{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 Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 16 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 32 "Periodic potential: Bloch states" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 163 "The objective of this worksheet is to explore a periodic potentia l that can represent a one-dimensional lattice of potential wells wrap ped around to form a circle." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 342 "In the Kronig-Penney model of conductivity Blo ch's theorem leads to a derivation of energy bands. The eigenvalues ar e continuous as they depend on a real parameter (the wave propagation \+ vector K). On the other hand it is possible to approximate the continu ous energy eigenvalue regions using a finite number of potential wells (we could have " }{TEXT 19 8 "Nw=10^24" }{TEXT -1 92 " as in a macros copic solid). This situation is presented, e.g., in the QM text of Gri ffiths." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 113 "Here we use the cosine-potential familiar from the quantum pendul um. We impose the boundary condition that after " }{TEXT 19 2 "Nw" } {TEXT -1 370 " potential wells the wavefunction (not just the probabil ity density) repeats itself. From well to well we should have periodic ity of the density (the wavefunctions are allowed to differ by a compl ex number of modulus 1 according to the Bloch theorem which utilizes t he fact that the operator which represents displacement by one cell wi dth commutes with the Hamiltonian." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 11 "We use the " }{TEXT 19 19 "QuantumPendulu m.mws" }{TEXT -1 354 " worksheet as a starting point. It was written i n such a way that other potential energy assemblies than the cosine po tential could be implemented for the eigenvalue calculations. We use, \+ however, the specific cosine potential for the construction of the eig enfunctions, as this permits us to use the known solutions to the Math ieu differential equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "restart; Digits:=15: with(LinearAlg ebra):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "a:=Pi;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG%#PiG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "V0:=2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#V0G\"\"# " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "V:=x->V0*(1-cos(Pi*x/a) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"VGj+6#%\"xG6\"6$%)operatorG% &arrowGF(*&%#V0G\"\"\",&F.F.-%$cosG6#*(%#PiGF.9$F.%\"aG!