Wave functions

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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's_formula

Furthermore, by using the complex exponential function, the probability amplitude becomes a complex-valued function (a + bi). Were sine or cosine used, the probability amplitude of the wave function would not be a complex-valued function. Is there a reason that the probability amplitude should be a complex-valued function?
 
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  • #2
Ben Niehoff
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You need complex-valued wavefunctions so that you can have standing waves of constant magnitude over time (such as in a potential well). Real-valued functions would have to oscillate in magnitude.
 
  • #3
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What do you mean by magnitude? Is it the same as amplitude?
 
  • #4
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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
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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's_formula

Furthermore, by using the complex exponential function, the probability amplitude becomes a complex-valued function (a + bi). Were sine or cosine used, the probability amplitude of the wave function would not be a complex-valued function. Is there a reason that the probability amplitude should be a complex-valued function?

My post https://www.physicsforums.com/showpost.php?p=1539835&postcount=20 and some other posts in that thread may be relevant.
 
  • #6
dx
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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
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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].

What properties of standard quantum mechanics require a complex wavefunction?
 
  • #8
malawi_glenn
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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].QUOTE]


What properties of standard quantum mechanics require a complex wavefunction?

well the physics of spin 1/2 systems for example, see Sakurai - Modern Quantum Mechanics chapter 1
 
  • #9
dx
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What properties of standard quantum mechanics require a complex wavefunction?

That's like asking what property of Newtonian physics requires the mechanical state of a particle to be position and momentum. It is possible to construct a theory where the time evolution of a particle depends only on initial position, but that's not the way nature is. It just happens to be so that the mechanical state of a quantum system needed to predict future probabilities is a complex function.
 

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