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Math
->Klein-Gordon equation<-
If the particle is charge-neutral and spinless, and relativistic effects cannot
be ignored, we may use the Klein-Gordon equation to describe the wave function. The Klein-Gordon equation for a free particle is written
- <amsmath>
\mathbf{\nabla}^2\psi-\frac{1}{c^2}\frac{\partial^2}{\partial t^2}\psi = \frac{m^2c^2}{\hbar^2}\psi </amsmath>
with the same solution as in the non-relativistic case:
- <amsmath>
\psi(\mathbf{r}, t) = e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)} </amsmath>
except with the constraint
- <amsmath>
-k^2+\frac{\omega^2}{c^2}=\frac{m^2c^2}{\hbar^2} </amsmath>
Just as with the non-relativistic particle, we have for energy and momentum:
- <amsmath>
\langle\mathbf{p}\rangle=\langle \psi |-i\hbar\mathbf{\nabla}|\psi\rangle = \hbar\mathbf{k} </amsmath>
- <amsmath>
\langle E\rangle=\langle \psi |i\hbar\frac{\partial}{\partial t}|\psi\rangle = \hbar\omega </amsmath>
Except that now when we solve for k and ω and substitute into the constraint equation, we recover the relationship between energy and momentum for relativistic massive particles:
- <amsmath>\left.\right.
\langle E \rangle^2=m^2c^4+\langle \mathbf{p} \rangle^2c^2 </amsmath>
For massless particles, we may set m=0 in the above equations. We then recover the relationship between energy and momentum for massless particles:
- <amsmath>\left.\right.
\langle E \rangle=\langle |\mathbf{p}| \rangle c </amsmath>
Plot
<plot> set samples 500 plot [-3:3][-10:10] 5*sin(3*x)+2*sin(23*x) </plot>
<amsmath>\frac{\sin\left(\sqrt{x^2+y^2}\right)}{\sqrt{x^2+y^2}}</amsmath>:<plot>
set border 4095 lt -1 lw 1.000
set view 130, 10, .5
set samples 50, 50
set isosamples 50, 50
unset surface
unset colorbox
set pm3d at s
set pm3d scansbackward flush begin noftriangles nohidden3d implicit corners2color mean
splot sin(sqrt(x**2+y**2))/sqrt(x**2+y**2)
</plot>
Classical Greek
<greek>
ABCDEFGHIJKLMNOPQRSTUVWXYZ
Ei)s ('Hfaiston; \begin{verse} \quad ('Hfaiston kluto'mhtin a)ei'deo Mou=sa li'geia,\\ o(`s met' )Aqhnai'hs glaukw'pidos a)glaa` e)'rga\\ a)nqrw'pous e)di'dacen e)pi` xqono's, oi(` to` pa'ros per\\ a)'ntrois naieta'askon e)n ou)'resin h)u'+te qh=res.\\ nu=n de` di) ('Hfaiston klutote'xnhn e)'rga dae'ntes\\ r(hi+di'ws ai)w=na telesfo'ron ei)s e)niauto`n\\ eu)'khloi dia'gousin e)ni` sfete'roisi do'moisin.\\ \quad )All' i('lhq' ('Hfaiste; di'dou d' a)reth'n te kai` o)'lbon. \end{verse} </greek>
Chemistry
1. Glycine
<chem> \tetrahedral{0==C;1==H;2==H$_3$N$^+$;3==H;4==COO$^-$} </chem>
2. Arginine
<chem> \tetrahedral{0==C;1==H;2==H$_3$N$^+$;3==\tetrahedral{0==(CH$_2$)$_3$;1==(yl);3==\tetrahedral{0==NH;1==(yl);3==\tetrahedral{0==C;1==(yl);3==NH$_2$;4D==NH}}};4==COO$^-$} </chem>
3. Glutamine
<chem> \tetrahedral{0==C;1==H;2==H$_3$N$^+$;3==\tetrahedral{0==(CH$_2$)$_2$;1==(yl);3==\tetrahedral{0==NH;1==(yl);3==\tetrahedral{0==C;1==(yl);3==NH$_2$;4D==O}}};4==COO$^-$} </chem>
<chem> \bzdrv{1==NH2;5==CH$_{3}$;4==OC$_{16}$H$_{33}$;% 2==\ryl(4==NH--SO$_{2}$){4==\bzdrh{1==(yl);2==OCH$_{2}$CH$_{2}$OCH$_{3}$;% 5==\ryl(2==NH--SO$_{2}$){4==\bzdrh{1==(yl);% 5==\ryl(2==SO$_{2}$--NH){4==\naphdrh{1==(yl);5==OH;% 8==\lyl(4==N=N){4==\bzdrh{4==(yl);1==NO$_{2}$;5==SO$_{2}$CH$_{3}$}}}}}}}}} </chem>
Simple Music
<music> {
f' a' c e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e
} </music>
The 'F A C E' notes:
<music>
\new Staff {
\clef Violin
\time 3/4
\relative c'
\context Staff <<
\new Voice
{ \voiceOne
f1 a1 c1 e1
} >> } </music>
<music> \new Staff { \clef Violin \time 3/4 \relative c \context Staff << \new Voice { \voiceOne
g1 e1
} >> } </music>
<music>
{ c'4^"This has got to be the most rad thing EVAR" d'2. e64 f g a b c d e f g \bar "|." }
</music>
Music
<music> \new Pianostaff << \new Staff { \time 3/4 \clef violin \context Staff << \new Voice { \voiceOne
c1 d1 e1.
} >> } \new Staff { \time 3/4 \clef bass \context Staff << \new Voice { \voiceOne
a1 a1 a1.
} >> } >> </music>
<music> \new Pianostaff << \new Staff { \time 2/2 \clef violin \key cis \minor \relative c \context Staff << \new Voice { \voiceOne
r4 cis8 dis e4 fis gis8 fis gis a gis fis e gis fis e fis gis fis e dis fis e dis e fis e d cis e d cis d e d cis b d cis b cis d cis b a cis b a b cis b a gis b a2 r cis2.
} \new Voice { \voiceTwo
e,8 gis a b cis dis bis cis dis4 r r2 r1 r1 r4 fis, b b b a8 gis a2 gis1~ gis8 gis fis eis fis2 gis2.
} \new Voice { \voiceThree \stemDown
s1 s s s s2. fis4 eis2 fis
} >> } \new Staff { \clef bass \time 2/2 \key cis \minor \relative c' \context Staff << \new Voice { \voiceOne
s1 r4 gis cis cis cis bis8 ais bis2 cis1 b2. s4 s1 b2 cis~ cis~ cis8 cis b a gis2.
} \new Voice { \voiceTwo
\stemUp cis,1 bis2 e dis1 \stemDown cis4 e a a a gis8 fis gis2~ \stemUp gis fis gis1 a2 fis~ fis8 fis e dis e4
} \new Voice { \voiceThree
\stemDown cis4 b a2 gis4 r4 g2\rest e1\rest e1\rest e1\rest r4 cis' fis fis fis eis8 dis eis2 fis r r
} >> } >> </music>
Chess
<chess> \largeboard \notationOff \styleC \newgame \mainline{1.e4 c6 2.d4 d5 3.Nc3 dxe4 4.Nxe4 Nd7 5.Ng5 Ngf6 6.Bd3 e6 7.N1f3 Bd6 8.Qe2 h6 9.Ne4} \showboard </chess>
