# Why is wavefunction complex-valued?

I'd like to know why wave functions are taken to be complex-valued in general, and where the i in the Schrödinger equation comes from and what it means. I've seen plenty of we-use-this-because-it-works type arguments. What I'm hoping for is a meaningful explanation of how the complex numbers arise out of the physics and what physical quantities (or changes of physical quantities, etc) are modeled by the real and imaginary parts of the wave function (in particular, not just the familiar interpretation of the modulus of this function). I'd appreciate any insight or perhaps just a reference to a book where this is discussed. Right now I'm reading Shankar. The physics is pulled out of nowhere.

I'm a PhD math student trying to pick up quantum mechanics. In particular, I'm a harmonic analyst if that means anything to you, but my background in physics is negligible

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I have read a paper called something like "A Classical context to Schroedinger equation" in Am. J of Phys. In this paper the author explains what may be the answer you are searching for. He says SE having complex nature allows the formalism to contain both difusive and anti-difusive behavior, and quantum mechanics of a particle subjected to a given potential needs such anti-difusive behavior besides the difusive one. Notice that it (SE) resembles to much the difusion equation.

May the paper itself be of more help.

Best Regards,

DaTario

I'd like to know why wave functions are taken to be complex-valued in general, and where the i in the Schrödinger equation comes from and what it means. I've seen plenty of we-use-this-because-it-works type arguments. What I'm hoping for is a meaningful explanation of how the complex numbers arise out of the physics and what physical quantities (or changes of physical quantities, etc) are modeled by the real and imaginary parts of the wave function (in particular, not just the familiar interpretation of the modulus of this function). I'd appreciate any insight or perhaps just a reference to a book where this is discussed. Right now I'm reading Shankar. The physics is pulled out of nowhere.

I'm a PhD math student trying to pick up quantum mechanics. In particular, I'm a harmonic analyst if that means anything to you, but my background in physics is negligible
See, e.g., https://www.physicsforums.com/showpost.php?p=1415621&postcount=13

Regarding the post at https://www.physicsforums.com/showpos...1&postcount=13 [Broken], I had read that before posting my question but I was hoping for more--not just that it can be done without reference to the complex numbers, but *why* the complex numbers are appropriate, and why we choose to use them.

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Some discussion can be found in previous posts like

I have collected a lot of mathematical articles about the complex amplitude which I haven't had the time to read. But if you suggest a way to transfer files, then I can send them to you.

Regarding the post at https://www.physicsforums.com/showpos...1&postcount=13 [Broken], I had read that before posting my question but I was hoping for more--not just that it can be done without reference to the complex numbers, but *why* the complex numbers are appropriate, and why we choose to use them.
The physicist M. Nussenzveig once has said that he bet if one can build quantum mechanism without complex numbers.

Best Wishes

DaTario

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but *why* the complex numbers are appropriate, and why we choose to use them.
You could instead express QM by a set of coupled differential equations. However, this set included the Cauchy-Riemann equations, so it becomes quite natural to use complex numbers because those equations are automatically satisfied.

Thanks for your replies. I'm finding the discussion at https://www.physicsforums.com/showthread.php?t=287383&page=2 very helpful, and it didn't previously turn up in my searches. As I'm beginning to understand, we need a function that represents a probability density and is also wavelike, and wanting it to be wavelike suggests using the complex plane, where harmonic oscillators most naturally live.