File:Hydrogen eigenstate n4 l3 m1.png

Summary

Calculated 4f₁<a href="https://en.wikipedia.org/wiki/Atomic_orbital" class="extiw" title="en:Atomic orbital">orbital</a> of an electron's eigenstate in the Coulomb-field of a hydrogen nucleus. An <a href="https://en.wikipedia.org/wiki/Eigenfunction" class="extiw" title="en:Eigenfunction">eigenstate</a> is a state which keeps it's shape except for a complex phase when the <a href="https://en.wikipedia.org/wiki/Hamiltonian_(quantum_mechanics)" class="extiw" title="en:Hamiltonian (quantum mechanics)">Hamilton operator</a> is applied, thus being invariant in time while obeying the <a href="https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation" class="extiw" title="en:Schrödinger equation">Schrödinger equation</a>. The orbital is aligned around the z-axis, but remains an eigenfunction if rotated to any direction.

The <a href="https://en.wikipedia.org/wiki/Hydrogen_atom#Wavefunction" class="extiw" title="en:Hydrogen atom">wavefunction</a> is:

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): <semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><msub><mi>ψ<!-- &psi; --></mi><mrow class="MJX-TeXAtom-ORD"><mn>4</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>r</mi><mo>,</mo><mi>ϑ<!-- &thetasym; --></mi><mo>,</mo><mi>φ<!-- &phi; --></mi><mo stretchy="false">)</mo><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><msqrt><msup><mrow class="MJX-TeXAtom-ORD"><mrow><mo>(</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>1</mn><mrow><mn>2</mn><mspace width="thinmathspace"></mspace><msub><mi>a</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mn>3</mn></mrow></msup><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>1</mn><mrow><mn>8</mn><mo>⋅<!-- &sdot; --></mo><mn>7</mn><mo>!</mo></mrow></mfrac></mrow></msqrt></mrow><mo>⋅<!-- &sdot; --></mo><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="false" scriptlevel="0"><mo>−<!-- &minus; --></mo><mi>r</mi><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mo stretchy="false">(</mo><mn>4</mn><mspace width="thinmathspace"></mspace><msub><mi>a</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo stretchy="false">)</mo></mstyle></mrow></msup><mo>⋅<!-- &sdot; --></mo><msup><mrow><mo>(</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mrow><mn>1</mn><mspace width="thinmathspace"></mspace><mi>r</mi></mrow><mrow><mn>2</mn><mspace width="thinmathspace"></mspace><msub><mi>a</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></mrow></mfrac></mrow><mo>)</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mn>3</mn></mrow></msup><mo>⋅<!-- &sdot; --></mo><msubsup><mi>Y</mi><mrow class="MJX-TeXAtom-ORD"><mn>3</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>ϑ<!-- &thetasym; --></mi><mo>,</mo><mi>φ<!-- &phi; --></mi><mo stretchy="false">)</mo></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle \psi _{4,3,1}(r,\vartheta ,\varphi )={\sqrt {{\left({\frac {1}{2\,a_{0}}}\right)}^{3}{\frac {1}{8\cdot 7!}}}}\cdot e^{\textstyle -r/(4\,a_{0})}\cdot \left({\frac {1\,r}{2\,a_{0}}}\right)^{3}\cdot Y_{3}^{1}(\vartheta ,\varphi )}</annotation></semantics> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47b502dac8a333011a5de8ab5ec06453601c0338" class="mwe-math-fallback-image-inline mw-math-element" aria-hidden="true" style="vertical-align: -3.005ex; width:67.341ex; height:7.676ex;" alt="{\displaystyle \psi _{4,3,1}(r,\vartheta ,\varphi )={\sqrt {{\left({\frac {1}{2\,a_{0}}}\right)}^{3}{\frac {1}{8\cdot 7!}}}}\cdot e^{\textstyle -r/(4\,a_{0})}\cdot \left({\frac {1\,r}{2\,a_{0}}}\right)^{3}\cdot Y_{3}^{1}(\vartheta ,\varphi )}">

