{VERSION 4 0 "Windows Vista" "4.0" } {USTYLETAB {PSTYLE "Ordered List 1" -1 200 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 } {PSTYLE "Ordered List 2" -1 201 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 36 2 0 2 2 -1 1 }{PSTYLE "Ordered List 3" -1 202 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 72 2 0 2 2 -1 1 }{PSTYLE "Ordered List 4" -1 203 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 108 2 0 2 2 -1 1 }{PSTYLE "Ordered List 5" -1 204 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 144 2 0 2 2 -1 1 }{PSTYLE "Author" -1 19 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 8 8 2 0 2 0 2 2 -1 1 }{PSTYLE "Annotatio n Title" -1 205 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }3 1 0 0 12 12 2 0 2 0 2 2 -1 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Fixed Width" -1 17 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Line Prin ted Output" -1 6 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Normal256" -1 206 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Cou rier" 1 10 0 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Diagnostic" -1 9 1 {CSTYLE "" -1 -1 "Courier" 1 10 64 128 64 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Help" -1 10 1 {CSTYLE "" -1 -1 "Courier" 1 9 0 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Dash \+ Item" -1 16 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 0 0 1 }3 1 0 0 12 12 2 0 2 0 2 2 -1 1 }{PSTYLE "Error" -1 8 1 {CSTYLE "" -1 -1 "Courier" 1 10 255 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Head ing 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 8 4 2 0 2 0 2 2 -1 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 }{PSTYLE "Heading 4" -1 20 1 {CSTYLE "" -1 -1 "Times" 1 10 0 0 0 1 1 1 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "He ading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 8 2 2 0 2 0 2 2 -1 1 }{PSTYLE "Left Justified Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "List Item" -1 14 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 }{CSTYLE "Annotati on Text" -1 200 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Hel p Maple Name" -1 35 "Times" 1 12 104 64 92 1 2 1 2 2 2 2 0 0 0 1 } {CSTYLE "2D Math Bold" -1 5 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 } {CSTYLE "Help Menus" -1 36 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 } {CSTYLE "2D Math Italic" -1 3 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Normal" -1 30 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Plot Text" -1 28 "Times" 1 8 0 0 0 1 2 2 2 2 2 2 0 0 0 1 } {CSTYLE "Help Nonterminal" -1 24 "Courier" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help Heading" -1 26 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help Italic" -1 42 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Variable" -1 25 "Courier" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Italic Bold" -1 40 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Default" -1 38 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Underlined Italic" -1 43 "Times" 1 12 0 0 0 1 1 2 1 2 2 2 0 0 0 1 }{CSTYLE "Maple Input" -1 0 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "2D Output" -1 20 "Times" 1 12 0 0 255 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Bold Small" -1 10 "Times" 1 1 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Small" -1 7 "Times" 1 1 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Page Number" -1 33 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Output Labels" -1 29 "Times" 1 8 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Plot Title" -1 27 "Times" 1 10 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help Emphasized" -1 22 "Time s" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Symbol 2" -1 16 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Maple Comment" -1 21 "Courier" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Maple Input Pl aceholder" -1 201 "Courier" 1 12 200 0 200 1 2 1 2 2 1 2 0 0 0 1 } {CSTYLE "Code" -1 202 "Courier" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 } {CSTYLE "2D Inert Output" -1 203 "Times" 1 12 144 144 144 1 2 2 2 2 1 2 0 0 0 1 }{CSTYLE "2D Math Italic Small" -1 204 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Comment" -1 18 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Underlined Bold" -1 41 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 0 0 0 1 }{CSTYLE "Copyright" -1 34 "Times" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "Times" 1 12 0 128 128 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "Help Underlined" -1 44 "Times" 1 12 0 0 0 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "Prompt" -1 1 "Courier" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Input" -1 19 "Times" 1 12 255 0 0 1 2 2 2 2 1 2 0 0 0 1 }{CSTYLE "Header and Footer" -1 205 "Times" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Text" -1 206 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Notes" -1 37 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help Bold" -1 39 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Equation Label" -1 207 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "LaTeX" -1 32 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Popup" -1 31 "Times" 1 12 0 128 128 1 1 2 1 2 2 2 0 0 0 1 }{CSTYLE "Dictionary Hyperlink" -1 45 "Times" 1 12 147 0 15 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "Help Fixed" -1 23 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{PSTYLE "" -1 207 1 {CSTYLE "" -1 -1 "Tim es" 1 16 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 } {PSTYLE "" -1 208 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 1 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "" -1 208 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{PSTYLE "" -1 209 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 1 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }} {SECT 0 {EXCHG {PARA 207 "" 0 "" {TEXT 206 57 "Feynman's sum