User:Cuzkatzimhut/sandbox

«Wilst du ins Unendliche schreiten, Geh nur im Endlichen nach allen Seiten.»

[If you want to reach the infinite, explore every aspect of the finite.] • 𝔍𝔬𝔥𝔞𝔫𝔫 𝔚𝔬𝔩𝔣𝔤𝔞𝔫𝔤 𝔳𝔬𝔫 𝔊𝔬𝔢𝔱𝔥𝔢

Je suis bon encyclopédiste,
Je connais le mal et le bien,
Je suis Diderot a la piste;
Je connais tout et ne crois rien.
• Denis Diderot

Ich sitze am Straßenrand.
Der Fahrer wechselt das Rad.
Ich bin nicht gern, wo ich herkomme.
Ich bin nicht gern, wo ich hinfahre.
Warum sehe ich den Radwechsel mit Ungeduld?
• Bertolt Brecht

Other mistakes may perchance...await the penetrating glance of some critical reader, to whom the joy of discovery, and the intellectual superiority which he will thus discern, in himself, to the author of this little book, will, I hope, repay to some extent the time and trouble its perusal may have cost him! • Lewis Carroll

Das Licht schimmerte hell durch die Vorhänge der hohen Fensterfront, vermehrt durch den Schein der Autorität und Vornehmheit, den die wartenden Wagen dazutaten, und durch die Blicke der Gaffer, die im Vorbeigehen stehenblieben und eine Weile hinaufsahen, ohne daß sie recht wußten, warum. Es hätte Diotima gefreut, wenn sie das wahrgenommen haben würde. Es standen immer Leute in der Halbhelle, die das Fest auf die Straße streute, und hinter ihren Rücken begann die große Dunkelheit, die in einiger Entfernung rasch undurchdringlich wurde. • dMoE . 72

Reality is that which, when you stop believing in it, doesn't go away. • Philip K. Dick

»Ah dau grober Eselkopp«, repliziert' er hinwieder, »dau bleiwest dein Lewelang a Narr, geit meich wunner, was aus dir wera wird, bist schun su a grußer Dölpel, un waist noch neit, was der Wolf für a veirfeußiger Schelm is.«  • SimSim

„𝕴𝖈𝖍 𝖘𝖎𝖙𝖟𝖊 𝖆𝖒 𝕾𝖙𝖗𝖆ß𝖊𝖓𝖗𝖆𝖓𝖉. 𝕯𝖊𝖗 𝕱𝖆𝖍𝖗𝖊𝖗 𝖜𝖊𝖈𝖍𝖘𝖊𝖑𝖙 𝖉𝖆𝖘 𝕽𝖆𝖉. 𝕴𝖈𝖍 𝖇𝖎𝖓 𝖓𝖎𝖈𝖍𝖙 𝖌𝖊𝖗𝖓, 𝖜𝖔 𝖎𝖈𝖍 𝖍𝖊𝖗𝖐𝖔𝖒𝖒𝖊. 𝕴𝖈𝖍 𝖇𝖎𝖓 𝖓𝖎𝖈𝖍𝖙 𝖌𝖊𝖗𝖓, 𝖜𝖔 𝖎𝖈𝖍 𝖍𝖎𝖓𝖋𝖆𝖍𝖗𝖊. 𝖂𝖆𝖗𝖚𝖒 𝖘𝖊𝖍𝖊 𝖎𝖈𝖍 𝖉𝖊𝖓 𝕽𝖆𝖉𝖜𝖊𝖈𝖍𝖘𝖊𝖑 𝖒𝖎𝖙 𝖀𝖓𝖌𝖊𝖉𝖚𝖑𝖉?“ ―𝕭𝖊𝖗𝖙𝖔𝖑𝖙 𝕭𝖗𝖊𝖈𝖍𝖙

“Never interrupt someone doing something you said couldn't be done.” • Amelia Earhart

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Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103.

Stats

${\frac {1}{\sin(2\pi /7)}}+{\frac {1}{\sin(3\pi /7)}}={\frac {1}{\sin(\pi /7)}}~~~~{\overset {\rightarrow }{\partial _{x}}}~~~\nearrow ~~~\searrow$ 📖 🎥 spukhafter Fernwirker — ❍

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×H F GzsAplqtke ‖ ‖ ‖ ‖ ‖ ‖ × ℘ ℑ ℜ ℂ ℤ ℚ ℝ ∝ m² m³ φ ϑ ↯↯ ℓ ϵ ε  §  · • — • — – ℏ ⤸ ⤹― • ¶ a_k^† ♫ □ ○ ∞ □ ° ♦ ƒ̂ χ̂ Ĥ x̂ p̂ ★ € ¤ ₴ ƒ ₠ ⅞ √ √n+12ℓ ∂ ∫ ^d⁄_dt ∑ ∏ ∞ ∇ ∇ ² ½⅓⅔¼ ¾ ⅜ ⅝ ⅝ ¹³²⁄₁₃₇ 13⁄27 ⟨ ⟩ ⟨|⟩ ∞ |z⟩ ⟨ψ| ⟨a|b⟩ 〈 〉 m³ √2 ≠ ≤ ≥ ± ≈ ≠ ≈ ħ→0 ħ ħ ħ ← ↔ ⇀ ⇄ ⟶ ⟹ ⟼ ↻ ↳ ↦ ↝«   » a ⅐ c ⅑ d ⅒ e ⅓ ⅔ ⅕ ⅖ ⅗ ⅘ ⅙ ⅚ ⅛ ⅜ ⅝ ⅞ f ⁰g ¹h ²m ³n ⁴p ⁵q ⁶ ⁷ ⁸ ⁹ r₀ s₁ t₂ u₃ v₄ w₅ ₆ ₇ ₈ ₉ ₊ 𝔄𝔅ℭ𝔇𝔈𝔉𝔊ℌℑ𝔍𝔎𝔏𝔐𝔑𝔒𝔓𝔔ℜ𝔖𝔗𝔘𝔙𝔚𝔛𝔜ℨ 𝔞𝔟𝔠𝔡𝔢𝔣𝔤𝔥𝔦𝔧𝔨𝔩𝔪𝔫𝔬𝔭𝔮𝔯𝔰𝔱𝔲𝔳𝔴𝔵𝔶𝔷

