How find the exact intersections in the graphs of wave equations

Questions
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1

(1/(Sqrt[Pi])(1/a0)^(3/2)Exp[-r /(a0)])

(1/(8Sqrt[Pi])(1/a0)^(3/2)(2-(r/a0))Exp[-r /(2 a0)])

(81/(Sqrt[3Pi])(1/a0)^(3/2)(27-(18r/a0)+2(r^(2)/a0^2))

These are three wave equations that I need to find where they intersect all at the same point. However every time I set them equal they say one is protected. I don’t know what I need to do to allow me to do what I need done Can someone help.

Sincerely, Zachary Nichols

2

Zachary,
It will be helpful if you cut and paste the code you are using in a code box (see the <> icon at top left) and indicate whether r is two- or three-dimensional, etc.
WCC

3

Set a0 = 1 and define 3 functions. These are evidently the 1 s, 2 s and 3 s wavefunctions of the hydrogen atom.

With corrections :

f1[r_] := (1/(Sqrt[Pi]) Exp[-r])

f2[r_] := (1/4 Sqrt[2 Pi]) (2 - r) Exp[-r/2 ]

f3[r_] := (1/(81 Sqrt[3 Pi])) (27 - 18 r + 2 r^2) Exp[-r/3 ]

Plot[{f1[r], f2[r], f3[r]}, {r, 0, 10}]

The plot will show that there are no triple intersections (except at r = Infinity).

4

Thank you S M Blinder; however now I’m working with three dimensional functions and trying to find the the exact intersections how would I do that

This is the graph of the wave equations i want to find the intersections of

Manipulate[SphericalPlot3D[{(1/(8Sqrt[Pi])(1/a0)^(3/2)(r/a0)Exp[-r /(2 a0)]Sin[[Theta]]Cos[[Phi]])^2,(1/(Sqrt[81Pi])(1/a0)^(3/2)(6-(r/a0))(r/a0)Exp[-r /(3a0)]Sin[[Theta]]Cos[[Phi]])^2,(1/(4Sqrt[2Pi])(1/a0)^(3/2)(r/a0)Exp[-r /(2 a0)]Cos[[Theta]])^2},{[Theta],0,2Pi},{[Phi],0,2Pi}],{r,0.001,1}]

a0=1

Sincerely, Zachary Nichols