每日彙整: 2016-10-09

樹莓派、樹莓派之學習、樹莓派之教育

光的世界︰【□○閱讀】反射式望遠鏡《一》

因為光之折射容易發生色散現象，而且集光力越大透鏡越重，因此大型天文望遠鏡通常以『反射鏡』為主作設計。切莫認為『反射』就像打乒乓球那樣的簡單明白，所以光子不過被邊界面轉向而已。也許一段量子力學物質波反射之描述︰

Quantum reflection

Quantum reflection is a physical phenomenon involving the reflection of a matter wave from an attractive potential. In classical mechanics, such a phenomenon is not possible; for instance when one magnet is pulled toward another, the observer does not expect one of the magnets to suddenly (i.e. before the magnets ‘touch’) turn around and retreat in the opposite direction.

Definition

Quantum reflection became an important branch of physics in the 21st century. In a workshop about quantum reflection,[1] the following definition of quantum reflection was suggested:

Quantum reflection is a classically counterintuitive phenomenon whereby the motion of particles is reverted “against the force” acting on them. This effect manifests the wave nature of particles and influences collisions of ultracold atoms and interaction of atoms with solid surfaces.

Observation of quantum reflection has become possible thanks to recent advances in trapping and cooling atoms.

Reflection of slow atoms

Although the principles of quantum mechanics apply to any particles, usually the term “quantum reflection” means reflection of atoms from a surface of condensed matter (liquid or solid). The full potential experienced by the incident atom does become repulsive at a very small distance from the surface (of order of size of atoms). This is when the atom becomes aware of the discrete character of material. This repulsion is responsible for the classical scattering one would expect for particles incident on a surface. Such scattering is diffuse rather than specular, and so this component of the reflection is easy to distinguish. Indeed to reduce this part of the physical process, a grazing angle of incidence is used; this enhances the quantum reflection. This requirement of small incident velocities for the particles means that the non-relativistic approximation to quantum mechanics is all that is required.

 

足以說明現今科學的前線疆域。如同將『歐拉』所講的『可加性』說法︰

十二因緣

《緣起經》玄奘譯

佛言，云何名緣起初義？謂：依此有故彼有，此生故彼生。所謂：無明緣行，行緣識，識緣名色，名色緣六處，六處緣觸，觸緣受，受緣愛，愛緣取，取緣有，有緣生，生緣老死，起愁、歎、苦、憂、惱，是名為純大苦蘊集，如是名為緣起初義。

『邏輯學』上說『有□則有○，無○則無□』，既已『有□』又想『無○』，哪裡能夠不矛盾的啊！過去魏晉時『王弼』講︰一，數之始而物之極也。謂之為妙有者，欲言有，不見其形，則非有，故謂之妙；欲言其無，物由之以生，則非無，故謂之有也。斯乃無中之有，謂之妙有也。假使用『恆等式』 1 - x^n = (1 - x)(1 + x + \cdots + x^{n-1}) 來計算 \frac{1 + x + \cdots + x^{m-1}}{1 + x + \cdots + x^{n-1}}，將等於 \frac{1 - x^m}{1 - x^n} = (1 - x^m) \left[1 + (x^n) + { (x^n) }^2 + { (x^n) } ^3 + \cdots \right] = 1 - x^m + x^n - x^{n+m} + x^{2n} - \cdots，那麼 1 - 1 + 1 - 1 + \cdots 難道不應該『等於』 \frac{m}{n} 的嗎？一七四三年時，『伯努利』正因此而反對『歐拉』所講的『可加性』說法，『同』一個級數怎麼可能有『不同』的『和』的呢？？作者不知如果在太空裡，乘坐著『加速度』是 g 的太空船，在上面用著『樹莓派』控制的『奈米手』來擲『骰子』，是否一定能得到『相同點數』呢？難道說『牛頓力學』不是只要『初始態』是『相同』的話，那個『骰子』的『軌跡』必然就是『一樣』的嗎？？據聞，法國籍義大利裔大數學家『約瑟夫‧拉格朗日』伯爵 Joseph Lagrange 倒是有個『說法』︰事實上，對於『不同』的 m,n 來講， 從『幂級數』來看，那個 = 1 - x^m + x^n - x^{n+m} + x^{2n} - \cdots 是有『零的間隙』的 1 + 0 + 0 + \cdots - 1 + 0 + 0 + \cdots，這就與 1 - 1 + 1 - 1 + \cdots 『形式』上『不同』，我們怎麼能『先驗』的『期望』結果會是『相同』的呢！！

應用於『真空』︰

220px-Casimir_plates_bubbles.svg

220px-Casimir_plates.svg

220px--Water_wave_analogue_of_Casimir_effect.ogv

一九四八年時，荷蘭物理學家『亨德里克‧卡西米爾』 Hendrik Casimir 提出了『真空不空』的『議論』。因為依據『量子場論』，『真空』也得有『最低能階』，因此『真空能量』不論因不因其『實虛』粒子之『生滅』，總得有一個『量子態』。由於已知『原子』與『分子』的『主要結合力』是『電磁力』，那麼該『如何』說『真空』之『量化』與『物質』的『實際』是怎麽來『配合』的呢？因此他『計算』了這個『可能效應』之『大小』，然而無論是哪種『震盪』所引起的，他總是得要面臨『無窮共振態』 \langle E \rangle = \frac{1}{2} \sum \limits_{n}^{\infty} E_n 的『問題』，這也就是說『平均』有『多少』各種能量的『光子？』所參與 h\nu + 2h\nu + 3h\nu + \cdots 的『問題』？據知『卡西米爾』用『歐拉』等之『可加法』，得到了 {F_c \over A} = -\frac {\hbar c \pi^2} {240 a^4}。

此處之『負』 - 代表『吸引力』，而今早也已經『證實』的了，真不知『宇宙』是果真先就有『計畫』的嗎？還是說『人們』自己還在『幻想』的呢？？
─── 摘自《【Sonic π】電聲學之電路學《四》之《 V!》‧下》

 

讓我們大開了眼界一般。

且先列出曲面反射的矩陣光學形制︰

【WiKi】

Reflection from a curved mirror {\displaystyle {\begin{pmatrix}1&0\\-{\frac {2}{R_{e}}}&1\end{pmatrix}}} {\displaystyle R_{e}=R\cos \theta } effective radius of curvature in tangential plane (horizontal direction)
{\displaystyle R_{e}=R/\cos \theta } effective radius of curvature in the sagittal plane (vertical direction)
R = radius of curvature, R > 0 for concave, valid in the paraxial approximation
\theta is the mirror angle of incidence in the horizontal plane.

 

【SymPy】

class sympy.physics.optics.gaussopt.CurvedMirror

Ray Transfer Matrix for reflection from curved surface.

Parameters : R : radius of curvature (positive for concave)

See also

RayTransferMatrix

Examples

>>> from sympy.physics.optics import CurvedMirror
>>> from sympy import symbols
>>> R = symbols('R')
>>> CurvedMirror(R)
Matrix([
[   1, 0],
[-2/R, 1]])

 

然後踏上探索之旅耶！！