[compuAI (OP)]

It is not clear what complex wave function means in Schrödinger, Pauli, Dirac equations. Is it always two-component (complex), or can it be real, or are both variants possible in different situations?
For example, how to understand:
-i • h/(2•π) • ∂ψ/∂t = h²/(8•π²•m) • div grad ψ
(for simplicity in absence of potential multiplied by function).
The imaginary unit “i” simply shows that quantum operator is used instead of classical derivative, or function must be divided into two components:
ψ = ψ1 + i • ψ2
and then in reality there are two equations
∂ψ1/∂t ~ div grad ψ2
∂ψ2/∂t ~ div grad ψ1
(~ symbol means is proportional with a constant multiplier).
In this case, the question arises how this relates to de Broglie equation, because it turns out to be
∂²ψ1/∂t² ~ div grad (div grad ψ1)
∂²ψ2/∂t² ~ div grad (div grad ψ2)
instead of traditional ∂²ψ/∂t² ~ div grad ψ
or ∂²ψ/∂t² ~ rot rot ψ for different kinds of waves.
Or is function real (should be, or can be)?
If Maxwell's equation is written as one formula, there are two components, electric field and magnetic, but instead of squared nabla single nabla (curl) is used, and this is consistent as de Broglie wave.
Do Pauli and Dirac equations follow the same principle as Schrödinger equation with respect to the complexity of function, or there are differences?

[Zer0]

I have zero Understanding on the Topic...

But I'm responding so that your OP goes back to the Top of the page list..

& Then Hopefully someone shall take Notice of it & perhaps try to Answer.

Thanks!

[compuAI (OP)]

It seems the topic is solved. I have gotten answers in another way.
As it turned out, within the framework of quantum mechanics wave function is always complex,
and both components are important. The name "wave" was assigned for historical reasons,
since in some special cases stationary function is really similar to standing waves.
But in reality these are "diffusion" equations, not "wave" equations, where first time
derivative corresponds to second spatial derivative (div grad).

[evan_au]

both (real & imaginary) components are important

As I understand it, the magnitude of a wave function indicates the probability of detecting the particle in a particular place.
- The magnitude includes both real and imaginary components

[Eternal Student]

Hi.

As I understand it, the magnitude of a wave function indicates the probability of detecting the particle in a particular place.

   Yes.      At a particular place and also a particular time  if  ψ = ψ(x,t) = a function of space AND time,    which is how  CompuAI seemed to be representing the wave function.   
    Also,   it's a probability density rather than a probability on it's own.        Given a small element of space  δx    and  of time δt    then  you do need to multiply by the space-time volume element  δt δx,    so that    |ψ(x,t)|² . δx . δt    =  probability of finding the particle in the element of space  between  x and x + δx    and  at a time between   t and t + δt.
     Similar idea  if  it's 3-dimensional space instead of just one dimensional along the x-axis.    Then  we use the space element δ³x  =  δx δy δz.

    Anyway,  the first part of what CompuAI had written looks OK.    You CAN consider the Schrodinger equation as a some differential equations involving only a real valued  function(s) if you want to.    In that case,  you have a PAIR  of simulatenous differential equations linking time derivatives of  Ψ₁ (x,t)  to  space derivatives of Ψ₂ (x,t)    (and vice versa....  time derivatives of Ψ₂   are linked to space derivatives of Ψ₁ )   but   both  Ψ₁    and  Ψ₂    are  real-valued functions.

     In general, it's NOT easier that way.    In solving that pair of simultaneous differential equations you probably would start by just writing it as one combined differential equation with a complex valued function.    However, it's of some relevance because there is always the discussion point about whether Complex numbers are in some sense "real",  i.e. that they have some deep real world meaning because they are essential in Quantum Mechanics.    The sort of thing where you replace the Schrodinger equation with a pair of real valued differential equations suggests that,  no you don't need Complex numbers,   you can keep all wave functions real valued  BUT it's harder that way.

Best Wishes.

[Zer0]

Compu...Oddly you Always end up Answering your Own questions.
Simply Brilliant!
👌
& Thanks for making it a point to Post Answers to your Own OPs once you find them out.
I'm Sure future readers & generations shall Benefit from it.
👍