Hydrogen eigenstate n2 l0 m0

The wavefunction is:

${\displaystyle \psi _{2,0,0}(r,\vartheta ,\varphi )={\sqrt {{\left({\frac {1}{a_{0}}}\right)}^{3}{\frac {1}{8}}}}\cdot e^{\textstyle -r/(2\,a_{0})}\cdot L_{1}^{1}(r/a_{0})\cdot {\frac {1}{\sqrt {4\,\pi }}}}$

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 = 2: Energy: ${\displaystyle E=-1\,\mathrm {Ry} /n^{2}=-13.6\,\mathrm {eV} /4}$
• l = 0: Angular momentum: ${\displaystyle |L|={\sqrt {l\,(l+1)}}\,\hbar =0}$
• m = 0: Angular momentum in z-direction: ${\displaystyle L_{z}=m\,\hbar =0}$

Since l=0, this is called a s-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.

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