Research on the Root of Quantum Mechanics

Abstract: This context is to analyze the obscure flaw of fundamental concepts in QM and to propose solutions to them. The concept of “electric wave” would be initiated. The real physical meaning of some mathematical operations would be made clear.
Author: Liqiang Chen (only)

Analysis of the Speed of Probability Wave

I don’t know how to display the full context of this chapter here because I am not able to use the software to show some mathematical calculation. What I can do is to show its abstract.

The beginning of the context is:
According to Duc de Broglie’s original idea, the frequency of probability wave γ = E / h, the wavelength of probability wave λ = h / p.

The speed of probability wave of a free particle could be calculated as below:
If v is the moving speed of the particle, then, the speed of the corresponding probability wave:
u = γλ = (E / h)( h / p) = E / p =…

The conclusion of the context is:
When v → 0, Lim u = 0,
When v → c, Lim u = c,
When 0 ＜ v ＜ c, v ＞ u

There are two hard problems as below:
1. What does the inconformity between “v” and “u” in case of 0 ＜ v ＜ c mean exactly in physics? It represents the wave state separating from the particle state? How lame it all sound. It’s too abstract and not understandable. It just means the concept of wavicle (wave – particle duality) in QM can’t establish in this situation. Why?
Seems there is a flaw here in QM.
2. And what does the conformity between “v” and “u” in case of “v → c, Lim u = c” mean exactly in physics? It should mean the concept of wavicle established in this situation. Why?

(If a guy wants a copy of “Analysis of the Speed of Probability Wave”, he can pm piggy.)

What’s the Meaning of the Amplitude of Wave Function

In QM, we really don’t know what physics meaning is the amplitude of probability wave. The square of (absolute value of) amplitude is probability density.
If * is non-sense in physics, how can the square of * make sense in physics? How lame it all sounds… “gangster’s logic”?
That just mean there is a flaw here too in QM.

Strictly speaking in physics, if people can’t find out the exact physics meaning of the basic, then the whole theoretical building is doubtful. QM seems a “tower floating in the air” no matter how delicate its mathematical system is.

Perhaps people should touch the elephant from an alternative angle and see whether there could be a better solution to make sense.
Strictly speaking, physics has to tackle such specific problem.

Simple 3D Space Mathematical Model of Unit Charge.

Let’s see the unit step function:
μ(t) = 0 (t＜0) μ(t) = 1 (t＞0)
It could be interpreted here as below:
Because t＜0 meaningless, so the signal does not exist.
When t＞0, the signal exist.
Next, analysis with Fourier transformation (detailed calculation omitted), we got:

1 = (1/2) + (1/π)∫0→+∞（1/ω）sinωt dω ①
Then, replace variables, we got:
1 = (1/2) + (1/π)∫0→+∞（1/ r）sin r p dr ②
Namely:
1 = (1/2) + (1/2) = (1/π)∫0→+∞（1/r）sin r p dr + (1/π)∫0→+∞（1/r）sin r p dr ③

It represents the superposition of two same waves in adjacent to the origin of coordinate system.

According to the wave function Ψ(r ) =（1 / r）sin r p , its amplitude only has something to do with distance (∝1/r). It looks like the effect of electric interaction in classical electromagnetism.
Since the value of momentum “p” does not affect the result of integration in equation ③, it looks like the fact of movement of elementary particles do not affect the amount of unit electric charge.

Okay, we just take this as mathematical model of unit charge. “1” represents unit charge carried by an elementary particle. (In my physics model of unit charge, “1/2” above could be considered as the half unit charge carried by a “layer”. The physics model is not the emphasis in this context. So, the details would not be provided.)
The three dimension coordinate space wave function Ψ(r ) =（1 / r）sin r p describes “the wave characteristic of electric interaction” (abbreviated as “electric wave”) activated by relative movement of “layer” with momentum p. (Those two same waves could be considered as individual “electric wave” activated by the movement of “layer”.)

And we adopt Duc de Broglie’s original idea, the frequency of “electric wave” γ = E / h, the wavelength of “electric wave” λ = h / p.
(Note: If we use “p / ħ” in variable replacement, then, the “electric wave” function above would be Ψ(r ) =（1 / r）sin (r p / ħ). But for the sake of simplicity and easy observation, we ignore Plank constant in all equations in this context.)