\"\"F7F.F(F(F( 6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Vmax:=V(a);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%VmaxG\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "We pick an odd number of wells:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 6 "Nw:=3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#Nw G\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "Vmw:=piecewise(se q(op([x>a*(2*j-1) and x%$VmwG-%*PIECEWISEG6%7$,&\"\"#\" \"\"*&F*F+-%$cosG6#%\"xGF+!\"\"32,$*&\"\"$F+%#PiGF+F1F02F0,$F7F17$F)32 F9F02F0F77$F)32F7F02F0,$*&F6F+F7F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "PTS:=[seq((2*j-1)*a,j=-(Nw-1)/2..(Nw+1)/2)];" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$PTSG7&,$*&\"\"$\"\"\"%#PiGF)!\"\",$ F*F+F*,$*&F(F)F*F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "We want t o have periodicity with " }{TEXT 19 2 "2a" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 54 "Pmw:=plot(Vmw,x=PTS[1]..PTS[nops(PTS)],numpoints=50 0):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "A periodic basis allows fo r left- and right-travelling states with periodicity conditions at " } {TEXT 19 1 "a" }{TEXT -1 6 ", and " }{TEXT 19 2 "-a" }{TEXT -1 1 "." } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "psi:=n->eval(exp(I*Pi*n*x/ (Nw*a)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$psiGj+6#%\"nG6\"6$%)op eratorG%&arrowGF(-%%evalG6#-%$expG6#*.%#PiG\"\"\"9$F4%\"xGF4%#NwG!\"\" %\"aGF8^#F4F4F(F(F(6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "int(conjugate(psi(0))*psi(2),x=PTS[1]..PTS[nops(PTS)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "L:=int(conjugate(psi(1))*psi(1),x=PTS[1]..PTS[nops(PTS)]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG,$*&\"\"'\"\"\"%#PiGF(F(" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 122 "The kinetic energy for the plane- wave basis functions can be calculated in closed form. As an example w e list the one for " }{TEXT 19 3 "n=3" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "int(-diff(psi(3),x$2)*conjugate(psi (3)),x=PTS[1]..PTS[nops(PTS)])/L;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# \"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "4*3^2*Pi^2/(L)^2; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 131 "Here we select the basis size. Note that the index label ing the plane-wave basis states ranges over positive and negative inte gers " }{TEXT 19 1 "n" }{TEXT -1 63 " (corresponding to right-travelli ng and left-travelling waves)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "n_max:=20; N:=2*n_max+1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% &n_maxG\"#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"#T" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "VME:=unapply(int(V(x)*exp(Pi*I*dn*x /(Nw*a)),x=x1..x2),x1,x2,dn);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$VM EGj+6%%#x1G%#x2G%#dnG6\"6$%)operatorG%&arrowGF*,$**\"\"'\"\"\",2*(^#! \"\"F1-%$expG6#*(^##F1\"\"$F19&F19$F1F1)F=\"\"#F1F1*&^#\"\"*F1F6F1F1** F?F1F6F1-%$cosG6#F>F1^#F1F1F1**F " 0 "" {MPLTEXT 1 0 23 "#VME(PTS[1],PTS[2],21);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "#limit(VME(PTS[1],PTS[2],s),s=0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "kh:=1; # for the cosine p otential we put the strength into V(x)." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#khG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "H:=Ma trix(N,N,shape=hermitian,datatype=complex[8]):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "We should have a " }{TEXT 19 23 "1/sqrt(2*a) = 1/sqr t(L)" }{TEXT -1 105 " normalization factor in front of the plane-wave \+ basis states! We can fix this by dividing the matrix by " }{TEXT 19 7 "L=Nw*2a" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "for i1 from 1 to N do: n1:=-n_max+i1-1: for i2 from 1 to i1 do: n 2:=-n_max+i2-1: if i2=i1 or n1-n2=Nw then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 209 "H[i1,i2]:=evalf(1/L*kh*add(limit(VME(PTS[i0],PTS[i0+ 1],s),s=n1-n2),i0=1..Nw)): else H[i1,i2]:=evalf(1/L*kh*add(VME(PTS[i0] ,PTS[i0+1],n1-n2),i0=1..Nw)): fi: od: H[i1,i1]:=H[i1,i1]+evalf(4*n1^2* Pi^2/(L)^2): od: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Eigenvalues(H) ;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "EV:=convert(%,list);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6*\")wAlC%'sfloatG&%'VectorG6#%'co lumnG%,rectangularG%.Fortran_orderG7\"\"\"\";F/\"#T" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#>%#EVG7K$\"0xTU&Hq)H*!#:$\"0PsA&ofQ$*F($\"0xtA&ofQ$*F ($\"0?#*z8wIg#!#9F-$\"03)zc-s'o#F/$\"0$=k3(osq$F/$\"0?bw\"[anSF/$\"0cb w\"[anSF/$\"0^DO;u1+&F/$\"0*eijTn+]F/$\"0wXD#)3I6'F/$\"0&zrEZXihF/$\"0 #R3I'pAa(F/$\"0=%3I'pAa(F/$\"0)4zcuf%=*F/$\"0Z\"zcuf%=*F/$\"0(y*eGNd5 \"!#8$\"0-eH\")[d5\"FJ$\"0%=_l'GdJ\"FJ$\"0(=_l'GdJ\"FJ$\"0w]+^T#[:FJ$ \"0!305:C[:FJ$\"0k*>r!zJ!=FJ$\"0*R**y!zJ!=FJ$\"0-Mo9z/3#FJ$\"09Mo9z/3# FJ$\"0,v!z?5!Q#FJ$\"03v!z?5!Q#FJ$\"0=tWMb?q#FJ$\"0quWMb?q#FJ$\"0va0HZi /$FJ$\"0&[b!HZi/$FJ$\"0f^CO0FT$FJ$\"0i^CO0FT$FJ$\"0$\\\\!Rd!4QFJ$\"0+& \\!Rd!4QFJ$\"0t;WAa'>UFJ$\"0(oTCUl>UFJ$\"0JY3UGDl%FJ$\"0YY3UGDl%FJ" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 118 "Compared to the single-well ener gy spectrum we observe that the lowest oscillator states are approxima tely replicated " }{TEXT 19 2 "Nw" }{TEXT -1 95 " times. There is a un ique ground state, and a pair of eigenvalues that appear to be degener ate." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 213 " We could construct the eigenfunctions from the Eigenvectors command. T he calculations with these states are tedious (slow). We prefer to con nect with the Mathieu differential equation. This requires us to resca le " }{TEXT 19 9 "xi=2*x*Nw" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "q:=2*V0*Nw^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"qG\"#O" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "MDE:=a->diff(u (x),x$2)+(a+2*q*cos(2*Nw*x))*u(x)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%$MDEGj+6#%\"aG6\"6$%)operatorG%&arrowGF(/,&-%%diffG6$-%\"uG6#%\"xG -%\"$G6$F4\"\"#\"\"\"*&,&9$F9*(F8F9%\"qGF9-%$cosG6#,$*(F8F9%#NwGF9F4F9 F9F9F9F9F1F9F9\"\"!F(F(F(6#FE" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "H ere " }{TEXT 19 1 "a" }{TEXT -1 39 " is the eigenvalue scaled accordin g to " }{TEXT 19 11 "a=4*(EV-V0)" }{TEXT -1 39 ", and the potential st rength parameter " }{TEXT 19 6 "q=2*V0" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "aM:=e- >Nw^2*4*(e-V0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#aMGj+6#%\"eG6\"6 $%)operatorG%&arrowGF(,$*(\"\"%\"\"\")%#NwG\"\"#F/,&9$F/%#V0G!\"\"F/F/ F(F(F(6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "sol:=dsolv e(MDE(aM(EV[1])));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG/-%\"uG6# %\"xG,&*&%$_C1G\"\"\"-%)MathieuCG6%#!/f%H(Q$\\q(\"/++++++=!\"%-%'arcco sG6#-%$absG6#-%$cosG6#,$*&\"\"$F-F)F-F-F-F-*&%$_C2GF--%)MathieuSGF0F-F -" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "sol:=u(x) = _C1*Mathi euC(-77049338729459/18000000000000,-4,3*x)+_C2*MathieuS(-7704933872945 9/18000000000000,-4,3*x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG/- %\"uG6#%\"xG,&*&%$_C1G\"\"\"-%)MathieuCG6%#!/f%H(Q$\\q(\"/++++++=!\"%, $*&\"\"$F-F)F-F-F-F-*&%$_C2GF--%)MathieuSGF0F-F-" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 4 "The " }{TEXT 19 8 "MathieuC" }{TEXT -1 5 " and " } {TEXT 19 8 "MathieuS" }{TEXT -1 272 " functions are symmetric and anti symmetric respectively, i.e., behave like sine and cosine. We invoke a physics argument which states that the eigenstates of the problem are either symmetric or antisymmetric, as the Hamiltonian is symmetric. T he ground state is symmetric." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "plot([eval(rhs(sol),\{_C1=1,_C2=0\}),eval(rhs(sol),\{_C2=1,_C1= 0\})],x=-Pi/2..Pi/2,color=[red,blue],view=[-Pi/2..Pi/2,-q/10..q/10]); " }}{PARA 13 "" 1 "" {GLPLOT2D 882 331 331 {PLOTDATA 2 "6&-%'CURVESG6$ 7gt7$$!0([zEjzq:!