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{SECT 0 {PARA 0 "" 0 "" {TEXT 216 0 "" }}{PARA 210 "" 0 "" {TEXT 207 
47 " Semi-Classical Mechanics: The Simple Bohr Atom" }}{PARA 19 "" 0 "
" {TEXT 217 46 "by Tony Scott, Darren Redfern, Richard Pavelle" }}
{PARA 211 "" 0 "" {TEXT 211 9 "Abstract:" }{TEXT 207 120 " Starting wi
th the classical relations for momentum, force, and energy, the quanti
zation rule is prescribed to momentum." }}{PARA 212 "" 0 "" {TEXT 212 
28 "Application Areas/Subjects: " }{TEXT 213 0 "" }{TEXT 207 55 " Scie
nce, Education, Physics, Applied Math, Chemistry, " }}{PARA 213 "" 0 "
" {TEXT 214 9 "Keywords:" }{TEXT 207 43 "  education, Bohr, atom, phys
ics, chemistry" }}{PARA 214 "" 0 "" {TEXT 215 9 "See Also:" }{TEXT 
207 45 " MapleTech vol 1 no 1, 1994 (& issue 7 1992)." }}{PARA 0 "" 0 
"" {TEXT 216 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:"
 }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 218 14 " The Bohr Atom" }}{PARA 0 ""
 0 "" {TEXT 216 119 "Starting with the classical relations for momentu
m, force, and energy, the quantization rule is prescribed to momentum.
" }}{PARA 0 "" 0 "" {TEXT 216 50 "                 2*Pi*hbar  =  Planc
k's constant  " }}{PARA 0 "" 0 "" {TEXT 216 44 "                 n =  \+
orbital quantum number" }}{PARA 0 "" 0 "" {TEXT 216 55 "              \+
    z  =  atomic charge number           " }}{PARA 0 "" 0 "" {TEXT 
216 39 "                  e  =  electron charge" }}{PARA 0 "" 0 "" 
{TEXT 216 55 "                  r   =  orbit radius                  "
 }}{PARA 0 "" 0 "" {TEXT 216 38 "                  v  =  particle spee
d" }}{PARA 0 "" 0 "" {TEXT 216 50 "                  m =  electron mas
s              " }}{PARA 0 "" 0 "" {TEXT 216 37 "                  En \+
 =  orbit energy" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Momentum
 := m*v*r = n*hbar;\n" }}}{PARA 0 "" 0 "" {TEXT 216 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Force := m*v^2/r = 1/(4*Pi*eps0)*Z*
e^2/r^2;\n" }}}{PARA 0 "" 0 "" {TEXT 216 0 "" }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 50 "Energy := En = (1/2)*m*v^2-1/(4*Pi*eps0)*Z*e^2/r;
\n" }}}{PARA 0 "" 0 "" {TEXT 216 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 51 "ba_sol := solve(\{Momentum,Force,Energy\},\{r,v,En\})
;\n" }}}{PARA 0 "" 0 "" {TEXT 216 80 "This yields the quantized energy
 En proportional to 1/n^2 for the Hydrogen atom." }}{EXCHG {PARA 0 "> 
" 0 "" {MPLTEXT 1 0 26 "with(ScientificConstants);" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 24 "Ene := eval(En, ba_sol);" }}}{EXCHG {PARA 0
 "> " 0 "" {MPLTEXT 1 0 111 "En_ba := evalf(eval(Ene/e, \{e=Constant(e
), m=Constant(m[e]), eps0=Constant(epsilon[0]), hbar=Constant(hbar)\})
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "evalf(eval(Ene/e, \{
e=Constant(e,units), m=Constant(m[e],units), eps0=Constant(epsilon[0],
units), hbar=Constant(hbar,units)\})):" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 12 "simplify(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "r_
ba := evalf(eval(eval(r, ba_sol), \{e=Constant(e), m=Constant(m[e]), e
ps0=Constant(epsilon[0]), hbar=Constant(hbar)\}));" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 120 "v_ba := evalf(eval(eval(v, ba_sol), \{e=Co
nstant(e), m=Constant(m[e]), eps0=Constant(epsilon[0]), hbar=Constant(
hbar)\}));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "evalf(eval(v_
ba/c, c=Constant(c)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "G
etConstant(alpha);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "evalf
(1/Constant(alpha));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "lam
da := (ni,nf) -> evalf(1/(Constant(R[infinity])*(1/nf^2-1/ni^2))*10^9)
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