Unilateral Conception of Field vs Bilateral Conception of Field

In the mathematical model of unit charge, the “electric wave” can be represented by the equation: Ψ(r) =（1 / r）sin r p. The amplitude of wave function naturally means the intensity of electric field E.
But it’s just a mathematical model. In order to describe real electric effect in physics, we need to introduce a physical parameter “k”. (Note: here “k” is a lower case letter.)
So, the equation for the intensity of electric field of any charge could be E = kQ / r.
(Note: The lower letter “k” is a proportionate constant.)

The magnitude of electric force for point charge in classical electromagnetism is F = KQq / r².
The intensity of electric field for point charge in classical electromagnetism is E = KQ / r².
(The upper letter “K” is the electric constant.)

Seems there is a bit different. What happens? Next we try to find a way to interconnect them.

We transform the equation F = KQq / r² in math.
Let K = K1 K2
Then the electric force formula can be transformed as F = (K1Q / r) × (K2q / r)
(Note: the symbol “×” means multiply only here.)

At this moment, smart guys who are sensitive in physics would be aware of this point: there might be two conceptions for the intensity of electric field.
So, we initiate the unilateral conception of field E = KQ / r² as well as the bilateral conception of field E = K1Q / r or E = K2q / r.

In unilateral conception of field: F = (KQ / r²) × q
It means field deals with particle.

In bilateral conception of field: F = (K1Q / r) × (K2q / r) = E1 × E2
It means field deals with field.

I supposes in unilateral conception of field, the electric constant should be denoted with upper letter K, while in bilateral conception of field, the electric constant should be denoted with lower letter k. K = k².

Then,
In unilateral conception of field, electric force F = (KQ / r²) × q;
In bilateral conception of field, electric force F = (K1Q / r) × (K2q / r) = (kQ / r) × (kq / r)

In unilateral conception of field, intensity of electric field E = KQ / r²
In bilateral conception of field, intensity of electric field E = kQ / r or E = kq / r

Now, we see the consistency of Ψ(r) =（1 / r）sin r p and the electricity in classical electromagnetism.
The conception of “electric wave” has a sound base.
(We should notice the dynamic electric field actually is wave characteristic. It’s a bit different from classical electromagnetism. We will analyze this point in details in the Theory of “electric wave” in the future.)

The Real Physics Meaning of “probability density”

In QM, the square of (absolute value of) amplitude of wave function is “probability density”, which is calculated with wave function multiplies its conjugate wave function:
Formula quoted from QM textbook: ω =︱ Ψ ︱² = ΨΨ*
Ψ（r, t）= N exp[i(p•r – Et)] note: Plank constant and vector symbol ignored for easy observation.
Ψ*（r, t）= N exp[i (p•r + Et)]
Then, ω =︱ Ψ ︱² = ΨΨ* = N exp[i(p•r)] • N exp[i(p•r)] = N²exp[i(p•r)] exp[i(p•r)]

Next, we go to explore the real physics meaning of this mathematical operation in QM.

In the mathematical model of unit charge, the “electric wave” can be represented by the equation:
Ψ(r ) =（1 / r）sin r p. The amplitude of wave function naturally means the intensity of electric field E.

Now we add the Et item to make a full wave function and associates it with its conjugate wave function. Denote them in the form of complex number.
Ψ(r, t ) =（1 / r）exp [i (r p – Et)] ① note: Plank constant and vector symbol ignored for easy observation.

Ψ*(r, t ) =（1 / r）exp [i (r p + Et)] ②

We notice the item “r p” actually is the doc product of vector “r” and vector “p” in case of angle θ equals to zero.
Design a simplest system of proton – electron: Assume the proton on the left side while the electron on the right side. They move away from each other relatively in a straight line.



If we consider ① is the “electric wave” activated by the “layer” in the proton (namely “p” is the momentum of the “layer” in the proton) and propagates rightward, then, ② can be considered as the conjugate “electric wave” responded by the “layer” in the electron and propagates leftward. (Note: thinking from the concept of “interaction”, it’s understandable why the “electric wave” propagates that way.)
If we consider ② is the “electric wave” activated by the “layer” in the electron (namely “p” is the momentum of the “layer” in the electron) and propagates leftward, then, ① can be considered as the conjugate “electric wave” responded by the “layer” in the proton and propagates rightward.
This is the real physics meaning of conjugate wave functions.
The conjugate wave functions reflect the spirit of interaction.

