We assume that the answer to the question of whether the quantum wave function Ψ is a scalar, a vector or not is that it is none of these.
Ψ is an abstract mathematical operator that has no physical meaning in itself.
The currently accepted procedure for describing Ψ in a given isolated quantum system is to construct the time-dependent Schrödinger SE equation for that particular system and then solve it by the method of separation of variables.
The importance of Ψ is that it conveys the phase of the system in x-t space and therefore can explain its wave properties such as interference, diffraction, degeneracy, etc.
We propose a simple alternative to solve for Ψ^2 (and not Ψ itself) which has a physical meaning of quantum particle energy per unit volume (or probability of finding the particle in unit volume of space unified x-t), then in a final solution last step to find Ψ as the square root of Ψ^2.
It is clear that the proposed technique completely neglects SE as if it never existed and proceeds to solve the Ψ^2 quantum system via a well-established probability diffusion equation.
Prof. Ismail Abbas
I will follow a book on Quantum Mechanics, that I like very much to try an answer to your thread :
Lectures On Quantum Mechanics by Prof. G. Baym, W.A. Benjamin, Inc., New York, 1969.
Chapter three gives a simplified but practical concept of a wave function as a complex function Ψ(r, t) that gives the probability of finding a particle at point r and time t.
It complements by saying that is the module | Ψ(r, t) |2 d3r in the volume d3r that gives such a probability (page 46, chapter 3).
We can complement for example by defining the space used in non-relativistic quantum mechanics, as the configuration space (see for example, Prof. A. Davydov book on quantum mechanics).
The same wave function is found in the Ginsburg-Landau theory, but in a nonlinear equation.
What I like the most is that Ψ(r, t) in a quasi-classical context contains a damping term.
Best Regards.
Prof. Ismail Abbas
Good Morning.
To compliment the previous answer to your inquiry, if you read from the same Prof. G. Baum book page 47, of same chapter 3; he wisely writes: I quote it verbatim "The amplitudes of Ψi(r, t) are like the components of a giant vector"
I hope this helps. Best Regards.
This question shows, in a way, that we are in great difficulty because current physics and mathematics are incomplete.
However, the short answer to this question is that Ψ(x,t) lives in 4D Schrödinger x-t space which can be described numerically by an nxn square matrix.
Additionally, Ψ^2 has two connected (non-independent) interpretations, namely quantum energy density and quantum particle probability density.
If we go back to the history of current science, it started with Newton who considered the universe to be three-dimensional to have a non-curved 3D spatial Euclidean geometry, and Newtons time was an external parameter or external controller.
He introduced his famous three laws of motion and, using these laws, was able to establish his dynamic theory of the universe.
On the other hand, Einstein, along with space, considered time as a fourth dimension in a four-dimensional universe in his theories of special and general relativity.
Time t is the fourth dimension woven into a 3D geometric space to form a unitary and inseparable 4D x-t space. Schwarzschild solved Einstein's field equations of general relativity around enormous masses such as a star in space, proving their accuracy and precision.
It is no longer possible to return to the 3D+t world, but once again we are in great difficulty because almost all physical and mathematical sciences must be reformulated.