Comments about "Schrödinger equation" in Wikipedia

This document contains comments about the article "Schrödinger equation" in Wikipedia
In the last paragraph I explain my own opinion.



The article starts with the following sentence.
In quantum mechanics, the Schrödinger equation is a mathematical equation that describes the evolution over time of a physical system in which quantum effects, such as wave–particle duality, are significant.
In this context it is very important which the defination of what a physical system is.
The equation is a type of differential equation known as a wave-equation, which serves as a mathematical model of the movement of waves.
The equation as such makes sense. The issue are its broader implementations.
In quantum mechanics, the analogue of Newton's law is Schrödinger's equation for a quantum system (usually atoms, molecules, and subatomic particles whether free, bound, or localised).
Which immediatly narrows its use.
In the Copenhagen interpretation of quantum mechanics, the wave function is the most complete description that can be given of a physical system.
Again that is to wide.
Solutions to Schrödinger's equation describe not only molecular, atomic, and subatomic systems, but also macroscopic systems, possibly even the whole universe.
To describe the whole universe by a single wave function does not make sense

1. Equation

1.1 Time-dependent equation

The form of the Schrödinger equation depends on the physical situation (see below for special cases). The most general form is the time-dependent Schrödinger equation, which gives a description of a system evolving with time:
The most famous example is the nonrelativistic Schrödinger equation for a single particle moving in an electric field (but not a magnetic field; see the Pauli equation):
To apply the Schrödinger equation, the Hamiltonian operator is set up for the system, accounting for the kinetic and potential energy of the particles constituting the system, then inserted into the Schrödinger equation. The resulting partial differential equation is solved for the wave function, which contains information about the system.
How do you do that in practice, for an electron, for an elephant, for the total universe

1.2 Time-independent equation
Same issues as above.

2 Derivation

So far, H is only an abstract Hermitian operator.
The issue is what is this H for any real application
However using the correspondence principle it is possible to show that, in the classical limit, the expectation value of H is indeed the classical energy.
The correspondence principle does not completely fix the form of the quantum Hamiltonian due to the uncertainty principle and therefore the precise form of the quantum Hamiltonian must be fixed empirically.
The uncertainty principle is not a physical laws. The uncertainty principle is in fact prove of our human limitations.
In order to calculate the Hamiltonian you have to perform experiments. Using the results of these experiments (which are not always the same) immediate are prove that the Hamiltonian is not rock bottom science.

3. Implications

3.1 Total, kinetic, and potential energy

3.2 Quantization

The Schrödinger equation predicts that if certain properties of a system are measured, the result may be quantized, meaning that only specific discrete values can occur.
First you have to perform the experiments to measure these properties.
There is "no" issue if the Schrödinger equation agrees with these results.
Another result of the Schrödinger equation is that not every measurement gives a quantized result in quantum mechanics.
I expect that most measurements at atomic or subatomic level fall in this cathegory. The question is what is the Schrödinger equation in these cases.

3.3 Measurement and uncertainty

In classical mechanics, a particle has, at every moment, an exact position and an exact momentum.
That is correct. The problem is it is impossible to measure.
These values change deterministically as the particle moves according to Newton's laws.
In reality it is impossible to use Newton's Law to calculate the position of elementary particles in time.
Under the Copenhagen interpretation of quantum mechanics, particles do not have exactly determined properties, and when they are measured, the result is randomly drawn from a probability distribution.
There exist only a problem when you want to measure the position of a particle. The result is like a normal distribution. Each particular experiment has its own probability distribution.
All of this is not in conflict that each particle at any moment has an exact position. The point is that this position can not be established, because the position changes continu.
The Schrödinger equation predicts what the probability distributions are, but fundamentally cannot predict the exact result of each measurement.
The only thing that is wrong is the word predict. A better word is describe.
THe issue is that for every experiment first the probability distribution has to be calculated based on many experiments. Secondly the individual parameters of the Schrödinger equation can be calculated.
It is similar as Newton's law. First the masses of the individual objects have to be calculated based on observations using Newton's Law. The more observations the better. Secondly, when these masses are know, you can predict the future.

3.4 Quantum tunneling

In classical physics, when a ball is rolled slowly up a large hill, it will come to a stop and roll back, because it doesn't have enough energy to get over the top of the hill to the other side.
This has nothing to do with classical physics. In general when there are two hills of the same height and when you give the ball just enough energy (at the top) that the ball starts rolling down towards the next hill, the ball will stop before the top and will roll down. The reason is friction.
However, the Schrödinger equation predicts that there is a small probability that the ball will get to the other side of the hill, even if it has too little energy to reach the top.
If the Schrödinger equation describes otherwise then the Schrödinger equation is "wrong" or you have an other experiment in mind.

3.5 Particles as waves

Two-slit diffraction is a famous example of the strange behaviors that waves regularly display, that are not intuitively associated with particles.
It is better to write: "etc behavior etc, that is not associated with single particles."
Intuitively, one would not expect this pattern from firing a single particle at the slits, because the particle should pass through one slit or the other, not a complex overlap of both.
It is better to write: At first glance etc.
However, since the Schrödinger equation is a wave equation, a single particle fired through a double-slit does show this same pattern (figure on right).
It should be remembered that the Schrödinger equation is not an explanation of this diffraction pattern. The explanation should answer the question "exactly" what is a particle (electron or photon) physical.
Related to diffraction, particles also display superposition and interference.
A clear defintion from superposition is required, based on an actual experiment.

3.6 Multiverse

It was that, when his Nobel equations seem to be describing several different histories, they are "not alternatives but all really happen simultaneously". This is the earliest known reference to the multiverse.
I think what his equation describe are different possiblities. Anyway his descriptions should be based on actual experiments and if these experiments involve simultaneous actions than his equations indicate the same.

4. Interpretation of the wave function

The Schrödinger equation provides a way to calculate the wave function of a system and how it changes dynamically in time. However, the Schrödinger equation does not directly say what, exactly, the wave function is.
It is more important to know what a system is

13. See also

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Created: 28 January 2017

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