# Why psi is complex!

1. Jan 17, 2005

### imran

why psi is complex!
why can't it is a real ?

2. Jan 17, 2005

### Sterj

because it is easier to write the wave function in complex numbers.

3. Jan 17, 2005

### inha

It's not neccessarily complex. Take the infinite square well for an example. What your hamiltonian is like determines what the wave function turns out to be. As far as I know the only restrictions on psi are that it must be a continuous, single valued function, normalizable and that it's first derivative must be continuous excluding points where the potential is infinite.

4. Jan 17, 2005

### Staff: Mentor

Because that's the only way people have figured out how to get predictions that agree with experiment.

OK, you can say it's because of the way the Schrödinger Equation is set up. Solutions to the S.E. have to be complex. But that just changes the question to "why is the Schrödinger Equation the way it is?" You can't win.

That's true for E&M. You can do electromagnetic waves using only real sines and cosines, but the math is easier sometimes if you use complex exponentials instead. In QM, on the other hand, you need those i's. The only way around them is to write the real and imaginary parts of psi as separate functions, and turn the Schrödinger Equation into a pair of coupled differential equations that mix the two functions together. But that just rephrases imran's question to, "why do we have to do that?" Again, you can't win.

You're probably thinking of the time-independent stationary state wave function psi(x). Include the time dependence to make it psi(x,t) and you'll make it complex. I tried to include an example, but the TeX parser doesn't seem to be working right now... I get the equations I posted in another message over the weekend, on another subject entirely!

5. Jan 17, 2005

### inha

Ah yes. I should have mentioned the time independence. It seems that I type faster than it's good for me.

6. Jan 17, 2005

### nrqed

If you start from the de Broglie's relations for the energy and the momentum, and you assume that they are related by the classical relation $E = { {\vec p}^2 \over 2 m} + V({\vec r})$, and you assume that that plane waves solutions must exist, and you assume linearity, then you are forced to introduced complex numbers. The key point is that the equation mixes a first derivative (to get the energy) and second derivatives (to get the p^2). In contrast, the classical wave equation contains only second derivatives.

Pat

7. Jan 19, 2005

### vanesch

Staff Emeritus
Gee, that's smart

Patrick.

8. Jan 19, 2005

### marlon

NRQED, that is indeed a very nice description...one to remember

marlon

9. Jan 22, 2005

### reggepole

If somebody of you is interested in reading what contemporary physicist thinks about the problem try www.quniverse.sk/buzek/zaujimave/p343_s.pdf[/URL] . As an attracting example let me give you a very small citation:
"But what of psi (r) itself? Why is it a complex number,
composed of real’ and imaginary’ parts, Re psi(r)+ i Im psi(r)?
The surprising truth, I believe, is that at this point we do
not know why psi has two parts."

Last edited by a moderator: Apr 21, 2017