This question cannot be resolved except by looking at the whole history of wave-particle duality. For example a ray with particles is a good aproximation to shadows, and interference a good starting point to believe in waves.

Then if you cannot explain the same thing as wave and particle, you must get creative and establish at least a relation between them. So that is a reasonable, but not an inescapable assumption to make.

QM is just all empirical. No logical proof by rationalists is around. We must remember that. "Do not demand from something more than it is"

I think the right approach to understanding this very interesting point is to consider the real fundamental nature of the entities described by the wave function. See

Conference Paper THE UNIFICATION OF PHYSICS

Data Prerecording of Conference Presentation on the Unification of Physics

If you take a look at the video you will see that the interpretation of the electron as a point particle with a probability distribution (Copenhagen interpretation) is equivalent for the purposes of the Schrodinger equation to the description of the electron as a looped wave in spacetime (Spacetime Wave theory).

Based on this you could complete a mathematical analysis to show this to be the case and then understand how the squared modulus of the wavefunction leads to the probability of a measurement giving a result at a specific position.

Richard

the interpretation of Born refers to a nonrelativistic theory, Schrödingers wavemechanics. in QFT, , the realistic, relativistic invariant theory, this question ist obsolete this refers also to the question, whether special relativity is violated by wave f unction reduction it is a ''relativistic'' question posed in a non relativistic theory, schrödinger wave mechanics. Hence it is a senseless question.

Werner Heisenberg was the founder and discoverer of Quntum Mechanics. And his formulation of Quantum Mechanics, thhe Matrizenmechanik, does not need Borns interpretation.

Dear Juan Weisz

You said:" and interference a good starting point to believe in waves. "

and

" Then if you cannot explain the same thing as wave and particle, you must get creative and establish at least a relation between them. So that is a reasonable, but not an inescapable assumption to make. "

There is a very important question here, and if we understand it, we understand, with the help of God, the origin of all the oddities of quantum mechanics! Rather, these oddities become understandable to us, and even logically justified!

In classical mechanics: "In an inertial reference frame, an object either remains at rest or continues to move at a constant velocity, unless acted upon by a force"

Well, why does this law not remain valid and applied in quantum mechanics?

I mean that the particle remains affected by the forces that influence it.

No physicals answer for this question in quantum mechanics only that the tiny object works like this!

but the following idea: " the continuous trajectory of the particle can not exist, so the motion is a sequence of appearances and disappearances events in space and time, so the particle does always jump to move from one position to another " leads us to (with the help of God):

If the particle is affected only by the forces applied to its position, then it jumps from its position to the next position then it simply exposes itself to the unknown! Because it did not take into account the forces and fields in this next point! An electron may suddenly find itself in the core of the nucleus, or in the core of a proton, or even another electron! Like someone who jumps out of his place without seeing or knowing where he will fall!

so for a huge number of particles that jump in the subatomic level, the Newton law may put our universe in an unstable situation, and this might happen specifically when the length of the jump is close (or greater) to the length of the Field's fluctuations.

So we conclude this result (physical idea):

To maintain our universe stable the particle needs to take into consideration not only the forces (fields) that applied to it (as in classical mechanics) but all the forces (fields) that exist in all space.

So the particle needs something to explore the space and inform it where it can be existing or not.

So I want to retain both concepts particle and wave-function.

Now, who did the role of leads the particle in space?

I mean who provides to the particle the probability distribution of existence (wave-function)?

To be honest, I don't know who leads the particle but I think space itself may do that, or perhaps some universal field exists in all space, space has an important role in the relativity theory (curvature, metric..), is it impossible to also it has an important role in quantum mechanics?

And then when the particle appears in one position the role of the old wave-function is terminated (wave collapse) because the particle in fact chose its position, so from this moment another wave-function starts to leads the particle to the next position, etc...

One thing, here we have two major questions: is particle jumps?

if yes who leads it?

I think we can answer each question separately, and I think the first question is easier than the second...

And I think that the derive of lamb shift from the quantum jump is an important argument about the validity of quantum jump (please read it in the article)

With best regards.

Dear Richard Lewis

Please can you summarize the physicals main idea that leads to the probability of a measurement?

With best regards.

Dear Hinnerk Albert

The "Double-slit experiment" for the electron is a good example of the probability density of the existence of the particle, so I think my question is important to know why the electron behaves like this?

With best regards.

Dear Mazen Khoder . I can summarise the explanation in the context of the Spacetime Wave theory:

Conference Paper THE UNIFICATION OF PHYSICS

Data Prerecording of Conference Presentation on the Unification of Physics

This explanation would be different if you used the Copenhagen interpretation of QM or the Many Worlds interpretation.

Let's stay with the dual slit experiment as described in the video linked above. The wave quantum (photon) passes through both slits as a wave disturbance of spacetime. Interference takes place so that the magnitude and phase of the interfering physical wave is affected at each point in space. When this dispersed wave reaches the screen there is a physical interaction between the wave and an atomic electron of the detector screen. An atomic electron is a looped wave in spacetime. The detector screen is designed so that there are many atoms, each of which could detect this particular wave quantum but conservation of energy means that only one atom can do the detecting. The probability of a particular atom detecting the wave quantum (photon) depends on the magnitude and phase of the wave at each point on the screen.

From this description it is clear that we are talking about a real physical wave and probability effects only come into action at the point of detection.

Richard

i did not want to criticize borns interpretation. that would be ridiculous but the ''philosophical'' discussion, that followed also the whole superfluous discussion about violation of causality, ''exchange of information with velocity higher than speed of light', which is also often wrong.

Hinnerk

About the founder of QM I would ascribe to Max Planck

around 1900

Both Schrodinger and Heisenberg were later appearances , around 1924, even after Bohr and the old QM.

Relativistic QM has a probability density very similar to that of Born and Schrodinger, except it is a sum of 4 such terms.