I think we were all surprised at the first time we got to know quantum mechanics that the squared modulus of the wave function is the probability density of the existence of the particle?

The role of the complex numbers here is strange, but the question here is:

Is there an idea that is deeper and easier to understand, so that entering the squared modulus of the complex number becomes a mathematical result from this idea only in order to facilitate the calculations?

Do you share with me my astonishment and my question?

God willing, I think we can find something deeper and even simple, actually I put this reason in my paper:


In summary:

The main idea of this paper is that 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.

So when the particle is in position p1 at time t1, where would it be in time t2?

In classical mechanics, the trajectory exists so the least action principle state that:

The path taken by the particle between times t1 and t2 is the one for which the action is stationary.

So what is the situation in quantum mechanics?

fortunately, we have a principle that is very close to the classical principle, but in this case, we didn't have any path, we have potential new positions, so in general, the particle has some preferred destinations based on a new quantum action principle named "alike action principle" that ensures the existence of physical harmony within our universe, like for example preventing the particle from easily reaching forbidden locations (guarded by fields of great forces).

Therefore, in general, this new constraint in motion could be valid at multiple positions at the same time, so in general, we have multiple acceptable positions at time t2.

Thus the probability of existence came up in our description of the movement in the quantum world.

We suppose that we have a preferred value of action that we call h (Plank constant), the new action principle called the "alike action principle" states:

"The preferred appearance destination position took by the particle at time t is the one for which all the remainders due to S/h (for all imaginary paths which lead to this destination) are stationary".

In other words, it is as having the same (or close to each other) remainder after dividing them by h.

For example, if we have two actions (for two paths) to one destination position, the natural function that verifies this principle is:

sin2((π/h)(S1 − S2)).

So after some steps of the calculation, we derive the relation between the probability density of the existence and the squared modulus of the wave function by deriving the path integral formulation of quantum mechanics, for more details please see the paper.


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