Wave Function: Real vs Imaginary Part

[LarryS]

Wave functions are, of course, almost always complex-valued. In all of the examples that I have seen (infinite square well, etc.), the real part of the wave function and the imaginary part of the wave function are basically the same function (except for a phase difference and possibly a sign difference).

Do you know of an example of a wave function, that is complex-valued, for which the real and imaginary parts are fundamentally different functions?

(To be honest, I have not seen that many examples).

Thanks in advance.

 

[jfizzix]

This could be totally off base, but if the wavefunction is analytic, then its real and imaginary parts should be related by a Hilbert transform. I think this implies that the real and imaginary parts of the wavefunction cannot be completely arbitrary.

Alternatively, the evolution of a wavefunction as determined by a hamiltonian, which also forces a relationship between the real and imaginary components due to the wavefunction needing to be continuous in all physical situations.

In particular, the phase of the wavefunction is continuous, so as the phase of the wavefunction changes, the real and imaginary parts change accordingly, again restricting the possible functions that the real and imaginary parts of the wavefunction can be.

 

[Jazzdude]

Due to the nature of complex vector spaces, splitting up the wavefunction into real and imaginary parts is arbitrary. So if you say that a wavefunction is real, then it really means that you can multiply it with a non-zero complex factor so that its imaginary part vanishes.

The usual examples of "real wavefunctions" are produced by 1-dimensional momentum symmetric problems in position expansion. If you want a wavefunction with a fundamentally different real and imaginary part, simply take two orthogonal real wavefunctions given by the stationary solutions of such a system and add them with a relative phase of pi/2.

 

[PhilDSP]

Hi referFrame,

In the analysis of waves in dispersive media, the real and imaginary parts of the signal have very different functions. They are used to classify either the absorption characteristics (the real part) or the dispersive characteristics (the imaginary part).

http://www-keeler.ch.cam.ac.uk/lectures/understanding/chapter_4.pdf (page 4-3)
http://www.inmr.net/Help/pgs/fid.html (fifth paragraph)

Also as a matter of general interest, waves traveling through dispersive media may invoke the necessity of non-locality. There is a section of Jackson's "Classical Electrodynamics" textbook that deals with that. If I remember correctly that is where Jackson discusses the Kramers–Kronig dispersion relations.

A more thorough description of how non-locality arises is given in Thomas H. Stix's "Waves in Plasmas".

 

[Nugatory]

I'm not complaining because these are good answers to a good question... But you guys do realize that you've just set some sort of record for white-hat thread necromancy here?