# Wave functions

by redtree
Tags: functions, wave
 P: 64 Why are wave functions, e.g., Schrodinger's, based on the complex exponential function (e$$^{}ix$$) 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 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?
 Sci Advisor P: 1,593 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.
 P: 64 What do you mean by magnitude? Is it the same as amplitude?
 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$$^{it}$$ e$$^{it}$$=1.
P: 594
 Quote by redtree Why are wave functions, e.g., Schrodinger's, based on the complex exponential function (e$$^{}ix$$) 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 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 http://www.physicsforums.com/showpos...5&postcount=20 and some other posts in that thread may be relevant.
 HW Helper PF Gold P: 1,961 The simple answer is that standard quantum mechanics as we understand it requires a complex wavefunction $$\psi$$ defined on the configuration space. You can replace this with a function to $$\mathbb{R} \times \mathbb{R}$$ and change the equations accordingly, since $$\mathbb{C}$$ and $$\mathbb{R} \times \mathbb{R}$$ are isomorphic. But you can't replace $$\psi$$ with a function to just $$\mathbb{R}$$. Also, Euler's formula doesn't turn a complex number into a real number. $$\cos \theta + i \sin \theta$$ is still a complex number, and the probability amplitude will still be a complex valued function.
P: 64
 Quote by dx The simple answer is that standard quantum mechanics as we understand it requires a complex wavefunction $$\psi$$ defined on the configuration space. You can replace this with a function to $$\mathbb{R} \times \mathbb{R}$$ and change the equations accordingly, since $$\mathbb{C}$$ and $$\mathbb{R} \times \mathbb{R}$$ are isomorphic. But you can't replace $$\psi$$ with a function to just $$\mathbb{R}$$.
What properties of standard quantum mechanics require a complex wavefunction?
HW Helper
P: 4,738
[QUOTE=redtree;1912396]
 Quote by dx The simple answer is that standard quantum mechanics as we understand it requires a complex wavefunction $$\psi$$ defined on the configuration space. You can replace this with a function to $$\mathbb{R} \times \mathbb{R}$$ and change the equations accordingly, since $$\mathbb{C}$$ and $$\mathbb{R} \times \mathbb{R}$$ are isomorphic. But you can't replace $$\psi$$ with a function to just $$\mathbb{R}$$.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
HW Helper
PF Gold
P: 1,961
 Quote by redtree 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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