Wave Function: Real vs Imaginary Part

In summary, Wave functions are, of course, almost always complex-valued. Wave functions can be real or imaginary, but they can also be complex. The real and imaginary parts of the wave function are related by a Hilbert transform, and the phase of the wave function is continuous.
  • #1
LarryS
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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.
 
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  • #2
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.
 
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  • #3
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.
 
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  • #4
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".
 
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  • #5
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?
 

1. What is the difference between the real and imaginary part of a wave function?

The real part of a wave function represents the physical quantity or observable of a system, such as position or momentum. The imaginary part, on the other hand, is related to the probability of finding the system in a particular state or energy level.

2. How are the real and imaginary parts of a wave function related?

The real and imaginary parts of a wave function are related through the complex conjugate. This means that the imaginary part is the negative of the complex conjugate of the real part.

3. Can the imaginary part of a wave function have physical significance?

Yes, the imaginary part of a wave function can have physical significance in certain cases. For example, in quantum mechanics, the imaginary part is related to the probability amplitude of finding a particle in a specific state. It is also used to calculate the phase of a wave function.

4. How do the real and imaginary parts of a wave function affect the overall wave function?

The real and imaginary parts of a wave function combine to form the overall wave function. The real part determines the physical quantity of the system, while the imaginary part contributes to the overall probability of finding the system in a particular state.

5. Can the real and imaginary parts of a wave function be separated or observed separately?

No, the real and imaginary parts of a wave function cannot be separated or observed separately. They are always interconnected and cannot exist independently. This is due to the mathematical relationship between the two parts through the complex conjugate.

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