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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "PHYSICS RESOURCE PACKETS P
ROJECT                               File: LEVELSQR.MS" }}{PARA 0 "" 
0 "" {TEXT -1 109 "Rose-Hulman Institute of Technology                \+
                       Authors: Perry Peters & Dr. Moloney" }}{PARA 0 
"" 0 "" {TEXT -1 97 "http://www.rose-hulman.edu/~moloney              \+
                     Software: Maple V Release 4" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Search engine for energy \+
levels in a potential well." }}{PARA 0 "" 0 "" {TEXT -1 34 "Solves for
 even and odd solutions." }}{PARA 0 "" 0 "" {TEXT -1 120 "Other potent
ial functions can be used in the program, but the program is not able \+
to normalize every potential function." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 85 "h-bar^2 /2m is coefficient of first t
erm.  ( x' = ax, where x in m, x' in nm. a=10^9)" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "d/dx = dx'/dx d/dx';  x'=
1nm" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
20 "restart;with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 
"hbar:=6.6260755E-34;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Me
:=9.1093987E-31;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "ec:=1.6
0217733E-19;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "ck:=evalf((
hbar*1E9/(2*Pi))^2 /(2*Me * ec));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "En
ter in maximum width(nm) and depth(eV) of well" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "depth:=4; width:=2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 38 "Enter in a function for the well shape" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 42 "V := -depth*(1-Heaviside(abs(x)-width/2));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "g1:=plot(V,x=-width/2-1..wi
dth/2+1,color=blue,axes=framed,title=`V vs x`,labels=[nm,eV]): #see if
  V looks ok" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "g1;" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "Sc
hrodinger's steady-state equation." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
41 "eq1:=-ck*diff(y(x),x,x)+V*y(x) = eV*y(x);" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Solve Schrodinger's eq
uation numerically as a function of eV." }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 68 "s_odd:= k -> dsolve(\{subs(eV=k,eq1),D(y)(0)=1,y(0)=0\},y(x),n
umeric);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "s_even:= k -> d
solve(\{subs(eV=k,eq1),D(y)(0)=0,y(0)=1\},y(x),numeric);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 150 "Look at what the equation does outside t
he barrier.  The goal is to find the energy levels at which the wave f
unction decays to 0 outside the barrier." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "test_point:=width/2+4;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 44 "r_odd:= p -> (rhs(s_odd(p)(test_point)[2]));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "r_even:= p -> (rhs(s_even(p)
(test_point)[2]));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 189 "Draw vertical lines where the function outside t
he barrier changes signs, i.e. goes from negative to positive, or posi
tive to negative.  This tells us what the allowable values for eV are.
 " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "r_oddl:= p -> 3*sign(rhs(s_odd
(p)(test_point)[2]));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "r_
evenl:= p -> 3*sign(rhs(s_even(p)(test_point)[2]));" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 31 "Dummy function for plot command" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 8 "f:=p->p;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
59 "Plot an energy band diagram of the allowable energy levels." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "plot([r_oddl,f,-depth..0],view=[0.
