{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" 18 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 0 2 2 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Helvetica" 1 18 0 0 0 0 2 1 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 18 0 0 0 0 2 2 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT 256 19 "EFFECTIVE POTENTIAL" }} {PARA 0 "" 0 "" {TEXT 257 18 "by Corinne Manogue" }}{PARA 0 "" 0 "" {TEXT -1 30 "Copyright 2007 Corinne Manogue" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 114 "In this worksheet you will l ook at the effective potential for a number of different values of var ious parameters." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart:with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "Range:=0..1.5:View:=-5..5:setoptions(numpoints=200): " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "mu;" "6#%#m uG" }{TEXT -1 23 " is the reduced mass\n " }{XPPEDIT 18 0 "U(r)=-k/r " "6#/-%\"UG6#%\"rG,$*&%\"kG\"\"\"F'!\"\"F," }{TEXT -1 21 " is the p otential\n " }{XPPEDIT 18 0 "l" "6#%\"lG" }{TEXT -1 45 " is the magn itude of the angular momentum\n " }{XPPEDIT 18 0 "E" "6#%\"EG" }{TEXT -1 22 " is the total energy:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "mu :=9; k:=1; el:=1; E:=-1;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 "\n" } {TEXT 258 28 "All on the same graph, plot:" }{TEXT -1 16 "\nthe potent ial " }{XPPEDIT 18 0 "U(r)=-k/r" "6#/-%\"UG6#%\"rG,$*&%\"kG\"\"\"F'! \"\"F," }{TEXT -1 51 " (blue), \nthe angular part of the kinetic ener gy " }{XPPEDIT 18 0 "l^2/(2*mu*r^2)" "6#*&%\"lG\"\"#*(F%\"\"\"%#muGF' %\"rGF%!\"\"" }{TEXT -1 35 " (red), \nthe effective potential " } {XPPEDIT 18 0 "V[eff](r)=-k/r+l^2/(2*mu*r^2)" "6#/-&%\"VG6#%$effG6#%\" rG,&*&%\"kG\"\"\"F*!\"\"F/*&%\"lG\"\"#*(F2F.%#muGF.F*F2F/F." }{TEXT -1 46 " (magenta), i.e. the sum, \nthe total energy " }{XPPEDIT 18 0 "E" "6#%\"EG" }{TEXT -1 10 " (green), " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "U:=plot(-k/r,r=Range,View,color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "elpart:=plot(el^2/(2*mu*r^2),r=Rang e,View,\ncolor=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "Vef f:=plot(-k/r+el^2/(2*mu*r^2),r=Range,View,\ncolor=magenta):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "Eplot:=plot(E,r=Range,View,c olor=green):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "display([U, elpart,Veff,Eplot]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 38 "\nIMPORT ANT: Try changing the value of" }{TEXT -1 1 " " }{XPPEDIT 261 0 "l;" "6#%\"lG" }{TEXT 262 3 ". \n" }{TEXT -1 86 "What happens to the shape \+ of the effective potential? \nWhere is the shape changing? " }{TEXT 259 1 "\n" }{TEXT -1 56 "\nTo answer these questions, it might be help ful to plot " }{XPPEDIT 18 0 "V[eff]" "6#&%\"VG6#%$effG" }{TEXT -1 29 " for two different values of " }{XPPEDIT 18 0 "l" "6#%\"lG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "el[small]:=1.0; \+ el[large]:=1.5;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "Veff[sma ll]:=plot(-k/r+el[small]^2/(2*mu*r^2),\nr=Range,View,color=blue):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "Veff[large]:=plot(-k/r+el[la rge]^2/(2*mu*r^2),\nr=Range,View,color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "display([Veff[small],Veff[large],Eplot]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 2 0" 14 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }