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at the classical solution of the Morse oscillator. The periodic soluti
ons for bound states are obtained numerically and decomposed into a Fo
urier series. One can see how for low energies the solutions are appro
ximated well by a harmonic oscillator solution which has negligible Fo
urier components. For energies below the dissociation limit, but well \+
above zero the anharmonicities are observed in the form of non-negligi
ble Fourier coefficients." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 356 "In this regime of intermediate energies the class
ical model predicts that charged particles undergoing this type of mot
ion can radiate directly at multiples of the fundamental frequency whi
ch is given by the inverse of the oscillation period. This corresponds
 in quantum mechanics to a direct spontaneous decay to an energy level
 below the neighboring one." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
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S\"F,7$$\"/e*[oc2u\"Fihl$!/ynwgl!4#F,7$$\"/n\"HYt7v\"Fihl$!/(R\"R&[Gr#
F,7$$\"/%ek6,1x\"Fihl$!/R6%*)*4,OF,7$$\"/++q(G**y\"Fihl$!/V'GqI3.%F,7$
$\"/3-)omaz\"Fihl$!/WUtnmaSF,7$$\"/</1Y+,=Fihl$!//'p<qD.%F,7$$\"/E1CDa
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7`]z/NF,7$$\"/n;9@BM=Fihl$!/Cdq/!4,$F,7$$\"/%eRZQT%=Fihl$!/RdvqnpCF,7$
$\"/+vL[/a=Fihl$!//zi!4!e=F,7$$\"/;a$>^R'=Fihl$!/b<!\\6z>\"F,7$$\"/LL`
v&Q(=Fihl$!/n\"**3rU4&F07$$\"/C1*z>W)=Fihl$\"/Z@)HZVO#F07$$\"/;zW?)\\*
=Fihl$\"/3&[bo]x*F07$$\"/3_!HWb!>Fihl$\"/.c=)****p\"F,7$$\"/+DOl5;>Fih
l$\"/0\"pa$G$R#F,7$$\"/]7`f@E>Fihl$\"/x5_D3AIF,7$$\"/++q`KO>Fihl$\"/ox
rxe6OF,7$$\"/](oyMk%>Fihl$\"/HP%p2!eTF,7$$\"/+v.Uac>Fihl$\"/m!\\85'eYF
,7$$\"/](=5s#y>Fihl$\"/8Ks7PrbF,7$$\"#?F)$\"/pR>Ki`iF,-%'COLOURG6&%$RG
BG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"t6\"Q!F\\hn-%*THICKNESSG6#\"\"#-%%V
IEWG6$;F(F]gn%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "Let u
s calculate the Fourier coefficients numerically. For the given energy
 the period is known to be:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
32 "T:=E->evalf(2*Pi/sqrt(2*(1-E)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%\"TGj+6#%\"EG6\"6$%)operatorG%&arrowGF(-%&evalfG6#,$*(\"\"#\"\"\"%
#PiGF2-%%sqrtG6#,&F1F2*&F1F29$F2!\"\"F:F2F(F(F(6#\"\"!" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "T0:=T(E0);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#T0G$\"/wk?$*>I^!#8" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 73 "a_n:=n->2/T0*evalf[10](Int(X(t)*cos(2*Pi*n*t/T0),t=0.
.T0,method=_Gquad));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$a_nGj+6#%\"
nG6\"6$%)operatorG%&arrowGF(,$*(\"\"#\"\"\"%#T0G!\"\"-&%&evalfG6#\"#56
#-%$IntG6%*&-%\"XG6#%\"tGF/-%$cosG6#,$*,F.F/%#PiGF/9$F/F?F/F0F1F/F//F?
;\"\"!F0/%'methodG%'_GquadGF/F/F(F(F(6#FI" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 7 "a_n(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"/]MEw
\"pO%!#9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "a_n(1);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"/%*=_&*)*e`!#9" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 25 "A:=[seq(a_n(n),n=0..10)];" }}{PARA 12 "" 1 "
" {XPPMATH 20 "6#>%\"AG7-$\"/]MEw\"pO%!#9$\"/%*=_&*)*e`F($!/CAoFozr!#:
$\"/wln5a#G\"F-$!/5lRFRxD!#;$\"/!y3ffX_&!#<$!/pNhkgL7F5$\"/5E*>'=PG!#=
$!/#>lB!yPm!#>$\"/H\"eZV(f:F=$!/wC\\\\LjP!#?" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 74 "The Fourier-cosine coefficients converge rapidly to zer
o for increasing n." }}{PARA 0 "" 0 "" {TEXT -1 90 "Now show that the \+
Fourier-sine coefficients vanish for our choice of periodicity interva
l:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 73 "b_n:=n->2/T0*evalf[10](Int(X(t)*sin(2*Pi*n*t/
T0),t=0..T0,method=_Gquad));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$b_n
Gj+6#%\"nG6\"6$%)operatorG%&arrowGF(,$*(\"\"#\"\"\"%#T0G!\"\"-&%&evalf
G6#\"#56#-%$IntG6%*&-%\"XG6#%\"tGF/-%$sinG6#,$*,F.F/%#PiGF/9$F/F?F/F0F
1F/F//F?;\"\"!F0/%'methodG%'_GquadGF/F/F(F(F(6#FI" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 7 "b_n(1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!
/y%=Z#H!f$!#?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "B:=[seq(b_
n(n),n=1..10)];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"BG7,$!/y%=Z#H!f
$!#?$!/1\"3xe'[ZF($!/?gy(Rq;#F($!/%yJ,L(z#*!#@$!/ydz)=<<\"F($!/wL)G^RK
\"F($!/as.oB1*)F/$!/qc$oD'>aF/$!/irkGfGvF/$!/OZ*pYt)))F/" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 70 "The results tell us something about the p
recision of the calculations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 44 "Thus, we assemble the Fourier-cosine seri
es:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "xF3:=t->0.5*A[1]+add
(A[n+1]*cos(2*Pi*n*t/T0),n=1..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
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{MPLTEXT 1 0 59 "plot([X(t),xF3(t)],t=0..2*T0,color=[red,blue],thickne
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0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve
 7" "Curve 8" "Curve 9" "Curve 10" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 43 "It looks like the harmonics are too strong?" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "why did I take the square
s?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "wha
t is the radiation proportional to?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT 256 11 "Exercise 1:" }}{PARA 0 "" 0 "" {TEXT 
-1 311 "Investigate the radiative spectrum for orbits of different ene
rgy. Observe how for low-energy orbits in the linear (nearly harmonic)
 regime the so-called harmonic motion involves oscillations at the fun
damental frequency only. Make sure to understand the frequency spectru
m also an energy spectrum according to " }{TEXT 19 5 "E=h*f" }{TEXT 
-1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{MARK "0 4 0" 355 }{VIEWOPTS 1 1 0 1 1 1803 1 1 
1 1 }{PAGENUMBERS 0 1 2 33 1 1 }