.2,-depth..0],color=blue,labels=[` `,eV],xtickmarks=0,title=`odd energ
y levels`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "plot([r_eve
nl,f,-depth..0],view=[0..2,-depth..0],color=blue,labels=[` `,eV],xtick
marks=0,title=`even energy levels`);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 157 "Enter in approximations of where the energy levels fall on the
 graphs produced above.  You need only enter the levels of the wave fu
nctions you wish to plot." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "levels
:=[even,-3.9],[odd,-3.7],[even,-3.3],[odd,-2.8],[even,-2.1],[odd,-1.3]
,[even,-0.4];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "This calulates t
he number of levels you have chosen to graph." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "numlevels:=nops([levels]);" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 146 "Definition of the derivative, used later to do Newton'
s method.  Newton's method is used to a more exact value of eV for the
 approximations above." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "dr:=x->(r
(x+0.000001)-r(x))/0.000001;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "P
lot the wave functions specified in \"levels\"" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "for j from 1 to num
levels do" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
36 "Pick an energy level out of \"levels\"" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "level:=levels[j][1];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 18 "mid:=levels[j][2];" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 45 "level is used to determine which DE to solve." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "if level='odd'" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 24 "then r:=r_odd; s:=s_odd;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "else r:=r_even; s:=s_even;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 3 "fi;" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 202 "Use Newton's method to find an energy level that decay
s to 0 outside the barrier.  \"mid\" is used as an initial guess.  The
 loop exits after 10 iterations, or after a certain precision has been
 obtained." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "mid0:=0: counter:=0:
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "while (abs(mid-mid0)>0.00000000
2) and (counter < 10) do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "mid0:=m
id:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "mid:=mid0-r(mid0)/dr(mid0):
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "print(mid);" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 21 "counter:=counter + 1:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 3 "od:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 72 "Plot the well, the wave function, and its energy level \+
on the same graph" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "energy:=conver
t(mid,string);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "g2:=odeplot(s(mid
),[x,y(x)],-width/2-1..width/2+1,axes=framed,title=`Ev = `.energy,labe
ls=[nm,eV]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "print(display([g2,g
1]));" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "
Print the energy levels to the screen." }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 16 "print(s(mid.j));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "mid.j:=
mid;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "s.j:=s(mid.j);" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 16 "print(Ev=mid.j);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Plot |psi|
^2 and find the area under |psi|^2" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
28 "for j from 1 to numlevels do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 
"s_abs.j:=(x) -> rhs(s.j(x)[2])^2;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
30 "energy:=convert(mid.j,string);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
79 "plot(s_abs.j,-width/2-1/2..width/2+1/2,title=`original |psi|^2    \+
Ev=`.energy);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "area.j:=evalf(Int(
s_abs.j,-width/2-1/2..width/2+1/2));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 3 "od;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Plot a the normalized
 wave functions" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "for j from 1 to \+
numlevels do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "s_norm.j:= (x) -> r
hs(s.j(x)[2])/sqrt(area.j);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "ener
gy:=convert(mid.j,string):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "g2:=
plot(s_norm.j,evalf(-width/2-1..width/2+1),labels=[nm,eV],title=`norma
lized psi   Ev=`.energy,axes=framed):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 24 "print(display([g2,g1]));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od
:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Animate several wave functio
ns in the well." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 40 "Value of hbar in this coordinate system." }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 15 "hbar1:=hbar/ec;" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 126 "Make the wave functions time dependent and find |Psi|^2 \+
of the sum of the wave functions.  Add or delete equations as desired.
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "y1:= (x,t) -> rhs(s1(x)[2])*exp
(-I*mid1*t/hbar1)/sqrt(area1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 61 "y2:= (x,t) -> rhs(s2(x)[2])*exp(-I*mid2*t/hbar1)/sqrt(area2);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "y3:= (x,t) -> rhs(s3(x)
[2])*exp(-I*mid3*t/hbar1)/sqrt(area3);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 61 "y4:= (x,t) -> rhs(s4(x)[2])*exp(-I*mid4*t/hbar1)/sqrt
(area4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "y5:= (x,t) -> r
hs(s5(x)[2])*exp(-I*mid5*t/hbar1)/sqrt(area5);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 61 "y6:= (x,t) -> rhs(s6(x)[2])*exp(-I*mid6*t/hbar
1)/sqrt(area6);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "y7:= (x,
t) -> rhs(s7(x)[2])*exp(-I*mid7*t/hbar1)/sqrt(area7);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "Y:=(x,t)->abs(y1(x,t)+y2(x,t)+y3(x,
t)+y4(x,t)+y5(x,t)+y6(x,t)+y7(x,t))^2;" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 53 "View value of function at any point and time.  Y(x,t)" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "Y(1.5,0);" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 22 "V1:= X -> subs(x=X,V);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 31 "g1:=plot(V1,-2..2,color=black):" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 151 "g2:=animate(Y,-2..2,0..4E-14,axes=framed,l
abels=[nm,eV],title=`Wave function sloshing in potential well`,color=r
ed,frames=40,numpoints=100,color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "display([g2,g1]);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{MARK "40 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }