Hydrogen eigenstate n2 l0 m0 wedgecut

Autor/Urheber:

Geek3

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https://dewiki.de/b/8764d

Quelle:

Wikimedia Commons

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2560 x 2560 Pixel (277892 Bytes)

Beschreibung:

Computed 2s₀orbital of an electron's eigenstate of the energy- and angular momentum operators in the Coulomb-field of a hydrogen nucleus. Such an eigenstate keeps its spatial shape over time while obeying the Schrödinger equation and only advances its complex phase.

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.

A wedge was cut away from the orbital to make the internal structure visible.

Lizenz:

CC BY-SA 4.0

Credit:

Eigenes Werk, Created with hydrogen 1.1

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