# 原子：電子篇

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subshell係空間量子化 .

Hartree approximation 唔考慮包立不相容原理,

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 tom.care VIP 帖子3838 積分3261 金幣0  註冊時間2015-8-29  發短消息 加為好友 64# 大 中 小 發表於 2017-10-3 10:41 PM  只看該作者     學習 Spherical Harmonics ，由第一頁睇起。 https://en.wikipedia.org/wiki/Spherical_harmonics In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations that commonly occur in science. The spherical harmonics are a complete set of orthogonal functions on the sphere, and thus may be used to represent functions defined on the surface of a sphere, just as circular functions (sines and cosines) are used to represent functions on a circle via Fourier series. Like the sines and cosines in Fourier series, the spherical harmonics may be organized by (spatial) angular frequency, as seen in the rows of functions in the illustration on the right. Further, spherical harmonics are basis functions for SO(3), the group of rotations in three dimensions, and thus play a central role in the group theoretic discussion of SO(3). 呢一段我大部份都睇得明，雖然佢有好多都係 state without proof 。 Complete set of orthogonal functions on the sphere 呢個概念好得意。 但係 SO(3) 係乜，我大概知道係 Lie group ，但係唔肯定，查下先。 .postmorevideochannel{margin-top:10px;display:block; width:160px; height:30px; background:url('/images/video_channel_more.png?v=20160829b') no-repeat; font-size:0;} .postmorevideochannel:hover,.postmorevideochannel:focus{background-position:0 -30px;} @media (-webkit-min-device-pixel-ratio: 2), (min-resolution: 192dpi) { .postmorevideochannel{background:url('/images/video_channel_more_2x.png?v=20160829b') no-repeat; background-size:100% auto;} } 回覆 引用 TOP
 tom.care VIP 帖子3838 積分3261 金幣0  註冊時間2015-8-29  發短消息 加為好友 65# 大 中 小 發表於 2017-10-4 12:35 AM  只看該作者     [按此打開] [隱藏] SO(3) 應該係 set of 3D rotation as function interpreted as a group。Closure, associativity, inverse 都 ok，冇錯，係 group。但係一個 basis function for a group 係咩意思呢？再查！ .postmorevideochannel{margin-top:10px;display:block; width:160px; height:30px; background:url('/images/video_channel_more.png?v=20160829b') no-repeat; font-size:0;} .postmorevideochannel:hover,.postmorevideochannel:focus{background-position:0 -30px;} @media (-webkit-min-device-pixel-ratio: 2), (min-resolution: 192dpi) { .postmorevideochannel{background:url('/images/video_channel_more_2x.png?v=20160829b') no-repeat; background-size:100% auto;} } 回覆 引用 被引用(1) TOP
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SO(3) 應該係 set of 3D rotation as function interpreted as a group。Closure, associativity, inverse 都 ok，冇錯，係 group。但係一個 basis function for a group 係咩意思呢？再查！

Despite their name, spherical harmonics take their simplest form in Cartesian coordinates, where they can be defined as homogeneous polynomials of degree {\displaystyle \ell }
in {\displaystyle (x,y,z)}
that obey Laplace's equation. Functions that satisfy Laplace's equation are often said to be harmonic, hence the name spherical harmonics. The connection with spherical coordinates arises immediately if one uses the homogeneity to extract a factor of {\displaystyle r^{\ell }}
from the above-mentioned polynomial of degree {\displaystyle \ell }
; the remaining factor can be regarded as a function of the spherical angular coordinates {\displaystyle \theta }
and {\displaystyle \varphi }
only, or equivalently of the orientational unit vector {\displaystyle {\mathbf {r} }}
specified by these angles. In this setting, they may be viewed as the angular portion of a set of solutions to Laplace's equation in three dimensions, and this viewpoint is often taken as an alternative definition.
A specific set of spherical harmonics, denoted {\displaystyle Y_{\ell }^{m}(\theta ,\varphi )}
or {\displaystyle Y_{\ell }^{m}({\mathbf {r} })}
, are called Laplace's spherical harmonics, as they were first introduced by Pierre Simon de Laplace in 1782.[1] These functions form an orthogonal system, and are thus basic to the expansion of a general function on the sphere as alluded to above.

Homogeneous 既話就可以抽 r^l factor ，所以可以factorize左個radical component，淨番另外兩個coordinate，就係theta 同 phi，暫時依然明。

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Pauli 原理 :
ψ = (1/root2) [ ψ1(1)ψ1 (2)  - ψ1(2) ψ1(1) ]  = 0

Hartree approximation 唔考慮包立不相容原理,主要係電子 ψ的設定:

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 tom.care VIP 帖子3838 積分3261 金幣0  註冊時間2015-8-29  發短消息 加為好友 69# 大 中 小 發表於 2017-10-6 10:09 AM  只看該作者     下一步要研究一下 history 裏面個個 expansion。 點解會突然間出左 Legendre's Polynomial。 .postmorevideochannel{margin-top:10px;display:block; width:160px; height:30px; background:url('/images/video_channel_more.png?v=20160829b') no-repeat; font-size:0;} .postmorevideochannel:hover,.postmorevideochannel:focus{background-position:0 -30px;} @media (-webkit-min-device-pixel-ratio: 2), (min-resolution: 192dpi) { .postmorevideochannel{background:url('/images/video_channel_more_2x.png?v=20160829b') no-repeat; background-size:100% auto;} } 回覆 引用 被引用(1) TOP
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 tom.care VIP 帖子3838 積分3261 金幣0  註冊時間2015-8-29  發短消息 加為好友 71# 大 中 小 發表於 2017-10-6 10:33 PM  只看該作者     睇明左 Legendre's polynomial 個 part 。 其實只不過係一個 Taylor Expansion ，把 1/distance 當成一個function of ratio of point norm 黎睇，再係 0 個個位開始展開 power series expansion，咁啱就係一套 orthogonal polynomials 。 科普解釋：研究呢堆數學，就係想明白 atomic orbital 既 shape 係點黎。 .postmorevideochannel{margin-top:10px;display:block; width:160px; height:30px; background:url('/images/video_channel_more.png?v=20160829b') no-repeat; font-size:0;} .postmorevideochannel:hover,.postmorevideochannel:focus{background-position:0 -30px;} @media (-webkit-min-device-pixel-ratio: 2), (min-resolution: 192dpi) { .postmorevideochannel{background:url('/images/video_channel_more_2x.png?v=20160829b') no-repeat; background-size:100% auto;} } 回覆 引用 被引用(1) TOP

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The Pauli exclusion principle is the quantum mechanical principle which states that two or more identical fermions (particles with half-integer spin) cannot occupy the same quantum state within a quantum system simultaneously。

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The Paul ...
The Pauli exclusion principle is the quantum mechanical principle which states that two or moreidentical fermions (particles with half-integer spin) cannot occupy the same quantum state within a quantum system simultaneously。

Bingo！等咗你這初步答案已很耐了，可惜無人即時答到，可惜！

[ 本帖最後由 LT3648 於 2017-10-7 02:14 PM 編輯 ]

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The Pauli exclusion principle is the quantum mechanical principle which states that two or moreidentical fermions (particles with half-integer spin) cannot occupy the same quantum state within ...

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