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Wave functions 
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#1
Oct1008, 02:51 AM

P: 64

Why are wave functions, e.g., Schrodinger's, based on the complex exponential function (e[tex]^{}ix[/tex]) and not trigonometric functions (sine or cosine)?
See Euler's formula for their relationship: http://en.wikipedia.org/wiki/Euler%27s_formula Furthermore, by using the complex exponential function, the probability amplitude becomes a complexvalued function (a + bi). Were sine or cosine used, the probability amplitude of the wave function would not be a complexvalued function. Is there a reason that the probability amplitude should be a complexvalued function? 


#2
Oct1008, 04:29 AM

Sci Advisor
P: 1,588

You need complexvalued wavefunctions so that you can have standing waves of constant magnitude over time (such as in a potential well). Realvalued functions would have to oscillate in magnitude.



#3
Oct1008, 04:34 AM

P: 64

What do you mean by magnitude? Is it the same as amplitude?



#4
Oct1008, 08:41 AM

P: 55

Wave functions
Nothing keeps you from using sine and cos. That what Euler's identity says.
The time evolution of a state vector/ wave function must conserve the norm of the state vector, and it always has to be one. Note e[tex]^{it}[/tex] e[tex]^{it}[/tex]=1. 


#5
Oct1008, 10:11 AM

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#6
Oct1008, 02:24 PM

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The simple answer is that standard quantum mechanics as we understand it requires a complex wavefunction [tex] \psi [/tex] defined on the configuration space. You can replace this with a function to [tex] \mathbb{R} \times \mathbb{R} [/tex] and change the equations accordingly, since [tex] \mathbb{C} [/tex] and [tex] \mathbb{R} \times \mathbb{R} [/tex] are isomorphic. But you can't replace [tex] \psi [/tex] with a function to just [tex] \mathbb{R} [/tex].
Also, Euler's formula doesn't turn a complex number into a real number. [tex] \cos \theta + i \sin \theta [/tex] is still a complex number, and the probability amplitude will still be a complex valued function. 


#7
Oct1208, 03:47 PM

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#8
Oct1208, 04:02 PM

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[QUOTE=redtree;1912396]



#9
Oct1208, 05:22 PM

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