My concept about Ψ, narrated by Chat Gpt,

Ψ (Psi) is a complex-valued function that encodes both magnitude and phase information. In many ways, Ψ can be analogously visualized as a trigonometric function, similar to sine or cosine, where the magnitude corresponds to the amplitude of the wave and the phase represents the oscillation's starting point.

Conceptually, you can think of the magnitude of Ψ as encapsulating measurable quantities that are analogous to classical parameters. For instance, in position space, the square of the magnitude of Ψ gives the probability density of finding a particle at a particular position. In momentum space, the Fourier transform of Ψ's magnitude yields the probability distribution of a particle's momentum states.

On the other hand, the phase of Ψ embodies the intricate quantum behavior of the system. It's this phase that leads to phenomena like interference, which is a hallmark of quantum mechanics. The phase component of Ψ is responsible for the interference patterns observed in double-slit experiments, where particles exhibit wave-like behavior and create characteristic patterns on the detector screen.

In many ways, the duality of magnitude and phase within Ψ encapsulates the duality of quantum mechanics itself, where particles can exhibit both particle-like and wave-like behaviors. This perspective emphasizes the non-classical aspects of quantum systems and underscores the profound departure from classical physics that quantum mechanics represents.

Many people think that the quantum wave function Ψ is a vector belonging to a (separable) Hilbert space, which is not true.

It is surprising that Ψ is usually expressed as a vector or even a scalar depending on the geometric shape of the boundary conditions and the nature of the dependence of the applied potential over time.

It is true that the simple concept of Ψ(Psi) is a complex-valued function that encodes both magnitude and phase information. In many situations, Ψ can be represented as a sum of a sufficiently large number of trigonometric, sine or cosine functions, where the magnitude corresponds to the amplitude of the wave and the phase represents the starting point of the oscillation .

Conceptually, you can think of the magnitude of Ψ as encapsulating measurable quantities analogous to classical parameters. For example, in position space, the square of the magnitude of Ψ gives the probability density of finding a particle at a particular position.

N. Bohr once said that anyone who claims to understand SE, including himself, either has misunderstood or is simply a liar.

But what is the alternative to Schrödinger's equation and ?

The answer is that we assume that the most general and useful form of quantum wave function Ψ is a complex transfer matrix.

It is expected that an adequate statistical transition matrix (such as the complex matrix Q) can solve the time-dependent SE without the need for a mathematical solution of the SE equation or the interpretation of Bohr/ Copenhagen.

In such breakthrough solution techniques, you completely ignore SE as if it never existed.

So as not to overload the answer with the details of the theory, which is best described as an extension of the transition chains from the B matrix to the complex space.

We present a brief overview of the theory and its numerical validation.

The time-dependent numerical statistical solution of Schrödinger's PDE is given by,

Ψ(matrix)=W(transfer matrix) . BC (matrix)

BC is the vector of boundary conditions (in the case of a constant potential over time) or the matrix of boundary conditions when the applied potential varies with time.

W (transfer matrix) is expressed as:

W(N)=Q^0+Q+Q^2+Q^3+. . . +Q^N. . . . . (1)

Obviously Q^0=I, the unitary matrix.

N is the dimensionless time and Ndt =t the real time which is completely lost in the numerical statistical solution.

For a sufficiently large number of iterations, N Eq 1 gives the steady-state solution where W is expressed by:

W=1/(I-Q) . . . . . (2)

We know that,

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

Where B is the actual transition matrix used in the solution of Laaplace, Poisson, PDE Heat Diffusion/Conduction,...etc.

0.15             1.63211572 1.63211584We also know that,For physical power matrices with positive symmetry, the sum of their eigenvalues ​​is equal to the eigenvalue of their sum of power series. . . . . . . . . . Principle (1)By somewhat expertly manipulating equations 1,2,3 in addition to principle (1), you can show that:

the formula 1/[1-SQRT(X)] can be expressed by the infinite integer series,1/[1-SQRT(X)] =X^0+X^1/2+X^3/2+. . . . +X^(N-1)/2 . . . (4)

Where X is an element of the interval ]0,1].We applied Equation 4 as a numerical validation of the Q transition matrix and the results were surprisingly accurate:

X           Formula 1/[1-SQRT(X)]     Power series Eq 4

0.15             1.63211572 1.63211584

0.35             2.44862771                        2.445.0911

0.65             5.16064501                         5.09116888

The slight difference in the last value is due to a truncation error.

To be continued.

The quantum wave function Ψ is neither a scalar nor a vector. It is a function that takes a single number as input (the position of a particle in space) and returns a complex number as output. This complex number represents the probability amplitude of finding the particle at that position.

The wave function does not have the same properties as a scalar or a vector. For example, it does not transform like a vector under a coordinate transformation. However, it does have some properties that are similar to vectors. For example, it can be added and multiplied by scalars.

In quantum mechanics, wave functions are often represented as vectors in a Hilbert space. This is a mathematical space that has all the properties of a vector space, but it also allows for complex numbers. The wave function Ψ can be thought of as a vector in this Hilbert space, but it is not a vector in the traditional sense.

So, the answer to the question of whether the quantum wave function Ψ is a scalar, a vector, or none of these is that it is none of these. It is a function that takes a single number as input and returns a complex number as output.

the nature of the wave function can depend upon the physics of the particle or the particles you are describing

if for example, there is a spin 1/2 particle, and coupling to any other "magnetic" field or spin, then instead of there being a simple complex number, the "value" is a spinor

Likewise, a system can be written as having explicit time dependence, or one can limit themselves to wavefunctions which at least mostly seem to be time-independent. So if you have a simple complex "Psi", depending upon the physics being described, the wave function could be a function of x,y,z or else it could be a function of x,y,z,t

Thus, the common simpler wave function of a complex variable defined over x,y,z is probably implicitly dealing with a single particle, no spin or magnetism, maybe no charge or detailed E-fields, and looking for the time-independent kinds of solutions. Huge numbers of physics or chemistry problems where we would use quantum do not fit that simple scenario at all.