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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "HEISENBERG UNCERTAINTY REL
ATION" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 367 
"According to Heisenberg the minimum uncertainties in a particle's pos
ition and its momentum are related.   The smallest value that is allow
ed the product of these two uncertainties is (roughly) Planck's consta
nt h divided by 2pi (called \"h bar\").   This means that the smaller \+
the uncertainty in its momentum the larger the uncertainty in its posi
tion and vice versa." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 1117 "To represent a moving particle of uncertain momentum,
 we can use De Broglie's idea that there is a wave of wavelength h/p a
ssociated with a particle whose momentum is p.  A particle of uncertai
n momentum can be represented by a superposition of  waves of various \+
wavelengths and wave numbers k, over a wavelength range which covers t
he uncertainty range of p, making up what is called a \"wave packet\".
  The purpose of this exercise is to illustrate how this superposition
 results in a wave function with a built-in position uncertainty, and \+
to show that decreasing the range of momentum uncertainty broadens the
 position uncertainty.   At the same time we would like to have a look
 at the \"phase\" and \"group\" velocities associated with the wave pa
cket.  [The latter is the speed at which the envelope moves, the forme
r is the speed at which the wave inside the envelope moves; the group \+
velocity is the most probable speed of the particle.]   The group velo
city is twice the phase velocity for nonrelativistic particles, so one
 should be able to see the envelope overtaking individual peaks as the
 packet moves along." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 308 "In order to build up a \"Gaussian\" wave packet, we st
art with a wave whose wave number ko represents the most probable part
icle momentum and add, over a range of wave numbers,  smaller amplitud
e waves whose numbers represent momenta of lesser probability.  The re
sults in a summation of waves whose forms are:" }}{PARA 0 "" 0 "" 
{TEXT -1 84 "                                     y:=exp(-c2*(k[i]-ko)
^2)*sin(k[i]*x-c1*k[i]^2*t)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 279 "Note that the amplitude of a wave of wave numb
er k[i] is smaller the farther it is fom ko, the wave number associate
d with the most probable particle momentum.  The constant c1 is (hbar)
/2m and constant c2 is 1/[2*delta_k)], where (delta_k) is the spacing \+
between the k values). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 81 "The following assumes a wave packet for an electron \+
with a speed of 5 x 10^5 m/s." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "res
tart; with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "v:=5*
10^5;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "m:= 9.1 * 10^(-31)
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "hbar:=1.055*10^(-34);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "ko := m*v/hbar;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "delta_k := ko/50;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "c1:=hbar/(2*m);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "c2:=1/(2*delta_k)^2;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "N:=40;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "kRange:=4*delta_k;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "for i from 1  to N do " }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 37 "k[i]:=ko-kRange/2 + (i-1)*(kRange/N);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "s:=0;
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "for i from 1  to N do \+
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "s:=s+exp(-c2*(k[i]-ko)^2)*sin(k
[i]*x-c1*k[i]^2*t);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "y:=subs(t=0,s):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "plot(y,x=0..4*10^(-8),title=`Range \+
= 4 ko / 50`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "animate(s
,x=0..3*10^(-8),t=0..4*10^(-14),frames=20,numpoints=200); " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "kRange2:=2*delta_k;" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for
 i from 1 to N do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "k[i]:=ko-kRang
e2/2+(i-1)*(kRange2/N);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "s2:=0;" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 20 "for i from 1 to N do" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 52 "s2:=s2+exp(-c2*(k[i]-ko)^2)*sin(k[i]*x-c1*k[i]^2*t);
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 3 "c1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "y:
=subs(t=0,s2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "s2:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "plot(y,x=0..4*10^(-8),title=
`Range = 2 ko / 50`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "an
imate(s2,x=0..3*10^(-8),t=0..4*10^(-14),frames=20,numpoints=200);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "25 0 0" 0 }
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