over pat hs method for Fraunhofer scattering" }}{PARA 0 "" 0 "" {TEXT 209 0 "" }}{PARA 0 "" 0 "" {TEXT 209 411 "The sum over paths method is used to \+ produce the single-slit diffraction pattern. The idea is that photons \+ can take all kinds of possible paths which connect the source with a p oint on the screen. We can illustrate this idea by considering straigh t-line paths between the source and a point y(i) somewhere in the aper ture. We subdivide the aperture in equally spaced intervals according \+ to the following scheme:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 " restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "N:=40; # the nu mber of paths" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "a:=10; # t he slit width in wavelengths" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "y_i:=i->evalf(a/2-(i-1/2)*a/N); # a subdivision of the aperture" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "y_i(1),y_i(N/2),y_i(N/2+ 1),y_i(N);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 209 366 "We have chosen N \+ paths through the aperture at equidistant heights. We choose a distanc e between source and aperture (called X), we use this same distance be tween aperture and screen, and use the variable Y to denote how far aw ay from the center position we are on the screen. All these distances \+ are measured in wavelength (this is of the order of microns or less)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "X:=500; # the source-scr een and screen-aperture distance" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "Y:='Y';" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 209 90 "The distance \+ travelled by a photon along one of these paths is given by two contrib utions:" }}{PARA 0 "" 0 "" {TEXT 209 37 "source to aperture: sqrt(X^2 \+ + y_i^2)" }}{PARA 0 "" 0 "" {TEXT 209 41 "aperture to screen: sqrt(X^2 + (y_i-Y)^2)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "d_i:=(i,Y) ->sqrt(X^2+y_i(i)^2)+sqrt(X^2+(y_i(i)-Y)^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "t_i:=i->d_i(i)/c;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "f:=c/lambda;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 209 540 "Feynman's idea to explain the phase: think of the photon travelli ng along the straight-line path with a clock whose handle shows how mu ch time has passed. The photon oscillates with a frequency which depen ds on the wavelength (color) of the light. All we need to know is the \+ orientation of the clock handle at the end of the path, when it hits t he screen. Technically this determines the phase (in phasor language w e determine whether the wavelet that travelled along this path arrives with a crest or a through, i.e., with what amplitude)." }}{PARA 0 "" 0 "" {TEXT 209 0 "" }}{PARA 0 "" 0 "" {TEXT 209 49 "To determine the p hase: we want exp(I*2*Pi*f*t_i)" }}{PARA 0 "" 0 "" {TEXT 209 139 "We r ealize that c drops out, and that we need a distance divided by wavele ngth, i.e., the distance measured in multiples of the wavelength " } {XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\1673\", italic = \+ \"false\", mathvariant = \"normal\"));" "-I%mrowG6#/I+modulenameG6\"I, TypesettingGI(_syslibGF'6#-I#miGF$6%Q)λF'/%'italicGQ&falseF'/%, mathvariantGQ'normalF'" }{TEXT 209 1 "." }}{PARA 0 "" 0 "" {TEXT 209 0 "" }}{PARA 0 "" 0 "" {TEXT 209 322 "Then we add up all the phases, i .e., we add the clock handles in the sense of vector addition. It is a n addition of complex numbers of equal modulus (magnitude). This gives us the probability amplitude with which the wave arrives. The squared modulus (magnitude) of this number gives us the light intensity at th e point Y." }}{PARA 0 "" 0 "" {TEXT 209 0 "" }}{PARA 0 "" 0 "" {TEXT 209 85 "Let us illustrate how this vector addition works using little \+ arrows (clock handles):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 " with(plots):" }{MPLTEXT 1 0 17 " with(plottools):" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 6 "Y:=70;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 " Lis:=[]: v0:=[0,0]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 197 "for i from \+ 1 to N do: w1:=evalf(exp(2*Pi*I*d_i(i,Y))); v1:=[Re(w1),Im(w1)]; Lis:= [op(Lis),[arrow(v0,v0+v1,.2,.4,.1,color=red)]]: v0:=v0+v1; od: Lis:=[o p(Lis),[arrow([0,0],v0,.2,.4,.1,color=blue)]]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "display(seq(Lis[j],j=1..nops(Lis)),scaling=constraine d);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 209 214 "Depending on the value o f Y the arrows add up to form a long arrow, or they add up in such a w ay as to cancel contributions with a smaller resultant vector arrow. E xplore the above diagram for different choices of " }{TEXT 19 1 "Y" } {TEXT 209 1 "!" }}{PARA 0 "" 0 "" {TEXT 209 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 209 81 "Finally, we can write a function which carries out \+ the phase (vectorial) addition" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "A:=Y->add(evalf(exp(2*Pi*I*d_i(i,Y))),i=1..N);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "A(10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "plot([seq([iY*2,abs(A(iY*2))^2],iY=-100..100)],style= point);" }}}{EXCHG {PARA 208 "" 0 "" {TEXT 206 11 "Exercise 1:" }} {PARA 0 "" 0 "" {TEXT 209 42 "Investigate different cases of slit widt h " }{TEXT 208 1 "a" }{TEXT 209 252 " to wavelength ratios. Can you de termine how the position of the first maximum scales? Consult a first- year physics textbook and check whether the results found in this work sheet agree with the formula for the diffraction maxima and minima giv en there." }}{PARA 0 "" 0 "" {TEXT 209 0 "" }}{PARA 209 "" 0 "" {TEXT 206 11 "Exercise 2:" }}{PARA 0 "" 0 "" {TEXT 209 117 "Investigate to w hat extent the results depend on the chosen discretization of the aper ture, i.e., vary the parameter " }{TEXT 19 1 "N" }{TEXT 209 1 "." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }