system, which contains all the information about the system's possible states and their probabilities. It is a mathematical equation that allows us to predict the behavior of quantum particles.

## 3. How does the Schrödinger Equation relate to the uncertainty principle?

The Schrödinger Equation is one of the fundamental equations of quantum mechanics, and it is used to describe the behavior of particles at the subatomic level. This equation is closely related to the uncertainty principle, which states that it is impossible to know both the position and momentum of a particle with absolute certainty.

## 4. Can the Schrödinger Equation be solved exactly?

In most cases, the Schrödinger Equation cannot be solved exactly. Instead, scientists use approximations and numerical methods to find solutions. However, there are some simple systems for which exact solutions can be found, such as the hydrogen atom.

## 5. What are the applications of the Schrödinger Equation?

The Schrödinger Equation has numerous applications in modern technology, including the development of transistors, lasers, and computer memory. It is also used in fields such as chemistry, material science, and quantum computing to understand and manipulate the behavior of particles at the atomic level.

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