[{WikipediaArticle oldid='195373293'}]



[{VerifyArticle user='eertl' template='Standard' date='07. November 2014' page-date='2014' comment='Überprüft, nach Jaros, A., Nussbaumer, A., Nussbaumer, P., Physik compact. Basiswissen 7RG; Mortimer, C.E., Müller, U., Chemie. Das Basiswissen der Chemie; Dobrinski, P., Krakau, G., Vogel, A., Physik für Ingenieure; bis auf die Quantentheorie' funder='44' }]
[{ALLOW edit Admin}][{ALLOW view All}][{ALLOW comment All}][{ALLOW verify eertl}]
%%information
%%(font-size: 80%;)
!!!License Information of Images on page
||Image Description||Credit||Artist||License Name||File
| | | | [{Image src='https://www.austria-forum.org/cc/images/slim/publicdomain.png' alt='Public domain' align='center' link='https://www.austria-forum.org/cc/public-domain-10.html' target='_blank'}]| Datei:AOs-3D-dots.png
| Atomic hydrogen -like single-electron orbital showing ( ψ n = 4 , l = 1 , m = − 1 − ψ n = 4 , l = 1 , m = 1 ) / 2 \displaystyle (\psi _n=4,l=1,m=-1-\psi _n=4,l=1,m=1)/\sqrt 2 , also called 4px-orbital for its alignment in x-direction. The image is a 3D rendering of the spatial density distribution of 𝜓² with the color depicting the phase of 𝜓. The spatial distribution is smooth and vanishes for large radii. The cloud is a more realistic representation of an orbital than the more common solid-body approximations. At full resolution, 1 Å =6.3px.| Diese   PNG- Rastergrafik wurde  mit Python erstellt.| Geek3| [{Image src='https://www.austria-forum.org/cc/images/slim/by-sa.png' alt='CC BY-SA 4.0' align='center' link='https://www.austria-forum.org/cc/by-sa-40.html' target='_blank'}]| Datei:Atomic-orbital-cloud n4 px.png
| The Wikimedia Commons logo, SVG version.| Original created by Reidab ( PNG version ) SVG version was created by Grunt and cleaned up by 3247 . Re-creation with SVG geometry features by Pumbaa , using a proper partial circle and SVG geometry features. (Former versions used to be slightly warped.)| Reidab , Grunt , 3247 , Pumbaa| [{Image src='https://www.austria-forum.org/cc/images/slim/by-sa.png' alt='CC BY-SA 3.0' align='center' link='https://www.austria-forum.org/cc/by-sa-30.html' target='_blank'}]| Datei:Commons-logo.svg
| Computed 1s 0 orbital 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: ψ 1 , 0 , 0 ( r , ϑ , φ ) = ( 2 a 0 ) 3 1 2 ⋅ e − r / a 0 ⋅ 1 4 π \displaystyle \psi _1,0,0(r,\vartheta ,\varphi )=\sqrt \left(\frac 2a_0\right)^3\frac 12\cdot e^\textstyle -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 = 1: Energy : E = − 1 R y / n 2 = − 13.6 e V \displaystyle E=-1\,\mathrm Ry /n^2=-13.6\,\mathrm eV  l = 0: Angular momentum :  L  = l ( l + 1 ) ℏ = 0 \displaystyle L=\sqrt l\,(l+1)\,\hbar =0 m = 0: Angular momentum in z-direction : L z = m ℏ = 0 \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.| Eigenes Werk , Created with hydrogen 1.1| Geek3| [{Image src='https://www.austria-forum.org/cc/images/slim/by-sa.png' alt='CC BY-SA 4.0' align='center' link='https://www.austria-forum.org/cc/by-sa-40.html' target='_blank'}]| Datei:Hydrogen eigenstate n1 l0 m0 wedgecut.png
| Computed 2s 0 orbital 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: ψ 2 , 0 , 0 ( r , ϑ , φ ) = ( 1 a 0 ) 3 1 8 ⋅ e − r / ( 2 a 0 ) ⋅ L 1 1 ( r / a 0 ) ⋅ 1 4 π \displaystyle \psi _2,0,0(r,\vartheta ,\varphi )=\sqrt \left(\frac 1a_0\right)^3\frac 18\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 : E = − 1 R y / n 2 = − 13.6 e V / 4 \displaystyle E=-1\,\mathrm Ry /n^2=-13.6\,\mathrm eV /4 l = 0: Angular momentum :  L  = l ( l + 1 ) ℏ = 0 \displaystyle L=\sqrt l\,(l+1)\,\hbar =0 m = 0: Angular momentum in z-direction : L z = m ℏ = 0 \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.| Eigenes Werk , Created with hydrogen 1.1| Geek3| [{Image src='https://www.austria-forum.org/cc/images/slim/by-sa.png' alt='CC BY-SA 4.0' align='center' link='https://www.austria-forum.org/cc/by-sa-40.html' target='_blank'}]| Datei:Hydrogen eigenstate n2 l0 m0 wedgecut.png
| Calculated 2p -1 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: ψ 2 , 1 , − 1 ( r , ϑ , φ ) = ( 1 a 0 ) 3 1 24 ⋅ e − r / ( 2 a 0 ) ⋅ ( r a 0 ) ⋅ Y 1 − 1 ( ϑ , φ ) \displaystyle \psi _2,1,-1(r,\vartheta ,\varphi )=\sqrt \left(\frac 1a_0\right)^3\frac 124\cdot e^\textstyle -r/(2\,a_0)\cdot \left(\frac ra_0\right)\cdot Y_1^-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 = 2: Energy : E = − 1 R y / n 2 = − 13.6 e V / 4 \displaystyle E=-1\,\mathrm Ry /n^2=-13.6\,\mathrm eV /4 l = 1: Angular momentum :  L  = l ( l + 1 ) ℏ = 2 ℏ \displaystyle L=\sqrt l\,(l+1)\,\hbar =\sqrt 2\,\hbar  m = -1: Angular momentum in z-direction : L z = m ℏ = − ℏ \displaystyle L_z=m\,\hbar =-\hbar  Since l=1, this is called a p-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.| Eigenes Werk , Created with hydrogen 1.0| Geek3| [{Image src='https://www.austria-forum.org/cc/images/slim/by-sa.png' alt='CC BY-SA 3.0' align='center' link='https://www.austria-forum.org/cc/by-sa-30.html' target='_blank'}]| Datei:Hydrogen eigenstate n2 l1 m-1.png
| Calculated 2p 0 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: ψ 2 , 1 , 0 ( r , ϑ , φ ) = ( 1 a 0 ) 3 1 24 ⋅ e − r / ( 2 a 0 ) ⋅ ( r a 0 ) ⋅ Y 1 0 ( ϑ , φ ) \displaystyle \psi _2,1,0(r,\vartheta ,\varphi )=\sqrt \left(\frac 1a_0\right)^3\frac 124\cdot e^\textstyle -r/(2\,a_0)\cdot \left(\frac ra_0\right)\cdot Y_1^0(\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 = 2: Energy : E = − 1 R y / n 2 = − 13.6 e V / 4 \displaystyle E=-1\,\mathrm Ry /n^2=-13.6\,\mathrm eV /4 l = 1: Angular momentum :  L  = l ( l + 1 ) ℏ = 2 ℏ \displaystyle L=\sqrt l\,(l+1)\,\hbar =\sqrt 2\,\hbar  m = 0: Angular momentum in z-direction : L z = m ℏ = 0 \displaystyle L_z=m\,\hbar =0 Since l=1, this is called a p-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.| Eigenes Werk , Created with hydrogen 1.0| Geek3| [{Image src='https://www.austria-forum.org/cc/images/slim/by-sa.png' alt='CC BY-SA 3.0' align='center' link='https://www.austria-forum.org/cc/by-sa-30.html' target='_blank'}]| Datei:Hydrogen eigenstate n2 l1 m0.png
