Tuesday, January 27, 2009

Quantum mechanics without imaginary numbers

I've been thinking about how the Schroedinger equation intimidates students because it is inherently complex:
-\frac{\hbar^2}{2m}\nabla^2 \Psi + V\Psi=\uc{red}{i} \hbar \frac{\partial \Psi}{\partial t}
(note the red i). This has led me in the past to say things like "quantum mechanics is weird because apparently the universe is both real and imaginary" and "we can only see the real parts". Now that last quote especially is suspect since what we can "see" is really the magnitude squared of the wavefunction:
which corresponds to the probability density of finding something in a particular location. But I'm coming around to a position where even the first quote above is suspect. It seems to me that you can rewrite the equation as two coupled equations:
-\frac{\hbar^2}{2m}\nabla^2 \Psi_1 + V\Psi_1=-\hbar \frac{\partial \Psi_2}{\partial t}
-\frac{\hbar^2}{2m}\nabla^2 \Psi_2 + V\Psi_2=\hbar \frac{\partial \Psi_1}{\partial t}
Note how the 1's and 2's switch places. What I've done here is renamed the real part of \Psi as \Psi_1 and the imaginary part of \Psi as \Psi_2. Only here I'm just numbering them and not really giving any preference to either. These two equations are totally equivalent to the original equation and what it says about the universe is that for every object you need to keep track of two things. In the end to make predictions about the object you'll need \Psi_1^2+\Psi_2^2 but note that I don't need imaginary numbers at all!

I think interpreting those two equation is of interest as well. Essentially the spatial curvature of one produces temporal changes in the other and vice versa. That's actually pretty cool as you could stare at a snapshot of both the two and predict what's going to happen in the next moment of time.


  1. I think your formulation of the Schroedinger equation as two equations is far better than condensing it through the use of imaginary number symbols, and than using imaginary numbers. I am one of those people that think that imaginary numbers are actually imaginary mathematical toys. The original formulation with imaginary numbers appears to be more concise at first, in that it is a single equation, but it is not really twice as concise because complex numbers have two components, so you end up with two of something anyway. Bravo to you for posting this.

  2. Hello Andy,
    Good thinking! i is no longer needed for relativity either. Einstein's original Pythagorean triangle made the traveling twin's time-trajectory the longer hypotenuse. That's because he drew the time-front as a straight line. He need i for a time-to-length conversion factor to show that the traveler's time-passage was shorter.
    Then Minkowski showed that the time-front is curved inward, which brings it nearer to the traveler, so that the stay-home twin's trajectory becomes the longer hypotenuse.
    If we accept that time passes because of the universe's expansion, and expansion's at the limiting speed, the conversion is simplified to 1 sec = 300,000 Km.
    To sum up, Einstein used i as as a ladder to climb where no man had gone before. Now that we can join him on the summit, it shows no disrespect if we espy another simpler route which wasn't obvious in 1915.
    With best regards, David.
    P.s. I'd be glad to send you the diagram but I can't fit it into this box.