Clear[cm, sm, smo, r]

cm := ((Cos[(μ/2)])^2) (* Simplify[(1 + Cos[μ])/2] *)
sm := ((Sin[(μ/2)])^2) (* Simplify[(1 - Cos[μ])/2] *)
smo := Sin[μ]/Sqrt[2]
r := { {cm, sm, smo}, {sm, cm, -smo}, {-smo, smo, Cos[μ]}}
r // MatrixForm
FullSimplify[Det[r]]

Now want to plot ((x-y)/Sqrt[2] Sin[mu] + z Cos[mu])^2 e^(-r).  Plot in the a = x-y, z plane.

	Note: I had trouble with this plot originally, but got help here:

		http://stackoverflow.com/questions/7954580/mathematica-plot-with-manipulate-shows-no-output

Clear[f, x, y, z, mu, range]
f[ x_, y_, z_, mu_] = ((x-y) Sin[mu]/Sqrt[2]+z Cos[mu])^2 E^(-Sqrt[x^2 + y^2 + z^2])
Manipulate[ Plot3D[ f[x, y, z, mu], {x, -range, range}, {z, -range, range}],{{range,1} ,0.1, 100}, {mu, 0, 2 π}, {y, 0, range}]