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原子:電子篇

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引用:
原帖由 bjj123 於 2017-10-3 08:40 AM 發表


subshell係空間量子化 .
能階分裂 => Fine structure , Hyperfine .....
多電子原子 , 應該是用 Hartree
Hartree approximation 唔考慮包立不相容原理,
哈特里-福克方程 approximation 考慮包立不相容原理
. ...
可否簡單分享多些?



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引用:
原帖由 LT3648 於 2017-10-3 03:01 PM 發表



學習下樓主對追求學問,不計榮辱的胸襟囉!米成日嬲豬豬啦!
唔覺你有追求學問 你只係將自己既弱點暴露 仲有呢度只有你一個關心嬲豬 其他鬼會得閒理你






引用:
原帖由 sylim 於 2017-10-3 07:10 PM 發表




唔覺你有追求學問 你只係將自己既弱點暴露 仲有呢度只有你一個關心嬲豬 其他鬼會得閒理你
經常得梳蕉齋嗡那個咪有弱點囉!






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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。






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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!


引用:
原帖由 tom.care 於 2017-10-4 12:35 AM 發表

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!
睇唔明,不過好似唔係好重要,skip,睇下一段。

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.

有幾個重點,Laplace Equation,Spherical Coordinates 和 Homogeneous Polynomial

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。

但係Wave Equation 唔係 Sine Cosine 就搞到咩,點解要搞另外一套basis?下次再睇睇。






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引用:
原帖由 LT3648 於 2017-10-3 06:06 PM 發表




可否簡單分享多些?
哈特里-福克方程 approximation 考慮包立不相容原理, 主要係電子 ψ的設定:
電子 是費米子   ψ = - ψ  
若 2個電子 ψ   ;  ψ1(1) ψ2 (2) = - ψ1(2) ψ2(1)
位置交換  ψ = (1/root2) [ ψ1(1)ψ2 (2)  - ψ1(2) ψ2(1) ]

Pauli 原理 :
ψ = (1/root2) [ ψ1(1)ψ1 (2)  - ψ1(2) ψ1(1) ]  = 0
或者 ψ = (1/root2) [ ψ2(1)ψ2 (2)  - ψ2(2) ψ2(1) ]  = 0

Hartree approximation 唔考慮包立不相容原理,主要係電子 ψ的設定:
簡單設定為 ψ(1,2, ........, N-1 , N ) = ψ1(1)ψ2(2)......  ψN-1(N-1)ψN(N)






引用:
原帖由 bjj123 於 2017-10-4 10:48 AM 發表





哈特里-福克方程 approximation 考慮包立不相容原理, 主要係電子 ψ的設定:
電子 是費米子   ψ = - ψ  
若 2個電子 ψ   ;  ψ1(1) ψ2 (2) = - ψ1(2) ψ2(1)
位置交換  ψ = (1/root2) [ ψ1(1)ψ2  ...
為了推廣科普,可否輔以文字簡述?






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下一步要研究一下 history 堶戚茩 expansion。
點解會突然間出左 Legendre's Polynomial。



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引用:
原帖由 tom.care 於 2017-10-6 10:09 AM 發表

下一步要研究一下 history 堶戚茩 expansion。
點解會突然間出左 Legendre's Polynomial。
請繼續,最好加點科普性解釋。







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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。







引用:
原帖由 tom.care 於 2017-10-6 10:33 PM 發表

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 ...
科普解釋:研究呢堆數學,就係想明白 atomic orbital 既 shape 係點黎<-----是否誤會了?我由頭到尾都係問不同orbital overlap與包利不相容原理有無抵觸。







引用:
原帖由 LT3648 於 2017-10-6 10:38 PM 發表

科普解釋:研究呢堆數學,就係想明白 atomic orbital 既 shape 係點黎
冇理解錯,你研究你的有冇抵觸問題,我理解我的Spherical Harmonics,冇抵觸 :p

題外話,你條問題真係好低能,你再睇一睇 Pauli Exclusion Principle 既 statement 就知道。Wikipedia 第一句就寫。

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。

關 atomic orbital 餐蛋治 …… 這個 principle 只管 quantum state,不管電子本身就哪堙C







引用:
原帖由 tom.care 於 2017-10-6 11:47 PM 發表


冇理解錯,你研究你的有冇抵觸問題,我理解我的Spherical Harmonics,冇抵觸 :p

題外話,你條問題真係好低能,你再睇一睇 Pauli Exclusion Principle 既 statement 就知道。Wikipedia 第一句就寫。

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!

再深入d,問你一個問題,電子cannot occupy the same quantum state ,咁可否occupy the same space or overlap?

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



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引用:
原帖由 LT3648 於 2017-10-7 12:46 PM 發表

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 ...
呢個問題一樣多舊魚,係 Quantum 堙A電子根本冇確定既 position,何來 occupy the same space ?



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