Since the time of N. Bohr, E. Schrdinger, W. Heisenberg and all the great scientists, physicists and mathematicians have called the Schrödinger equation (SE) the quantum wave equation.

If we agree on a definition of the wave as the propagation of oscillations in space and time as in emw (E and H oscillate), Sound wave (pressure and displacement oscillate), then the question arises :

Why did physicists and mathematicians work out the details of Schrödinger's equation in different situations up to the 10th digit, but forgot to specify what oscillates there.

McCarthyism era quantum mechanical command:

Shut up and calculate,

suppressed or closed the door to genuine research, at least partially.

However, statistical transition B-matrix chains give birth to complex Q-matrix chains, eligible for solving time-dependent SE numerically.

The real matrix B is well defined and the complex matrix Q is also well defined as follows:

Q=Sqrt (B).

Therefore, we assume that the real chains of the statistical matrix B can provide a concept for the oscillations or waving in SE.

It is still discussed if the wave function

Is real or not

Alternitivly you can use matrices

For most qm calculation

“…Since the time of N. Bohr, E.Schrdinger, W.Heisenberg and all the great scientists, physicists and mathematicians have called Schrödinger's equation the quantum wave equation….”

- that isn’t completely so. Really firstly de Brogile the hypothesis that electron is “a wave” appeared, when he pointed that analogously Bohr hypothesis that stable electron orbits exist provided electron angular momentum integer number of ћ, the stable orbits also can be formed by integer number of standing ways with lengths λ=h/p, p is momentum of electron. Including after this the Schrödinger “wave -function” equation was developed; and it is widely used in mainstream QM till now.

However that

“…..If we agree on a definition of wave as oscillations in space and time as in emw (E and H oscillate), Sound wave (pressure and displacement oscillate), then the question arises:

Have physicists and mathematicians worked out the details of Schrödinger's equation in different situations up to the 10th digit, but forgot to specify what oscillates there?.…..”

- is quite correct. Mainstream physics indeed really fundamentally doesn’t understand – why QM is as it is [and everything in physics else, though] because of that in the mainstream all really fundamental phenomena/notions, first of all in this case “Matter” “Consciousness”, “Space”, “Time”, “Energy”, “Information”, are fundamentally completely transcendent/uncertain/irrational,

– and so everything in Matter, i.e. “particles”, “fields”, etc., is/are fundamentally completely transcendent/uncertain/irrational phenomena, etc., as well. Correspondingly all what mainstream can is that at experiments discover some for/by some mystic reasons and ways existent stable links between some really also existent for/by some mystic reasons and ways “particles”, etc.,

- however yeah, vast majority of mainstream physicists don’t suspect that they do for a mystic reason adequate sometimes to the reality work; and just because of this the mainstream physics is as it is; from that only rare exclusions are; and seems the author of this thread is a one of “a non-lucky few”.

Any really non-transcendent scientific understanding of what exists and happens in Matter can be only in framework of the philosophical 2007 Shevchenko-Tokarevsky’s “The Information as Absolute” conception, recent version of the basic paper see

https://www.researchgate.net/publication/363645560_The_Information_as_Absolute_-_2022_ed

- where it is rigorously proven that there exist nothing else than some informational patterns/systems of the patterns that are elements of the absolutely fundamental and absolutely infinite “Information” Set, and so all the fundamental p0heniomena/notions above are rigorously scientifically defined;

- and more concretely in physics in the Shevchenko-Tokarevsky’s informational physical model , two main papers in this case are

https://www.researchgate.net/publication/354418793_The_Informational_Conception_and_the_Base_of_Physics, and

https://www.researchgate.net/publication/355361749_The_informational_physical_model_and_fundamental_problems_in_physics .

First of all in the model it is rigorously scientifically shown that everything that exists in Matter is/are disturbances in Matter’s ultimate base - primary elementary logical structures – (at least) [4+4+1]4D binary reversible fundamental logical elements [FLE], which compose the (at least) [4+4+1]4D dense lattice, which is placed in the Matter’s fundamentally absolute, fundamentally flat, and fundamentally “Cartesian”, (at least) [4+4+1]4D spacetime with metrics (at least) (cτ,X,Y,Z, g,w,e,s,ct).