#9$\"0A=C[X/T&!#;7$$!03X>)on`:F*$\"05gBYf#)\\&F-7$$!0 H&4PubO:F*$\"0lL$>6$Rw&F-7$$!0]XA*zV>:F*$\"0G2v#*4T@'F-7$$!0r&RZ&=B]\" F*$\"0)o9C0ufoF-7$$!0XUDZFDZ\"F*$\"0_F>QZf\\)F-7$$!0>*o(ROFW\"F*$\"0U: mD%f(3\"!#:7$$!0Xl#=KB49F*$\"0Zd*f::g9FL7$$!0rT)Q+tv8F*$\"0E&o2FQi>FL7 $$!0/(fyZ+U8F*$\"0g!=*)3P:EFL7$$!0P_$=&z#38F*$\"0Lf$\\6wBMFL7$$!0]nL.( \\\"H\"F*$\"0a@[a?@)QFL7$$!0i#Q[Xru7F*$\"03!oPGAuVFL7$$!0u(Rj?$zD\"F*$ \"0:j;,Lb*[FL7$$!0(GTy&\\6C\"F*$\"0!H:F.))RaFL7$$!0GCl1!fD7F*$\"03yxU[ $efFL7$$!0oNYbI+@\"F*$\"0cA`_nC['FL7$$!04ZF/rW>\"F*$\"0^'3=Cv.qFL7$$!0 ]e3`6*y6F*$\"0Suh@nG^(FL7$$!0'*Rvb+G;\"F*$\"0)Hq())4m,)FL7$$!0V@Ue*oY6 F*$\"0%\\HODK&[)FL7$$!0!H!4hy08\"F*$\"0U`clzx!*)FL7$$!0O%ePwY96F*$\"0^ [3QELF*FL7$$!0KyUlV63\"F*$\"0[Xn1k\"4)*FL7$$!0Hs4n>y/\"F*$\"0be1AN**** *FL7$$!01bFR:&R5F*$\"0#pifv7!***FL7$$!0#y`96@J5F*$\"0RAr0Dt&**FL7$$!0e ?j$o!H-\"F*$\"0&Q$*Gxx,**FL7$$!0N.\"eDg95F*$\"0yiEI2R#)*FL7$$!0!))o;+% *z**FL$\"0?PY/1Pg*FL7$$!08WBXaQ\")*FL$\"0_:H(yD.$*FL7$$!0v/\\.:Ik*FL$ \"0wY%4jm>*)FL7$$!0Sluhv@Z*FL$\"03=h7]GfFL7$$!0!\\5v\\ fG&)FL$\"0#Gus`:FaFL7$$!0g7jD&>f$)FL$\"0B)*QN^#y[FL7$$!0N?v`&z*=)FL$\" 0y*)>KYIN%FL7$$!05G(=eR?!)FL$\"0y?ABBy&QFL7$$!0$e$**4'*4&yFL$\"0dR)3)p rR$FL7$$!0S4$>b!3^(FL$\"015blFL7$$!0 qn%y#yF%oFL$\"0C7B(yt[9FL7$$!0T_#=;%\\^'FL$\"0'o2?F0'3\"FL7$$!0gjM_Ms@ 'FL$\"0%3:;OL'[)F-7$$!0![nGu_>fFL$\"0N_^-OV&oF-7$$!0]*Rjh_UdFL$\"0WX^> <=>'F-7$$!0?C\")*[_lbFL$\"0L^]0!pPdF-7$$!0b'[lU-xaFL$\"04Ep>))[e&F-7$$ !0!*[GjB&)Q&FL$\"09#=c.8![&F-7$$!0D6-+B+I&FL$\"0w-%4%4FU&F-7$$!/OdnB_6 _F*$\"0j$o-_B7aF-7$$!0D\"fy!z:1&FL$\"0%>*)3_f,bF-7$$!0!*3'*yN;\"\\FL$ \"0LT#4eRFdF-7$$!0bE1]#phZFL$\"0*ys;*zR4'F-7$$!/Uk6#\\K%)F-7$$!/cyZ/29RF*$\"0))*QL7=F6FL7$$!0 l>i[)G0OFL$\"0Q)eRT\")z9FL7$$!/PlCl]'H$F*$\"0SQ@)*RL%>FL7$$!0l2NxBx&HF L$\"0F/'>'GXf#FL7$$!/;OA5%*=EF*$\"0G))G3%\\-MFL7$$!0=X_PSwX#FL$\"02fHk B2%QFL7$$!0vG\"G(RjH#FL$\"0W#Gb)G/J%FL7$$!0K753R]8#FL$\"0m>KONx![FL7$$ !/f*QVQP(>F*$\"0eDvuytK&FL7$$!0)*y,8Qa!=FL$\"0ao>:Zh)eFL7$$!00ik#y8P;F L$\"0f,KA!z_kFL7$$!07XF_P)o9FL$\"0%4_CRp;qFL7$$!/#G!>s`+8F*$\"0QV47lgc (FL7$$!0]j1'yK9**F-$\"0n`/1(*\\\\)FL7$$!.X5`$GBoF*$\"0VB%y?E`#*FL7$$!0 +nfju\"*[$F-$\"08/.N/&)z*FL7$$!-*)3ul]:F*$\"0/a`a(f****FL7$$\"0+!\\gyM 2r!#<$\"075e,\\:***FL7$$\"0+()HJNld\"F-$\"0jxha%\\e**FL7$$\"0+D*ReLUCF -$\"0-\\m*zq+**FL7$$\"0+joOO\"3LF-$\"0PVCAm'=)*FL7$$\"0+R2UP(R]F-$\"0. 