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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "   The following worksheet
 solves Problem 1.5 in Griffiths Introduction to Quantum Mechanics" }}
{PARA 0 "" 0 "" {TEXT -1 89 "by Monte Carlo methods.   We simulate by \+
random number generation the dropping of a stick" }}{PARA 0 "" 0 "" 
{TEXT -1 91 "of unit length onto a floor of parallel lines with unit s
pacing.  The question asks what is" }}{PARA 0 "" 0 "" {TEXT -1 95 "the
 probability that the stick crosses a line.  To illustrate the interes
ting final answer, the" }}{PARA 0 "" 0 "" {TEXT -1 96 "routine stick_t
oss(n)  returns 2/probability of crossing a line.  For a large enough \+
statistical" }}{PARA 0 "" 0 "" {TEXT -1 92 "sample (n>1000 or so), thi
s number starts to resemble a very interesting one in mathematics." }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "      Include the statistics pack
age:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(stats);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7*%&anovaG%)describeG%$fitG%+importdat
aG%'randomG%*statevalfG%*statplotsG%*transformG" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 61 "      Start the random number generator with a 'rand
om' seed:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "randomize;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%*randomizeG" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 89 "     Construct the routine stick_toss(n).  Here x is th
e distance of one end of the stick" }}{PARA 0 "" 0 "" {TEXT -1 83 "fro
m an arbitrary line.  It is chosen as a random number from 0 to 1 acco
rding to a" }}{PARA 0 "" 0 "" {TEXT -1 86 "uniform distribution.   The
 angle theta specifies the orientation of the stick.  It is" }}{PARA 
0 "" 0 "" {TEXT -1 79 "chosen at random from a uniform distribution be
tween 0 and Pi.  The stick is of" }}{PARA 0 "" 0 "" {TEXT -1 91 "unit \+
length, so the top of the stick is sin(theta) above the bottom (at x) \+
in the direction" }}{PARA 0 "" 0 "" {TEXT -1 100 "perpedicular to the \+
parallel lines on the floor.  The stick intersects a line if x + sin(t
heta) > 1." }}{PARA 0 "" 0 "" {TEXT -1 86 "The routine simply counts t
he number of times this occurs in the variable ave and then" }}{PARA 
0 "" 0 "" {TEXT -1 91 "divides by the total number of samples to obtai
n the probability P.  However, it prints out" }}{PARA 0 "" 0 "" {TEXT 
-1 60 "2/P for reasons that you will discover by playing with this." }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "stick_toss := proc(n)" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "local i, ave, x, theta;" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 9 "ave := 0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
21 "for i from 1 to n do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "x := r
andom[uniform[0,1]]();" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "theta := \+
random[uniform[0,Pi]]();" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "if sin(
theta) + x > 1 then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "ave := ave +
 1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "fi;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 3 "od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "evalf(2*n/av
e);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 95 "    Simply giving the stick_toss command at this point
 for a given sample size will return 2/P," }}{PARA 0 "" 0 "" {TEXT -1 
73 "where P is the probability that the stick hits one of the parallel
 lines." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "stick_toss(100);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+D&)oyK!\"*" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "9 0 0" 0 }{VIEWOPTS 1 1 0 1 1 
1803 }