The state is an eigenstate of <a href="https://en.wikipedia.org/wiki/Hamiltonian_(quantum_mechanics)" class="extiw" title="en:Hamiltonian (quantum mechanics)">H</a>, <a href="https://en.wikipedia.org/wiki/Angular_momentum_operator" class="extiw" title="en:Angular momentum operator">L</a>² and <a href="https://en.wikipedia.org/wiki/Angular_momentum_operator" class="extiw" title="en:Angular momentum operator">L</a>_z, which constitute a <a href="https://en.wikipedia.org/wiki/complete_set_of_commuting_observables" class="extiw" title="en:complete set of commuting observables">complete set of commuting observables</a>.
The <a href="https://en.wikipedia.org/wiki/quantum_number" class="extiw" title="en:quantum number">quantum numbers</a> mean that the following quantities have a sharp certain value:

• n = 4: <a href="https://en.wikipedia.org/wiki/Principal_quantum_number" class="extiw" title="en:Principal quantum number">Energy</a>: Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): <semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>E</mi><mo>=</mo><mo>−<!-- &minus; --></mo><mn>1</mn><mspace width="thinmathspace"></mspace><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">R</mi><mi mathvariant="normal">y</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><msup><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msup><mo>=</mo><mo>−<!-- &minus; --></mo><mn>13.6</mn><mspace width="thinmathspace"></mspace><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">e</mi><mi mathvariant="normal">V</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>16</mn></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle E=-1\,\mathrm {Ry} /n^{2}=-13.6\,\mathrm {eV} /16}</annotation></semantics> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/412bc6a17b6be94e7e57a9399faed3c14e47c5df" class="mwe-math-fallback-image-inline mw-math-element" aria-hidden="true" style="vertical-align: -0.838ex; width:30.678ex; height:3.176ex;" alt="{\displaystyle E=-1\,\mathrm {Ry} /n^{2}=-13.6\,\mathrm {eV} /16}">
• l = 3: <a href="https://en.wikipedia.org/wiki/Azimuthal_quantum_number" class="extiw" title="en:Azimuthal quantum number">Angular momentum</a>: Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): <semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>L</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><msqrt><mi>l</mi><mspace width="thinmathspace"></mspace><mo stretchy="false">(</mo><mi>l</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></msqrt></mrow><mspace width="thinmathspace"></mspace><mi class="MJX-variant">ℏ<!-- &#8463; --></mi><mo>=</mo><mn>2</mn><mrow class="MJX-TeXAtom-ORD"><msqrt><mn>3</mn></msqrt></mrow><mspace width="thinmathspace"></mspace><mi class="MJX-variant">ℏ<!-- &#8463; --></mi></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle |L|={\sqrt {l\,(l+1)}}\,\hbar =2{\sqrt {3}}\,\hbar }</annotation></semantics> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8a8881670acaae7cb1bc338f2e11803449c8e2b" class="mwe-math-fallback-image-inline mw-math-element" aria-hidden="true" style="vertical-align: -1.838ex; width:26.809ex; height:4.843ex;" alt="{\displaystyle |L|={\sqrt {l\,(l+1)}}\,\hbar =2{\sqrt {3}}\,\hbar }">
• m = 1: <a href="https://en.wikipedia.org/wiki/Magnetic_quantum_number" class="extiw" title="en:Magnetic quantum number">Angular momentum in z-direction</a>: Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): <semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><msub><mi>L</mi><mrow class="MJX-TeXAtom-ORD"><mi>z</mi></mrow></msub><mo>=</mo><mi>m</mi><mspace width="thinmathspace"></mspace><mi class="MJX-variant">ℏ<!-- &#8463; --></mi><mo>=</mo><mi class="MJX-variant">ℏ<!-- &#8463; --></mi></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle L_{z}=m\,\hbar =\hbar }</annotation></semantics> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07d8c0c21ddbd43f3bef41b1f1fbd5ef455d0366" class="mwe-math-fallback-image-inline mw-math-element" aria-hidden="true" style="vertical-align: -0.671ex; width:13.892ex; height:2.509ex;" alt="{\displaystyle L_{z}=m\,\hbar =\hbar }">

Since l=3, this is called a f-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 <a href="https://en.wikipedia.org/wiki/Ray_tracing_(graphics)" class="extiw" title="en:Ray tracing (graphics)">raytraced</a> using modified <a href="https://en.wikipedia.org/wiki/Phong_reflection_model" class="extiw" title="en:Phong reflection model">Phong lighting</a>.

The <a href="https://en.wikipedia.org/wiki/fine_structure" class="extiw" title="en:fine structure">fine structure</a> is neglected.

Licensing

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