## >Quantum mechanics without imaginary numbers

>

I’ve been thinking about how the Schroedinger equation intimidates students because it is inherently complex:

(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:
$\left|\Psi(\vec{r})\right|^2$
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.

Associate professor of physics at Hamline.
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### 7 Responses to >Quantum mechanics without imaginary numbers

1. Anonymous says:

>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. bwfrank says:

Via Joe Redish: “Another way of doing this is to write the wave function as a magnitude and a phase and write coupled equations for them. I think it was David Bohm who did that.”

3. ibeatty says:

I shared this with a colleague — who used to be in theoretical nuclear physics, and has taught pretty much every level of QM course hat exists. He had this to say: “This isn’t easier. AND it buries the idea of psi as a probability amplitude. This idea is not new and it fails to enable an expression of many other QM concepts. Working with more than one particle requires new rules for adding psi’s, e.g add the real parts and add the imaginary parts and find probabilities by squaring each and adding. The physics of interference is in the cross terms. A mess . . . ”

FWIW.

• Andy "SuperFly" Rundquist says:

I agree with most if not all of that. I think my point is that imaginary numbers are not, strictly speaking, required for QM. I’m certainly not arguing that they are useless. In fact, I would guess that they make most if not all of the math easier.

4. deturing says:

Just one comment here: You say “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!”, but this really depends on the experiment. For example, in an Aharonov-Bohm type experiment, the phase of the wave-function of electrons is affected by the magnetic flux, and this effect can be measured via interference, so the complex nature is certainly important! Whether you call it complex, or two real numbers I guess is a matter of semantics, but the framework of complex numbers lends itself beautifully to the concept of unitarity of time evolution, the concept of analytic functions and so on… I honestly don’t know what one can do in QM without complex numbers!

• ibeatty says:

Agreed. You can set up an alternate mathematical formalism to avoid using “i”, but I argue that you’re still using complex numbers, just with a less compact notation. If it’s mathematically isomorphic, it’s equivalent.

And don’t let the term “imaginary” scare you. *ALL* numbers are “imaginary”, in the sense that they’re conceptual abstractions existing only in our minds, which serve as useful models for some aspect(s) of our experience. Numbers with “i” in them are no more or less “real” than numbers without them. The rules for combining and manipulating complex numbers — whether or not you use an “i” to do so — are no more or less arcane and sketchy than the rules for combining and rotating vectors. IMHO, of course.