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Is it safe to say that imaginary part of wavefunction (one that "located" in a Complex plane around it's real component) - always represents some physical entity, that cannot be simultaneously known as precise as whatever is represented by real part?

Say, if "real" component is position - then "imaginary" will be momentum, and vice versa (except for value conversion).

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It seems you've been bit by the idea that "imaginary numbers" are somehow "not real", as in "fictitious", because of their name. This name is just a historical artifact; it has no philosophical implication from a modern mathematical point of view. Imaginary/complex numbers are just one mathematical structure among many that are actually useful tools for describing things in the real world, many others of which are far more exotic.

Instead of looking at a wave function in terms of real and imaginary components, it is much better to look at it in magnitude-phase representation, or "polar form":

$$\psi_x(x) = \sqrt{f_x(x)} \cdot e^{i \phi_x(x)}$$

The magnitude of the wave function is $\sqrt{f_x(x)}$. This is the square root of the probability density function, or pdf, for the particle position. The other part of the product above is the phasor. This part is important in governing the evolution and also producing quantum interference effects. It tells us how that, when we superpose the wavefunctions of two possible circumstances, those circumstances will interfere with each other: if phases are aligned, they will interfere constructively, and if anti-aligned, then they will cancel each other out.

Indeed, the real/imaginary components have much less direct physical meaning individually.

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  • $\begingroup$ Thank you (and everyone else) for answer(s)! I'm not actually thinking that imaginary part "isn't real" (I was hoping that I emphasized that enough when I mentioned Complex plane and following assumptions). I'm just too... impatient :D to have made actual sense of it's math during last couple years of self-studying. $\endgroup$ Jun 22, 2021 at 7:04
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The real and imaginary components of a wavefunction can be interchanged by multiplying the entire wavefunction by a global phase $i$. Therefore, they cannot have different physical meaning from each other. The only physical information lies in the relative phases between components of the wavefunction.

If you are thinking of a quantum harmonic oscillator and plotting it in a phase space with real and imaginary axes corresponding to the $x$ and $p$ quadratures, then certainly the location of this distribution in phase space has physical meaning. That location is actually encoded directly in the relative phases between various superposed components of the wavefunction, so it exists regardless of what you choose to be the real or the imaginary axis.

In your example, there is no reason for position to be able to be more precisely measured than momentum.

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For the impatient man: There is nothing special about complex numbers. Although the standard postulate specifically defines a complex Hilbert space, they are really there because the equations/representations that arise are easiest to handle using complex numbers.

Other numbers/groups/mathematical structures arise in QM but complex numbers are the lingua franca for expressing these ideas. Forget about real and imaginary parts having any special meaning, it varies from situation to situation.

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  • $\begingroup$ Yeah, I know that. I guess it's just better for me to get more familiar with these situations you're referring to, to see this math in all it's (physically) meaningful glory and action. $\endgroup$ Jun 22, 2021 at 11:21
  • $\begingroup$ It's kind of like the same situation for me as with algebra and trigonometry back in my school days: it always was soo out of context and touch with reality (i.e., abstract), that my mind usually struggled to make freaking use of those things. And what a delightful surprise and satisfaction (on a contrary) it was for me to rediscover all these roots and integrals (to name a few) from inside an actual physical system! $\endgroup$ Jun 22, 2021 at 11:26

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