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