Particles are specific disturbances in the lattice - cyclic close-loop algorithms that move in the lattice, and just so the trajectory of a particle’s “FLE-flip point” at given time moment is some at least 4D “helix” in the 4D space with metrics (cτ,X,Y,Z). Since the observed directly interactions happen really in the 3DXYZ space, that happens in the 3D projection of the helix, which is a 3D “wave”.

That is the explanation of what is the “wave-point particle duality” - particles indeed interact, at least practically, only as points that have sizes ~ Planck length, while this point at given time moment accidentally can be in some space volume with size ~ћ/p.

More see the linked above papers, SS posts in

https://www.researchgate.net/post/Should_all_light-emitting_processes_be_described_by_the_same_equations/2 , and

https://www.researchgate.net/post/Is-it-true-that-physicists-do-not-want-to-understand-quantum-mechanics#view=642b89716f2a73457707a995/250 July 19, are relevant to this thread question as well.

Cheers

Answer II - continued

Schrödinger's quantum "wave" equation with the Bohr/Copenhagen interpretation was invented to describe the electronic transitions between energy levels of the hydrogen atom, which is a special case of the case of a particle in a central potential field.

In order to understand which ripples in SE, if any, we need to study the general case of a particle (quantum subatomic or not) in a central potential field.

Let us now turn to the case of the quantum particle in a central field and the misunderstanding of Bohr's theory of the hydrogen atom and its principal quantum number n.

N. Bohr presented his theory as being based on two assumptions:

i-Quantization of the orbital angular momentum of the orbiting electron, namely,

mv. 2 Pie r=n h . . . (1)

2- The electronic transition between two quantized energy levels E1 and E2 is associated with the emission (or absorption) of a photon of frequency f,viz,

h f=E2-E1 . . . (2)

*In fact, the two equations 1 and 2 are not independent since Eq 1 if completed by the laws of conservation of energy and Newton's laws of motion and gravity can lead to Eq 2 .

Similarly, Eq 2 can produce Eq 1 via inverse physical processing.

** Also, the second, Equation 2 proposed in 1913 is not completely new but rather a natural extension of Einstein's equation of the photoelectric effect in 1905, namely,

h f = W + 1/2 m v^2 (max) . . . . . (3)

In other words, Bohr extended the interaction between an emw photon and the free electron which had been well known theoretically and experimentally since 1905 to the interaction between photons and bound electrons at different energy levels of the hydrogen atom.

It should be mentioned that this is not to calculate from the great scientist Bohr and his brilliant achievements but to state that Bohr's only contribution is his famous Eq 2.

For the case of the hydrogen atom, the stationary circuit orbits satisfy the condition,

Centrifugal force = Force of attraction, i.e.

mv^2/r = e^2/[4 Pie Eps r^2] . . . . (4)

Bohr applied equations 1,4 to arrive at his equation relating the energy eigenvalues ​​of the quantized energy states n = 1, 2, 3, 4...etc with the principal quantum number n as,

E(n) = constant . n^2 . . . . . (5)

That is, if the ground state energy (n=1) of the hydrogen atom is E1= -13.4 ev then this implies that E2=-3.35 and E3= -1.49 ...etc.

Surprisingly enough, Bohr's conclusion Eq 5 does not belong exclusively to Bohr nor to the hydrogen atom but belongs in general to any particle in a central field as elucidated by the statistical theory of the transition matrix B:

It can be shown that the transfer matrix D of B-matrix for the potential of a particle in a central field is a diagonal matrix where the diagonal elements are equal to 2/n.(2,1,2/3,1/ 2 .. .etc)

The potential energy matrix is ​​the square of this matrix which is another diagonal matrix of 4/n^2 diagonal elements ( 4,1,4/9,1/4 ...etc).

We know that the eigenvalues ​​of a diagonal matrix are equal to the values ​​of its diagonal.

therefore, equation 5 can be predicted statistically without the need for two Bohr assumptions.

To be continued.

Answer III - continued

In short, we believe that SE can be either a diffusion-like equation or a wave-like equation depending on the nature of the potential applied to the quantum particle.

When the potential applied to the quantum particle is practically zero (the main diagonal elements of the complex transition matrix Q are all zero), it behaves like a wave.

In other words, the particle exhibits a wave nature which is basically interference and diffraction.

The answer to the question is that nothing ripples or oscillates in the Schrodinger equation, but it can describe a wave nature of quantum particles regarding interference and diffraction.

It all started in 1927 with the well-known experiment of Davisson and Germer where a beam of electrons showed diffraction and interference properties as they passed through atomic crystals.