| Calculated 2p 1 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: ψ 2 , 1 , 1 ( r , ϑ , φ ) = ( 1 a 0 ) 3 1 24 ⋅ e − r / ( 2 a 0 ) ⋅ ( r a 0 ) ⋅ Y 1 1 ( ϑ , φ ) \displaystyle \psi _2,1,1(r,\vartheta ,\varphi )=\sqrt \left(\frac 1a_0\right)^3\frac 124\cdot e^\textstyle -r/(2\,a_0)\cdot \left(\frac ra_0\right)\cdot Y_1^1(\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 = 2: Energy : E = − 1 R y / n 2 = − 13.6 e V / 4 \displaystyle E=-1\,\mathrm Ry /n^2=-13.6\,\mathrm eV /4 l = 1: Angular momentum :  L  = l ( l + 1 ) ℏ = 2 ℏ \displaystyle L=\sqrt l\,(l+1)\,\hbar =\sqrt 2\,\hbar  m = 1: Angular momentum in z-direction : L z = m ℏ = ℏ \displaystyle L_z=m\,\hbar =\hbar  Since l=1, this is called a p-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.| Eigenes Werk , Created with hydrogen 1.0| Geek3| [{Image src='https://www.austria-forum.org/cc/images/slim/by-sa.png' alt='CC BY-SA 3.0' align='center' link='https://www.austria-forum.org/cc/by-sa-30.html' target='_blank'}]| Datei:Hydrogen eigenstate n2 l1 m1.png
| Computed 3s 0 orbital 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: ψ 3 , 0 , 0 ( r , ϑ , φ ) = ( 2 3 a 0 ) 3 1 18 ⋅ e − r / ( 3 a 0 ) ⋅ L 2 1 ( 2 r / ( 3 a 0 ) ) ⋅ 1 4 π \displaystyle \psi _3,0,0(r,\vartheta ,\varphi )=\sqrt \left(\frac 23\,a_0\right)^3\frac 118\cdot e^\textstyle -r/(3\,a_0)\cdot L_2^1(2\,r/(3\,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 = 3: Energy : E = − 1 R y / n 2 = − 13.6 e V / 9 \displaystyle E=-1\,\mathrm Ry /n^2=-13.6\,\mathrm eV /9 l = 0: Angular momentum :  L  = l ( l + 1 ) ℏ = 0 \displaystyle L=\sqrt l\,(l+1)\,\hbar =0 m = 0: Angular momentum in z-direction : L z = m ℏ = 0 \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.| Eigenes Werk , Created with hydrogen 1.1| Geek3| [{Image src='https://www.austria-forum.org/cc/images/slim/by-sa.png' alt='CC BY-SA 4.0' align='center' link='https://www.austria-forum.org/cc/by-sa-40.html' target='_blank'}]| Datei:Hydrogen eigenstate n3 l0 m0 wedgecut.png
| Calculated 3p -1 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: ψ 3 , 1 , − 1 ( r , ϑ , φ ) = ( 2 3 a 0 ) 3 1 6 ⋅ 4 ! ⋅ e − r / ( 3 a 0 ) ⋅ ( 2 r 3 a 0 ) ⋅ L 1 3 ( 2 r / ( 3 a 0 ) ) ⋅ Y 1 − 1 ( ϑ , φ ) \displaystyle \psi _3,1,-1(r,\vartheta ,\varphi )=\sqrt \left(\frac 23\,a_0\right)^3\frac 16\cdot 4!\cdot e^\textstyle -r/(3\,a_0)\cdot \left(\frac 2\,r3\,a_0\right)\cdot L_1^3(2\,r/(3\,a_0))\cdot Y_1^-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 : E = − 1 R y / n 2 = − 13.6 e V / 9 \displaystyle E=-1\,\mathrm Ry /n^2=-13.6\,\mathrm eV /9 l = 1: Angular momentum :  L  = l ( l + 1 ) ℏ = 2 ℏ \displaystyle L=\sqrt l\,(l+1)\,\hbar =\sqrt 2\,\hbar  m = -1: Angular momentum in z-direction : L z = m ℏ = − ℏ \displaystyle L_z=m\,\hbar =-\hbar  Since l=1, this is called a p-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.| Eigenes Werk , Created with hydrogen 1.0| Geek3| [{Image src='https://www.austria-forum.org/cc/images/slim/by-sa.png' alt='CC BY-SA 3.0' align='center' link='https://www.austria-forum.org/cc/by-sa-30.html' target='_blank'}]| Datei:Hydrogen eigenstate n3 l1 m-1.png
%%
%%