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I recently watched this video. I'm trying to learn about the origin of the wave function and therefore understand its use in the Schrödinger Equation.

However at the end of the video I understood up to:

$$ψ = \cos(kx - ωt)$$

This would be the real part of the wave function wholly defined as:

$$\cos(αx) + i\sin(αx)$$

Or:

$$ψ = e^{iαx}$$

What I'm having trouble understanding is the imaginary part of the wave function. It was never explained why we needed the imaginary sine function added to the real part of the wave, or what $e$ is.

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It is misleading to consider the real and imaginary parts of the wave function separately. The wave function is a function of spacetime that returns a complex number. We interpret this as meaning that the wavefunction requires two components to describe it. You can think of this as an amplitude and a phase. However the split between the real and imaginary parts is arbitrary and can be changed by a coordinate transformation, so there is nothing special about about the real part or the imaginary part.

The wavefunction is not an observable, so the fact it is a complex quantity does not matter. Anything we can observe is given by acting on the wavefunction with a Hermitian operator, and these are guaranteed to return a real result.

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The "real" part of wave function is no more real than the imaginary part. Both these parts are equally real or equally imaginary. None of them can independently describe the physical reality. Only when both these part are taken together then they represent the physical reality.

Either one of them can be termed real or imaginary. Since complex numbers provide ready means of describing numbers with two dimensions, they come in handy to describe the wave function. But there is no reason why an alternate mathematical structure that provides two dimensions to represent a number can not be employed to describe wave function.

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The equation for adding sinusoidal functions which differ in phase resembles the equation for adding the x (or y or imaginary) components of vectors which are rotating in the xy or complex number plane. It is often more convenient to work with these (non-existent) vectors than with the functions. This especially true with vectors represented by imaginary exponential functions. Generally, only one component of the vectors are of interest.

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