Davisson's experiment was followed by similar experiments of interference and diffraction of a "coherent" beam of electrons. A stream of focused electrons from a hot cathode strikes a plate with two narrow slits separated by a small distance. The electrons transmitted through the slits are observed to form a typical diffraction and interference pattern on the screen behind the slit plate.

Surprisingly enough, the experimental wavelength of the interference patterns coincides precisely with that predicted by De Broglie:

Lambda = h / m v. . . . (1)

A few years later, after the Davisson experiment, in 1931, Max Knoll and Ernst Ruska produced the first electron microscope. It is clear that the electron microscope is based on the theory and the experiment of equation 1

Due to its versatility, this magnificent instrument has taken a revolutionary leap forward in the microscope industry with the highest magnification power.

To be continued.

Answer IV- continued

In this answer, we explain the numerical statistical solution to the time-dependent SE without needing the SE itself in the same way used for solve the transient heat equation without using the heat equation.

In other words, we completely ignore the SE and the Bohr/Copenhagen interpretation as if they never existed.

The numerical statistical solution to SE will be,

Ψ=W . b. . . ..(1)

where b is the vector of the real potential applied to the quantum particle and W is the complex quantum transfer matrix expressed by:

W=Q + Q^2+Q^3+ . . .Q^N . . . . (2)

It was shown before that,

Q=Sqrt (B).. . . (3)

where B is the well-known real transition matrix used to find time-dependent solutions for the Poisson, Laplace, and heat diffusion partial differential equations.

Equations 1,2 show that the solutions of Schrödinger's equation depend only on the shape of the potential and the boundary conditions as expected.

In order not to worry too much about the details of the theory, let's move on to the following illustrative specific application without loss of generality:

Consider the simplest Cartesian geometrical shape , a cube of length L and eight vertices that represent the nodes or quantum states.

