In quantum mechanics, we assume wavefunctions are complex valued, and that probability amplitudes are given by the modulus of the wavefunction squared. This formalism can correctly explain interference effects, e.g. the double-slit experiment. I understand all of this.

Now, let's consider classical optics. If we have a linearly polarized EM wave, the electric field is defined in all space by a single real number. If we want to find intensity of light at any given point, we time-average the square of the electric field at that point. Alternatively, instead of representing the electric field with a time-varying real number at every point, we can also represent the electric field by a complex phasor which rotates in time. Then then intensity of the light at any point is (up to a factor of $\sqrt{2}$) just the modulus of the phasor squared. And if the light interferes with itself (e.g. in an interferometer) the new phasor at the point of interference is simply the sum of the old phasors, identical to interference in quantum mechanics.

So my question is: if, instead of assuming we have a complex valued wavefunction, we assume we have a REAL wavefunction (which is just the real part of our normal wavefunction), and all our probability measurements are inherently time-averaged (like our intensity measurements for an EM wave), would we run into any contradictions with logic/experiment?

(I realize this wouldn't be a particularly USEFUL formulation of quantum mechanics. Complex numbers are obviously more elegant--we invented phasors for a reason, after all. I just want to know if it's equivalent.)

Edit: I realize complex numbers can be thought of as simply two real numbers. I'm interested in the case of representing the wavefunction as a SINGLE real number. I guess what I'm looking for is a concrete difference between wavefunctions and electric fields.

  • $\begingroup$ Related: physics.stackexchange.com/q/8062/80818 $\endgroup$
    – tok3rat0r
    Commented Jul 8, 2015 at 18:17
  • $\begingroup$ This is a good question, but I think it's been asked before and has good answers already... $\endgroup$
    – innisfree
    Commented Jul 8, 2015 at 18:19
  • $\begingroup$ @innisfree I like the question you linked too, but most of the answers seem to be focusing on how a wavefunction could be viewed as a pair of real numbers. I'm more interested if we could view the wavefunction as a single real number. But I will keep reading through it! $\endgroup$ Commented Jul 8, 2015 at 18:22
  • $\begingroup$ @JahanClaes - I think the piece you're missing is the importance of the phase information. In the case of the electric field of the EM wave, you can look at the amplitude at a given point but you can't define anything about the wave's behaviour, or its interaction with a particle, without simultaneously defining the phase at the same point. You therefore end up with two numbers to completely describe (locally) the electric field. $\endgroup$
    – tok3rat0r
    Commented Jul 8, 2015 at 18:33
  • 2
    $\begingroup$ (Equivalently, if you want to get as far as possible from the idea of complex phasors, you need to define both the amplitude and the time derivative of the electric field at each point in order to fully describe the wave. It still boils down to two essential numbers, though) $\endgroup$
    – tok3rat0r
    Commented Jul 8, 2015 at 18:36