Below is the focus of this chapter.
Now we multiplies ① and ②.
Then, Ω = Ψ(r, t ) Ψ*(r, t ) = (1 / r²) exp[i (r p)] exp[i (r p)]
We see that the factor (1/ r²) actually has something to do with the electric force of unit charge interaction. (For details, please see the chapter “Unilateral conception of field vs bilateral conception of field”.)

Compare ω and Ω.
We see that the real physics meaning of “probability density” ︱ Ψ ︱² in QM should be to describe the electric force.

In QM, the formula of ω ahead describes a “quantum mechanics stationary state” in case of conservation of energy, which has nothing to do with time.
Then by comparison, we can see the “electric wave” function of Ψ(r ) =（1 / r）sin r p (in the mathematical model of unit charge) which has nothing to do with time actually describes a “stationary state” too. Just it’s a bilateral conception of field. The difference is the condition of “stationary state” of “electric wave” is conservation and quantization of charge.

(We should notice the dynamic electric force actually is wave characteristic. It’s a bit different from classical electromagnetism. We will analyze this point in details in the Theory of “electric wave” in the future.)

QM vs the Mathematical Model of Unit Charge, touchy and feely

In QM, the probability of a particle presenting in the whole space should be “1”.
c∫0→+∞︱ Ψ ︱²d 𝛕 = 1

As I said ahead, the real physics meaning of “probability density” ︱ Ψ ︱² in QM should be the electric force.
Then, thinking in the new way, if the integration of electric force in the whole space describes the probability of a particle presenting in the whole space should be “1”, it sounds non - sense.

Now we can think about it in this way: what physical quantity in classical electromagnetism can be a constant of “1”? Unit charge could be the best candidate.
Then, thinking in the new way, the integration of electric force in the whole space describes unit charge.

Next, quoted a section from the mathematical model of unit charge ahead:
1 = (1/2) + (1/π)∫0→+∞（1/ω）sinωt dω
Then, replace variables, we got:
1 = (1/2) + (1/π)∫0→+∞（1/ r）sin r p dr
Namely:
1 = (1/2) + (1/2) = (1/π)∫0→+∞（1/r）sin r p dr + (1/π)∫0→+∞（1/r）sin r p dr

Compare them, we can see:
QM touched the elephant from wave function to unit charge, while the mathematical model of unit charge touched the elephant from unit charge to wave function.
QM employed the “unilateral conception of field”, while the mathematical model of unit charge employed the “bilateral conception of field”.

We hereby has interconnected QM and the mathematical model of unit charge with mathematical – physics method in general, although the math on both sides not yet matches smoothly and accurately enough due to some technical details.
We hereby have dig out the root of QM.

Brief Introduction of Physics Model of Unit Charge

In my physics model of unit charge or say the “Chen’s Physics Model of elementary particle”, elementary particles can be divided into two big categories depending on matter state:
Matter state 1: Spherical electromagnetic wave. It’s the physics model for such elementary particles as electron, positron, proton, etc. Such elementary particles are consisted of two “layer”s.
Matter state 2: A section of electromagnetic wave travels in straight line in the speed of light c. It is the physics model of a released photon.
The elementary particles have an inherent property of spiral chirality. The corresponding anti - elementary particles have an inherent property of contrary spiral chirality. Either left – handed or right – handed. The electric interaction or charge is directly reflected by the spiral chilarity of spherical electromagnetic wave. If an elementary particle is neutral, it should have no spiral chirality. (The physics model is not the emphasis in this context. So, the details would not be provided.)

The Solutions to Hard Problems about the Speed of Probability Wave

Now we have a try to solve those two hard problems about the speed of probability wave delivered in the first chapter with the new conception of “electric wave” and the “Chen’s Physics Model of elementary particle”.

1. The range of velocity of 0 ＜ v ＜ c is just the character of movement of spherical electromagnetic wave. (For details, please see my article of “Research on the Root of Special Relativity”.)
As a whole, spherical electromagnetic wave doesn’t demonstrate wave characteristic. It demonstrates particle characteristic only.
Wave is the wave characteristic of electric interaction, not the wave characteristic of the particle of matter state 1 such as electron itself. The “wave” and the “particle” are different independent things. It’s unnecessary for the speed of “electric wave” “u” to be in conformity with the moving speed of the particle “v”.
A vivid analogy: A boat moves on the water and activates water wave. The water wave and the boat are different independent things. It’s unnecessary for the speed of water wave to be in conformity with the moving speed of the boat.

