Plotting a function $\psi(\rho,\theta,\phi)$ in spherical coordinates

Question

If I want to plot the function

$\qquad \psi(\rho,\theta,\phi)=\frac{1}{8}\left(\frac{1}{\pi}\right)^{1/2}\rho e^{-\rho/2}sin(\theta)e^{-i\phi}$,

for which I wrote the code

ψ[ρ_, θ_, ϕ_] := (E^(-(ρ/2) - i ϕ)  ρ Sin[θ])/(8 Sqrt[π])

How would I go about making a plot? I'm thinking that I could convert these spherical coordinates to cartesian coordinates using a built-in function, but I haven't been able to get that to work.

Answer 1

You likely need to take the modulus-squared of the wave function first:

ψ[ρ_, θ_, ϕ_] := (E^(-(ρ/2) - I ϕ) ρ Sin[θ])/(8 Sqrt[π]);
Abs[ψ[ρ, θ, ϕ]]^2 // ComplexExpand
(* (E^-ρ ρ^2 Sin[θ]^2)/(64 π) *)

Then plot a slice of the function radially:

Plot[Abs[ψ[ρ, π/2, 0]]^2 // ComplexExpand, {ρ, 0, 10}]

Then plot the angular part at fixed radius (which is typically done for hydrogen-like wave functions like this):

SphericalPlot3D[Abs[ψ[1, θ, ϕ]]^2 // ComplexExpand, {θ, 0, π}, {ϕ, 0, 2 π}]

Answer 2

As you have it right now, the function itself cannot be plotted for a variable radius. That being said, you can just specify it. Also, the expression is complex, and thus will not output anything when plotted. So we can take the real and imaginary parts, and plot those.

SphericalPlot3D[
 {Re, Im}[
  ComplexExpand[
   (Exp[-(\[Rho]/2) - I \[Phi]] \[Rho] Sin[\[Theta]])/(8 Sqrt[\[Pi]]) /. \[Rho] -> 1,
   TargetFunctions -> {Re, Im}
  ]
 ] // Through // Simplify[#, {\[Theta], \[Phi]} \[Element] Reals] & // Evaluate,
 \[Theta], \[Phi],
 PlotStyle -> {{Opacity[.4], Red (*Real*)}, {Opacity[.4], Blue (*Imaginary*)}},
 PlotPoints -> 100
]

We can make the surfaces a little translucent and see that the real and imaginary parts overlap a little bit near the origin.