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Complex Wave functions?

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Wanted to know why do we use complex representation of wave functions?


for example, exp(i(kx-wt)) is a standard wave function. WHat is the significance of the imaginary part of the number. If it does not exist then why do we include them in all representations/calculations?


Specially wanted to clarify this from the point of view of the Schrodinger wave equation that has a real and imaginary part.


Please throw light on the same.



Archana Bahuguna

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From the point of view of classical wave equations, the use of complex numbers is simply convenient.


From the point of view of the wavefunctions of quantum mechanics the wavefunction is fundamentally complex. The wavefunction though doesn't have any direct physical significance, so Im(Psi) and Re(Psi) don't have any direct physical signifcance either.

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The probabilty is given by the square of the wave function (i.e. you multiply by the complex conjugate). There are things that you can represent easier by using a complex wave function - the wave function itself has no physical meaning, so it doesn't have to be real. As Aeschylus said, it's a matter of convenience.

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In chronological order, the complex nature of the wave function preceeded its interpretation. First thing to come up was the schrodinger equation which produced complex functions even for the simplest situations. The interpretation mentioned by swansont was initially proposed by Max Borne. It is a matter of mathematics that wave functions turn out to be complex, we only apply an interpretory skeletol model to it.

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I read this somewhere... please confirm if my understanding is correct...


QM requires wave functions to be complex because there is a need to distinguish between positive and negative frequencies ....for eg if we took wave function of the form Acos(kx-wt) and if we have another wave with k1 =-k and w1 = -w then in terms of cosine the wave function essentially remains same -

Acos((-k)x -(-w)t) = Acos(-)kx-wt)) = Acos(kx-wt).

This yields indistinguishable wave functions, thus yielding physically indistinguishable states.

So using exp(-i(kx-wt)) helps solve that problem.


Seems like for simplicity of mathematical calculations and stuff that we take wave functions as complex.

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That doesn't mean that they have to be complex. As Aeschylus said, taking the wavefunction as complex is just a matter of convenience. One could describe your situation equally well using a 2 companent wavefunction A (cos(kx-wt), sin(kx-wt)) with a rule on how to take products, without an 'i' in sight. (This is of course formally identical to using complex numbers....)


You see this sort of thing quite often in particle physics. For example a Higgs boson doublet is usually described by a 2d vector (in SU(2) space) where each component is a complex field, but it can equally well be defined as a 4d vector with each component a real field.

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