The B 8x8 transition matrix is given by,

0 1/6   0 1/6 1/6   0   0   0

1/6   0 1/6   0   0 1/6   0   0

0 1/6   0 1/6   0   0 1/6   0

1/6   0 1/6   0   0   0   0 1/6

1/6   0   0   0   0 1/6   0 1/6

0 1/6   0   0 1/6   0 1/6   0

0   0 1/6   0   0 1/6   0 1/6

0   0   0 1/6 1/6   0 1/6   0

and the complex quantum transition matrix Q=Sqrt (B)  would be,

[Note the appearance of 2^1/2 which has a special importance in quantum mechanics.]((1+i)*2^0.5+(1+i)*6^0.5)/16 ((3-3i)*2^0.5+(1-i)*6^0.5)/48 ((3+ 3i)*2^0.5-(1+i)*6^0.5)/48 ((3-3i)*2^0.5+(1-i)*6^0.5)/48 ((3-3i)*2 ^0.5+(1-i)*6^0.5)/48 ((3+3i)*2^0.5-(1+i)*6^0.5)/48 ((1-i)*2^0.5-( 1-i)*6^0.5)/16 ((3+3i)*2^0.5-(1+i)*6^0.5)/48((3-3i)*2^0.5+(1-i)*6^0.5)/48 ((1+i)*2^0.5+(1+i)*6^0.5)/16 ((3- 3i)*2^0.5+(1-i)*6^0.5)/48 ((3+3i)*2^0.5-(1+i)*6^0.5)/48 ((3+3i)*2 ^0.5-(1+i)*6^0.5)/48 ((3-3i)*2^0.5+(1-i)*6^0.5)/48 ((3+3i)*2^0.5-( 1+i)*6^0.5)/48 ((1-i)*2^0.5-(1-i)*6^0.5)/16((3+3i)*2^0.5-(1+i)*6^0.5)/48 ((3-3i)*2^0.5+(1-i)*6^0.5)/48 ((1+ i)*2^0.5+(1+i)*6^0.5)/16 ((3-3i)*2^0.5+(1-i)*6^0.5)/48 ((1-i)*2 ^0.5-(1-i)*6^0.5)/16 ((3+3i)*2^0.5-(1+i)*6^0.5)/48 ((3-3i)*2^0.5+( 1-i)*6^0.5)/48 ((3+3i)*2^0.5-(1+i)*6^0.5)/48((3-3i)*2^0.5+(1-i)*6^0.5)/48 ((3+3i)*2^0.5-(1+i)*6^0.5)/48 ((3- 3i)*2^0.5+(1-i)*6^0.5)/48 ((1+i)*2^0.5+(1+i)*6^0.5)/16 ((3+3i)*2 ^0.5-(1+i)*6^0.5)/48 ((1-i)*2^0.5-(1-i)*6^0.5)/16 ((3+3i)*2^0.5-( 1+i)*6^0.5)/48 ((3-3i)*2^0.5+(1-i)*6^0.5)/48((3-3i)*2^0.5+(1-i)*6^0.5)/48 ((3+3i)*2^0.5-(1+i)*6^0.5)/48 ((1- i)*2^0.5-(1-i)*6^0.5)/16 ((3+3i)*2^0.5-(1+i)*6^0.5)/48 ((1+i)*2 ^0.5+(1+i)*6^0.5)/16 ((3-3i)*2^0.5+(1-i)*6^0.5)/48 ((3+3i)*2^0.5-( 1+i)*6^0.5)/48 ((3-3i)*2^0.5+(1-i)*6^0.5)/48((3+3i)*2^0.5-(1+i)*6^0.5)/48 ((3-3i)*2^0.5+(1-i)*6^0.5)/48 ((3+ 3i)*2^0.5-(1+i)*6^0.5)/48 ((1-i)*2^0.5-(1-i)*6^0.5)/16 ((3-3i)*2 ^0.5+(1-i)*6^0.5)/48 ((1+i)*2^0.5+(1+i)*6^0.5)/16 ((3-3i)*2^0.5+( 1-i)*6^0.5)/48 ((3+3i)*2^0.5-(1+i)*6^0.5)/48((1-i)*2^0.5-(1-i)*6^0.5)/16 ((3+3i)*2^0.5-(1+i)*6^0.5)/48 ((3- 3i)*2^0.5+(1-i)*6^0.5)/48 ((3+3i)*2^0.5-(1+i)*6^0.5)/48 ((3+3i)*2 ^0.5-(1+i)*6^0.5)/48 ((3-3i)*2^0.5+(1-i)*6^0.5)/48 ((1+i)*2^0.5+( 1+i)*6^0.5)/16 ((3-3i)*2^0.5+(1-i)*6^0.5)/48((3+3i)*2^0.5-(1+i)*6^0.5)/48 ((1-i)*2^0.5-(1-i)*6^0.5)/16 ((3+ 3i)*2^0.5-(1+i)*6^0.5)/48 ((3-3i)*2^0.5+(1-i)*6^0.5)/48 ((3-3i)*2 ^0.5+(1-i)*6^0.5)/48 ((3+3i)*2^0.5-(1+i)*6^0.5)/48 ((3-3i)*2^0.5+( 1-i)*6^0.5)/48 ((1+i)*2^0.5+(1+i)*6^0.5)/16

It can be shown that equation 2 for a sufficiently large number of time jumps or iterations N which gives the stationary time-independent solution reduces to,

W=  1/ (I-Q) . . . . .(4)

Therefore the transfer matrix W is given by,

((105+35*i)*2^0.5+(63+45*i)*6^0.5+928)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5-(63-45*i)*6^0.5+32)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840

((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5+(63+45*i)*6^0.5+928)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5-(63-45*i)*6^0.5+32)/840

((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5+(63+45*i)*6^0.5+928)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105-35*i)*2^0.5-(63-45*i)*6^0.5+32)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840

((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5+(63+45*i)*6^0.5+928)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5-(63-45*i)*6^0.5+32)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840

((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5-(63-45*i)*6^0.5+32)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105+35*i)*2^0.5+(63+45*i)*6^0.5+928)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840

((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5-(63-45*i)*6^0.5+32)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5+(63+45*i)*6^0.5+928)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840

((105-35*i)*2^0.5-(63-45*i)*6^0.5+32)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5+(63+45*i)*6^0.5+928)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840

((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5-(63-45*i)*6^0.5+32)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5+(63+45*i)*6^0.5+928)/840

And the steady-state numerical solution Eps=(W-I) . b for the vector of unit boundary conditions,

b= [1,1,1,1,1,1,1,1] T, becomes,

2^0.5+1

2^0.5+1

2^0.5+1

2^0.5+1

2^0.5+1

2^0.5+1

2^0.5+1

2^0.5+1

where all eigenvalues ​​of energy are real as expected.

