# How does the hydrogen atom actually “look like”? [duplicate]

When deriving the solutions for the "simple" quantum mechanical hydrogen problem, one normally uses spherical coordinates $$(r,\theta,\phi)$$, since the problem has rotational symmetry. The solution has the form $$\psi_{lmn}(r,\theta,\phi) = R_{nl}(r)Y_{lm}(\theta,\phi)$$ where $$R(r)$$ describes the radial part and $$Y_{lm}(\theta,\phi)$$ are the spherical harmonics.
When deriving this solution, one normally aligns the problem to the $$z$$ coordinate and gets a representation for $$Y_{lm}(\theta,\phi)$$ with the angle $$\theta$$ measured from the $$z$$-axis.
My question is, how does this asymmetry and distinguishing the $$z$$-axis relate to the "image" of hydrogen? If I actually had a single hydrogen atom and would want o measure the probability density of the electron: To which axis would the wave function be aligned?