Can the complex nature of quantum wave arise from the fact that particle is represented as wave packet in spatial frequency and particle's total energy is represented as wave packet in time frequency?

Those wave packets are connected since $E=p^2/2m+V$, but since wave-like particle posses definite location and definite energy, they make sense. Now we have $\cos (\text{phase})+i\centerdot \sin (\text{phase})=e^{i \centerdot \text{phase}}$ and with proper summation two dimensional information could be emerged into one scalar valued function $\psi (x,t)$.


closed as unclear what you're asking by Danu, Kyle Kanos, Brandon Enright, David Z Sep 22 '14 at 22:34

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    $\begingroup$ Voting to close, since there does not appear to be a question here. Note that this is not a site to post expositions, unless you answer someone's question (possibly your own) in the process. $\endgroup$ – Danu Sep 22 '14 at 17:22
  • $\begingroup$ Stephen Gasiorowicz opens Chapter 2 with your same thought, though he touches on the fact superposition should apply (in order for self-interactions to result), hence the ability to add the two wave functions and still have a valid wave function. $\endgroup$ – Kyle Kanos Sep 22 '14 at 17:23
  • $\begingroup$ There is a question on first line. Possible answers could consider if this is valid argument for complex presentation. This is a profound issue in physics and we should be able to discuss this approach. $\endgroup$ – user59412 Sep 22 '14 at 17:27
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    $\begingroup$ "What do you think" type questions are also off-topic (as it's asking for opinions) $\endgroup$ – Kyle Kanos Sep 22 '14 at 17:35
  • $\begingroup$ Right! I'll change the first line. $\endgroup$ – user59412 Sep 22 '14 at 17:37

No, this is not a valid explanation. Pardon my possible simplifications, but I understand the reasoning in the following way:

  1. $k$ bears independent information
  2. $\omega$ bears independent information
  3. We can "store" only one piece of information in the real line, so we need "two folders", which is provided by the two parts of a complex number.

This is simply not how the whole thing works. The argument is completely detached from what the wavefunction actually is and what it is used for. The "information content" of $\omega$ and $k$ is not generally decomposable into separate $A_1(k), A_2(\omega)$, and it certainly isn't divided up the way you suggest. The full $A(k,\omega)$ is not imposed but determined by the Schrödinger equation.

But to give a simple counterexample to show the invalidity of the argument by it's own means: Actual wavefunctions occur in 3D space, so we have $\vec{k}=(k_x,k_y,k_z)$. With $\omega$ this means "four information folders". So complex numbers are not enough. You could pass to quaternions. But we do not do that in quantum mechanics. Why? Because it is not actually needed, the premises of the argument are incorrect.

Physics sometimes just postulates objects and they work - the postulation cannot be fully explained. Nevertheless, I like the argument for the complex wave-function by Jerry Schirmer: We "need" the formalism to produce actually observed wave phenomena such as interference, but also "flat" and "non-wavy" probability distributions, which however have the potential of interfering.

  • $\begingroup$ Well, maybe my idea was too simple. But I must stress that $k$ and $\omega$ are interconnected: $E=p^2 /2m +V(x,t)$. Every moment of time particle is "localized packet" in a sense of both space and energy, and that was my starting point. Now, if we could combine those two condition that would result in correct wave function $\psi (x,t)$ - maybe. Also, scalar product $ k \centerdot x= \text{(real number)}$ requires only one "information folder". I belive that combining spatial and temporal leads to complex presentation of the wave, but I don't know how :) $\endgroup$ – user59412 Sep 22 '14 at 20:56