[Note again the multiple appearance of 2^1/2 which has special significance in quantum mechanics.]

To be continued.

Answer V- continued

The iron guards of the Bohr/Copenhagen interpretation defend to the last breath all issues of this interpretation, whether true or false.

Thanks to the opposition of the new generation of theoretical physicists and mathematicians, the old iron guards are only a small remnant of the era of MacArthism where the mastery of quantum mechanics dominated: Shut up and calculate.

Going back to SE with Bohr's interpretation, I guess these are the three most common mistakes while I'm sure most Researchgate contributors can find more:

1- Units and Dimensions

Bohr/Copenhagen interpretation of the wave function Ψ

defines the units and dimensions of Ψ differently depending on the number of dimensions of the receiving space.

i-For a spatial wave function of position 1D Ψ(x)

the normalization condition would be ∫Ψ∗(x)Ψ(x)dx=1

so Ψ has inverse square root units of length, which is m−1/2

ii-Similarly, for a three-dimensional position, the spatial wave function Ψ(x) has its own normalization condition such as:

∫Ψ∗(x)Ψ(x)d^3x=1

Ψ(x) has units of square root of inverse volume, which is m−3/2

Moreover, theoretical physics does not allow it, it is not acceptable.

Units and dimensions must be raised to zero or a whole whole power, but never to a fraction.

* If we compare this to the case of replacing SE with the statistical matrices Q and W, we find that the dimensions of W and Q are always the square root of the energy density per unit volume and the same is true for 1D, 2D and 3D.

The normalization condition is a pure summation condition and not an integration condition which is the sum of the multiplication.

.

2- The problem of distance and infinity

Distance is still defined in essentially the same way as in classical physics. In other words, spacetime is a flat Euclidean space with the usual Euclidean distance.

If we talk about the distance between two quantum particles, they must have a definite position in order to measure the distance between them. This requires them to be in proper positional states.

We assume that the SE interpretation with Bohr is limited to the study of the hydrogen-like atom where the total diameter distance is a few angstroms (Å) (1 A = 1E-10 m) and the effective electron radius is estimated at E -2 angstrom (Å).

The n=infinity mentioned in SE relates to the quantum number of infinite principle n which again corresponds to the diameter of the hydrogen atom by a few angeströms.

Obviously, the units and dimensions of the distance are not well defined in the SE interpretation.

** If we compare this to the case of replacing SE by the statistical matrices Q and W, we see that in the latter case the distance is well defined in 1D, 2D and 3D geometric spaces simply by ordering and arranging the nodes in these as we did for the actual transition matrix B.

3- Rigorous derivation

There is no rigorous or even serious derivation of SE with the Bohr/Copenhagen interpretation.

E. Schrödinger's solid derivation in 1927 proved other concepts in his equation and he himself considered the Bohr/Copenhagen interpretation an unpleasant joke. He expressed his feelings and thoughts through his well-known paradox. (Schrödinger's cat dead or alive).

*** If we compare this to the case of replacing SE with the numerical solution of the temporal chains of the statistical matrix Q, we see that in the latter case the derivation emerges from the well-proven statistical transition matrix B. Moreover, the results of the B-strings and Q-strings are

numerically validated.

To be continued.

What is waving in the Schrödinger equation and why is it called the “wave” of the Schrödinger equation, especially when its phase is undefined?

It is well known that there is theoretical and experimental evidence for a causal relationship between the phase of the wave function and physical reality.

The Copenhagen interpretation of quantum mechanics, which only gives physical meaning to the magnitude of the wave function, cannot be considered complete on this basis.

* A new dynamic-statistical interpretation of quantum mechanics is needed [1,2].

Believe it or not, attaching a well-defined phase to the amplitude of the SE wave would no longer complicate it but on the contrary would make it more understandable and its solution more accessible.

However, we assume that defining a phase at the amplitude of SE can be done via two different approaches:

i-reform the Bohr/Copenhagen interpretation of the Schrödinger equation.

ii-Apply the complex transition matrix Q to find the statistical numerical solution of SE.

To be continued.

1-Ivan Georgiev Koprinkov, Phase Causation of the Wave Function or Can the Copenhagen Interpretation of Quantum Mechanics Be Considered Complete? Journal of Modern Physics Vol.7 No.4, February 2016.

2-I.Abbas,Numerical statistical resolution of the Schrödinger wave equation, Researchgate.

-