□

$\therefore \underbrace {{\bar {\epsilon }}{\bar {\sigma }}{\overline {\psi }}} _{\int {\overline {\sigma }}dx}\neq ~{\overline {\sigma }}~$ → √m³ ℤ +7 ℓ + √ħ + α→ + →b + ~a + ∧n = □ •c .

Evolution	Picture
of:	Heisenberg	Interaction	Schrödinger
Ket state	constant	$|\psi _{I}(t)\rangle =e^{iH_{0,S}~t/\hbar }|\psi _{S}(t)\rangle$	$|\psi _{S}(t)\rangle =e^{-iH_{S}~t/\hbar }|\psi _{S}(0)\rangle$
Observable	$A_{H}(t)=e^{iH_{S}~t/\hbar }A_{S}e^{-iH_{S}~t/\hbar }$	$A_{I}(t)=e^{iH_{0,S}~t/\hbar }A_{S}e^{-iH_{0,S}~t/\hbar }$	constant
Density matrix	constant	$\rho _{I}(t)=e^{iH_{0,S}~t/\hbar }\rho _{S}(t)e^{-iH_{0,S}~t/\hbar }$	$\rho _{S}(t)=e^{-iH_{S}~t/\hbar }\rho _{S}(0)e^{iH_{S}~t/\hbar }$

𝕬 𝕭 𝕮 𝕯 𝕰 𝕱 𝕲 𝕳 𝕴 𝕵 𝕶 𝕷 𝕸 𝕹 𝕺 𝕻 𝕼 𝕽 𝕾 𝕿 𝖀 𝖁 𝖂 𝖃 𝖄 𝖅
𝖆 𝖇 𝖈 𝖉 𝖊 𝖋 𝖌 𝖍 𝖎 𝖏 𝖐 𝖑 𝖒 𝖓 𝖔 𝖕 𝖖 𝖗 𝖘 𝖙 𝖚 𝖛 𝖜 𝖝 𝖞 𝖟

$(x+y+z)^{3}-(x^{3}+y^{3}+z^{3})=3(x+y)(x+z)(y+z)$

Rotations

This is also directly evident from the symplectic rotation structure of the kernel. The quadratic form in the exponent of K, up to a factor of −1/2, involves the simplest (unimodular, symmetric) symplectic matrix in Sp(2,ℝ). That is,

$(x,y)\cdot {\mathbf {M} }~{\begin{pmatrix}{x}\\{y}\end{pmatrix}}~,$

${\mathbf {M} }\equiv {\text{cosech}}(2t){\begin{pmatrix}\cosh(2t)&-1\\-1&\cosh(2t)\end{pmatrix}}~,$

so it preserves the symplectic metric,

${\mathbf {M} }~{\begin{pmatrix}0&1\\-1&0\end{pmatrix}}~{\mathbf {M} }={\begin{pmatrix}0&1\\-1&0\end{pmatrix}}~.$ (137)

NB Defining exp(θ)= tanh(t) , would reduce M to the more familiar form of a symplectic rotation, (137),

${\text{cosech}}(2t){\begin{pmatrix}\cosh(2t)&-1\\-1&\cosh(2t)\end{pmatrix}}={\begin{pmatrix}\cosh(\theta )&\sinh(\theta )\\\sinh(\theta )&\cosh(\theta )\end{pmatrix}}~.$

A less symmetric form, but handy for applications, is the following,

$\Phi [f]={\frac {2}{(2\pi \hbar )^{3/2}}}\iint \!\!\!\iint \!\!dqdpd{\tilde {x}}d{\tilde {p}}~e^{{\frac {i}{\hbar }}({\tilde {x}}{\tilde {p}}-2({\tilde {p}}-p)({\tilde {x}}-q))}~f(q,p)~|{\tilde {x}}\rangle \langle {\tilde {p}}|.$

${\frac {1}{A+B}}={\frac {1}{A}}-{\frac {1}{A}}B{\frac {1}{A}}+{\frac {1}{A}}B{\frac {1}{A}}B{\frac {1}{A}}+...={\frac {1}{A}}{\frac {1}{1+B{\frac {1}{A}}}},$ provable by right multiplication.

The generic SU(3) group element — generated Wikipedia talk:WikiProject Physics

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