What is the meaning of Wave function?

[Outrageous]

Wave function is a complex number , why do we have to consider it as a complex number?
Quote:– but there is nothing in the wave equation that restricts them
to being real numbers. If we temporarily suspend disbelief that physical
quantities may be associated with complex amplitudes, then we may explore
what happens when these constants take complex or imaginary values.
from http://phyweb.phys.soton.ac.uk/quantum/lectures/waves4.pdf

is that mean no necessary to have wave function as a complex number?
I can understand why do we use complex number in the reactance of a capacitance because the impedance of a capacitance varies with the frequency.
Can you please teach me what is the meaning of the complex number in wave function ? anything change that make the intensity change??
Thank you.

 

[jtbell]

For physical quantities that are real (e.g. displacement of a mass on a spring, electric field in an electromagnetic wave), using the complex form of a wave is purely a mathematical convenience. It's always understood that we have to take the real part of the complex form at the end of the calculation.

You can probably take any such calculation and start with e.g. $\cos(\omega t)$ instead of $e^{i\omega t}$ to represent an oscillating motion, but then you have to mess around a lot with trigonometric identities.

In quantum mechanics, the situation is different. There was recently a long discussion in the Quantum Physics forum about why the QM wave function is complex:

https://www.physicsforums.com/showthread.php?t=683821