The role of complex conjugation in QM

[bunburryist]

What is the purpose of complex conjugation in quantum physics? What is it that complex numbers allow us to do that can't be done otherwise, or at least cannot be done as easily? I understand what complex numbers are and how complex conjugation is done, yet I can't find a straightforward explanation of the role it plays and why quantum equations were framed in that context in the first place.

As simple example, at http://www4.ncsu.edu/unity/lockers/users/f/felder/public/kenny/papers/psi.html we find

"In order to keep this topic as simple as possible, we're going to start by living in a very simple universe. Our universe has only three points: x=1, x=2, and x=3. Our particle must be on exactly one of those points. It cannot be anywhere else, including in between them. Since those three points define the whole universe, the wavefunction itself is defined at all three of those points, and nowhere else. So Y is just three complex numbers, which might look something like this. " . . . and then lists the numbers. But why complex numbers?

 

[tiny-tim]

What is the purpose of complex conjugation in quantum physics? What is it that complex numbers allow us to do that can't be done otherwise, or at least cannot be done as easily?

Hi bunburryist!

I suppose the technical reason is that QM is in Hilbert space, which is a space over the complex numbers …

if it wan't in Hilbert space, it wouldn't be QM.

Perhaps the non-technical explanation is that since everything is made of waves, we need to multiply by a complex number to signify movement along the wave, and we need conjugates because combining with a conjugate is the only way to get back to real numbers, which is what probabilities have to be.

 

[Fredrik]

Section 3 in this article offers some motivation.

Tiny-tim, there are real Hilbert spaces, but you're of course right that QM needs a complex Hilbert space.

 

[bunburryist]

I found this, and it helps some as well.

http://physics.nmt.edu/~raymond/classes/ph13xbook/node93.html

In quantum mechanics the absolute square of the wave function at any point expresses the relative probability of finding the associated particle at that point. Thus, the probability of finding a particle represented by a plane wave is uniform in space. Contrast this with the relative probability associated with a sine wave: $\vert \sin (kx - \omega t) \vert^2 = \sin^2 (kx - \omega t )$. This varies from zero to one, depending on the phase of the wave. The ``waviness'' in a complex exponential plane wave resides in the phase rather than in the magnitude of the wave function.

 

[Manchot]

The simplest way to think about it is to remember that a complex number is nothing more than a pair of real numbers, so the complex wavefunction is just a pair of interdependent real functions. Conjugation is just a way to get the "other" function.