Undergrad Complex Numbers in Wave Function: QM Explained

[Zaya Bell]

I just need to know. Why exactly what's the complex number i=√–1 put in the wave function for matter. Couldn't it have just been exp(kx–wt)?

 

[hilbert2]

I just need to know. Why exactly what's the complex number i=√–1 put in the wave function for matter. Couldn't it have just been exp(kx–wt)?

It's not possible to normalize that kind of a real-valued exponential function. The Schroedinger equation is a complex diffusion equation, and the complex numbers make it possible to obtain wave-like behavior (wavepackets reflecting from walls, etc.) instead of the simple dissipative dynamics of ordinary diffusion where the solution is a real function.

 

[fresh_42]

Is it a periodic function without the $i$ which we would expect from a wave? You may want to look up the connection between the sine (cosine) function and the exponential function.

 

[Khashishi]

I just need to know. Why exactly what's the complex number i=√–1 put in the wave function for matter. Couldn't it have just been exp(kx–wt)?

Sure, you can have parts of the wavefunction look like exp(kx-wt), but not the whole thing. If you consider a WKB approximation for the time independent Schrodinger equation, you get solutions that look like
$\exp(\sqrt{2m(V-E)})$
which looks like an exponential where V > E and an oscillation where E > V. You aren't allowed to have V > E for the whole wavefunction, because, as hilbert2 noted, it is not possible to normalize. But, for a bound state of a delta potential (V < 0), you have a double exponential solution https://en.wikipedia.org/wiki/Delta_potential.

As far as the time part, you get exponential decay for various decay processes.

 

[Zaya Bell]

Thank you all.