Drawing the Wavefunction

Question

In lecture, my professor had drawn the wavefunction for a particle which encounters a potential energy barrier, whose energy it fails to exceed. The graph was highly similar to the one which appears hereafter:

Yet, if the wavefunction is complex (specified, for instance, in region I by $psi(x) = Ae^{ikx}+Be^{-ikx}$ and in region III by $psi(x) = Ce^{ikx}$), then how might we be able to thus draw the wavefunction? Are we only drawing its 'real part'? (Yet, what might this real part be, if the coefficients are themselves complex)?

Saptarshi Sarkar · Answer

Yes. When drawing a wavefunction we just draw the real part. You can also write the wave function in terms of sines and cosines and avoid the exponential notation altogether.
Remember the Euler's formula
$e^{ikx}=cos{(kx)}+itextrm{ }sin{(kx)}$
Typically waves are written with the exponential notation as it makes it easier to manipulate it mathematically. We only need to consider the real part.

Ruslan · Answer

The real part being drawn is indeed misleading. Actual wavefunction, if we are talking about a particle incident on a barrier, should be complex. To draw it, you can plot separately the real and imaginary part, like in the animation of the time dependence below:

Another option is to draw a 3D spiral where $x$ coordinate along the spiral axis is the spatial coordinate, and the $y$ and $z$ coordinates show respectively the real and imaginary part of the wavefunction:

Drawing the Wavefunction

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