2. The Chen’s physics model for a released photon is “a section of electromagnetic wave travelling in straight line”, “a section” is from the angle of “particle” while “electromagnetic wave” is from the angle of “wave”. Of course v = u = c. Perfect conformity. Notice, at this moment, “u” changes to be the speed of electromagnetic wave rather than the speed of the “electric wave”.

Addition:
In QM, for a free particle, the probability of presence of the particle is the same in the whole space. It’s not understandable. Even it’s not in conformity with fact. Now we have a try to employ the conception of “electric wave” explain it: As illustrated ahead, the real physics meaning of “probability density” ︱ Ψ ︱² in QM should be to describe the electric force. For a free particle or say ideal neutral particle, the electric force it subjected should be zero, the same in the whole space.

The Chen’s physics model of elementary particle and the conception of “electric wave” can be in perfect consistent with Duc de Broglie’s original idea of “γ = E / h, while λ = h / P”.
The old flawed conception of probability wave should be given up.
QM should be the science to research the wave characteristics of electric interaction, or say, wave function should be to describe “electric wave”.

The Real Physics Meaning of Wave Function of Free Particle

When electric interaction disappears, it’s a free particle. An ideal situation of this is the infinite distance. We can see that the amplitude 1 / r of the wave function Ψ(r) =（1 / r）sin r p in the mathematical model of unit charge trends to zero when distance r trends to infinite. Actually the wave function will no longer meaningful. How to maintain the uniform of representation in math? We can assume the amplitude of the wave function is a constant A and it means the situation of without electric interaction. That’s the wave function for free particle. It’s a mathematical form only. Actually the wave characteristic in physics lost.

Complicated Conception of Conjugate Wave Functions

Design a complicated system of any charge Q vs q: Assume positive charge “Q” on the left side while negative charge “q” on the right side. They move away from each other relatively.
If we consider Ψ(r, t) =（k Q / r）sin (r p – Et) is the “electric wave” activated by the particle “Q” (namely “p” is the momentum of the particle “Q”) and propagates rightward, then, its conjugate “electric wave” is Ψ* (r, t) =（k q / r）sin (r p + Et).
Denote them in the form of complex number.

Ψ(r, t ) =（k Q / r）exp[i (r p – Et)] note: Plank constant and vector symbol ignored for easy observation.

Ψ*(r, t ) =（k q / r）exp[i (r p + Et)]

Then, ΨΨ* =（k Q / r）（k q / r）exp[i(p•r)] •exp[i(p•r)] = (KQq / r²) exp[i(p•r)] exp[i(p•r)]
KQq / r² = F in classical electromagnetism. It’s the magnitude of electric force for point charge.

If we consider Ψ(r, t) =（k q / r）sin (r p + Et) is the “electric wave” activated by the particle “q” (namely “p” is the momentum of the particle “q”) and propagates leftward, then, its conjugate “electric wave” is Ψ* (r, t) =（k Q / r）sin (r p – Et).
Denote them in the form of complex number.
Ψ (r, t ) =（k q / r）exp[i (r p + Et)] note: Plank constant and vector symbol ignored for easy observation.

Ψ* (r, t ) =（k Q / r）exp[i (r p – Et)]

Then, ΨΨ* =（k q / r）（k Q / r）exp[i(p•r)] •exp[i(p•r)] = (KQq / r²) exp[i(p•r)] exp[i(p•r)]
KQq / r² = F in classical electromagnetism. It’s the magnitude of electric force for point charge.

The conjugate wave functions reflect the spirit of interaction.

(We should notice the dynamic electric force actually is wave characteristic. It’s a bit different from classical electromagnetism. We will analyze this point in details in the Theory of “electric wave” in the future.)


(Reference material: any official QM textbook)

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Thank you, guys.
Have a lovely day.

Piggy hereby asks the a2k committee to investigate who’s the thumb down monkey / stalker and ask him to explain.
Professional physicists around the world to review.

Have you performed any experiments to see if any of this is true?