English: Calculated 3d₁ orbital of an electron's eigenstate in the Coulomb-field of a hydrogen nucleus. An eigenstate is a state which keeps it's shape except for a complex phase when the Hamilton operator is applied, thus being invariant in time while obeying the Schrödinger equation. The orbital is aligned around the z-axis, but remains an eigenfunction if rotated to any direction.

The wavefunction is:

${\displaystyle \psi _{3,2,1}(r,\vartheta ,\varphi )={\sqrt {{\left({\frac {2}{3\,a_{0}}}\right)}^{3}{\frac {1}{6\cdot 5!}}}}\cdot e^{\textstyle -r/(3\,a_{0})}\cdot \left({\frac {2\,r}{3\,a_{0}}}\right)^{2}\cdot Y_{2}^{1}(\vartheta ,\varphi )}$

The state is an eigenstate of H, L² and L_z, which constitute a complete set of commuting observables.
The quantum numbers mean that the following quantities have a sharp certain value:

• n = 3: Energy: ${\displaystyle E=-1\,\mathrm {Ry} /n^{2}=-13.6\,\mathrm {eV} /9}$
• l = 2: Angular momentum: ${\displaystyle |L|={\sqrt {l\,(l+1)}}\,\hbar ={\sqrt {6}}\,\hbar }$
• m = 1: Angular momentum in z-direction: ${\displaystyle L_{z}=m\,\hbar =\hbar }$

Since l=2, this is called a d-orbital.

The depicted rigid body is where the probability density exceeds a certain value. The color shows the complex phase of the wavefunction, where blue means real positive, red means imaginary positive, yellow means real negative and green means imaginary negative. The image is raytraced using modified Phong lighting.

The fine structure is neglected.