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The ground \+ state is unique, it has a symmetric eigenfunction." }}{PARA 0 "" 0 "" {TEXT -1 122 "Let us plot a probability using the eigenenergy as a bas eline, and superimpose it on the potential as a function of theta." }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "P0:=plot([EV[1],EV[1]+(eva l(rhs(sol),\{_C1=1,_C2=0,x=s/(2*Nw)\}))^2],s=-Nw*Pi..Nw*Pi,color=[blac k,green]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "plots[display ](Pmw,P0);" }}{PARA 13 "" 1 "" {GLPLOT2D 1020 263 263 {PLOTDATA 2 "6'- %'CURVESG6$7`jl7$$!30\">p2'zxC%*!#<$\"\"%\"\"!7$$!3;>WM:dD&Q*F*$\"3wo$ Ga>Q%)*RF*7$$!3)*HY#4in3N*F*$\"3AlP+[(RX*RF*7$$!3s\\5@7Y>7$*F*$\"3?0H' >PQt)RF*7$$!3E]@p`_Et#*F*$\"3%z)f@\"y(3xRF*7$$!38f)='>4_M#*F*$\"3SQ\"R jJ6R'RF*7$$!3_HHJi**f)>*F*$\"3ySllB71\\RF*7$$!3'3pX;*eSh\"*F*$\"3-%)o5 %=N5$RF*7$$!3Lf\\bI(RH7*F*$\"3'H+]BG$e4RF*7$$!3?!pI4$pf%3*F*$\"3_\"fKz b)Q&)QF*7$$!3_pze[m:X!*F*$\"3!y^)4k+hdQF*7$$!3P>FVEwT5!*F*$\"3w@D-!3[2 $QF*7$$!3#)44S:(48(*)F*$\"3]c_8Xi'yz$F*7$$!3l***GNAT?$*)F*$\"3C(\\:K:$ 3iPF*7$$!3#*>eLb()>%*))F*$\"3*R`#HO#G]s$F*7$$!3AfJ%)oT$)f))F*$\"3%Rqhv QP#*o$F*7$$!3!3V*o^8(*=))F*$\"3jg$>H7'3WOF*7$$!3Ufcd&=bVy)F*$\"3HB6e8k o.OF*7$$!3_!\\;^N)3W()F*$\"3U[NMHyFaNF*7$$!3w!G3sQX%3()F*$\"3t20@DyV3N F*7$$!3G]kI-%R$p')F*$\"3+LK-o=%fX$F*7$$!3\"31?pD,@j)F*$\"3EZ+I#y#)QS$F *7$$!3!)HtuerC$f)F*$\"3)Qy\")eo!\\ZLF*7$$!3S+!)onpcd&)F*$\"3?I7_5:\"RH $F*7$$!3MSTSh53>&)F*$\"3E#Q#eTZFMKF*7$$!3%36U5-0\"z%)F*$\"3))>_phYRqJF *7$$!3@z,RgdIW%)F*$\"3yUR,Z1E8JF*7$$!3+5#Hbg@nS)F*$\"3k$Q`QJU+0$F*7$$! 3E>c%f\\$*yO)F*$\"3[q\"[!=^<$)HF*7$$!3`p*Qew2*H$)F*$\"3MHn]1PK;HF*7$$! 3Q]4x=Y:$H)F*$\"3]Jx@_@Q]GF*7$$!3'*R>>:lM_#)F*$\"3%4HVlbBex#F*7$$!3AqC 6>'yc@)F*$\"3.!3il&Qs2FF*7$$!3!3$>W8y_w\")F*$\"3!>_GT#\\'Rj#F*7$$!3n]0 bk80T\")F*$\"3m,sYxhGmDF*7$$!3y!=a`BmA5)F*$\"3y!on+0$[\"\\#F*7$$!3[*>w 8Xsd1)F*$\"3rCl%)y?U?CF*7$$!3I?L\">$fiF!)F*$\"3-J*4W'faXBF*7$$!315%*eO 2L!*zF*$\"3#)pHRJS&=F#F*7$$!3O!3CM7'G^zF*$\"3qY?(f--V>#F*7$$!3!)>NQ?:o 8zF*$\"31_*p/')G$>@F*7$$!3E!Q@CQD_(yF*$\"34L.A,V[U?F*7$$!3Pgwp!o(3PyF* $\"3)fOL1q8i'>F*7$$!3TI?k$4V?!yF*$\"3Qe)yjhph*=F*7$$!3Wqm4U!y=w(F*$\"3 ?+fDYJ0;=F*7$$!39SM=TR&fs(F*$\"39nI%eaVYu\"F*7$$!3;!>ToN^wo(F*$\"3YGh* )47()o;F*7$$!3eHgK'*3*4l(F*$\"3WpJ7!)3!of\"F*7$$!3(*4(Qj;%G5wF*$\"3S() RvJ]T<:F*7$$!31!)z@H^.vvF*$\"353r(Q%RJ\\9F*7$$!3#)H;!o:F]`(F*$\"37f.'o 6_GP\"F*7$$!3?I3@LUc)\\(F*$\"3d%f)Q\"eOSI\"F*7$$!3Qg+=D&z'euF*$\"3-)pa n'f#)H7F*7$$!3Wq4TndECuF*$\"3;rN\"R'\\xm6F*7$$!3'*pON9D2&Q(F*$\"3)3M/l :th4\"F*7$$!3O5DzmP@ZtF*$\"3r\\&p&[AHH5F*7$$!3()\\_V^(z$4tF*$\"3:@B_Ly WQ'*!#=7$$!3)*HXf?\\orsF*$\"3r=OS1z7,!*F`]l7$$!3mHe&*))>ZNsF*$\"3<*f@A /pNS)F`]l7$$!3+Sr#3&[K'>(F*$\"3Kli.`!>Zx(F`]l7$$!3/qIm*y9!frF*$\"3/:QS B0!G>(F`]l7$$!3fpP;2xu>rF*$\"3Y>dhani*f'F`]l7$$!3?+btT:?%3(F*$\"3C*oLm /`/3'F`]l7$$!39grhuj\"\\/(F*$\"3D)ptK)G:FbF`]l7$$!3Pg??I!)G2qF*$\"3qFI ykyFF`]l7$$!3#)f0=bYkznF*$\"3N E&G)o#4XT#F`]l7$$!3BgSo8W$=u'F*$\"3e%y9J(=(p1#F`]l7$$!3.!=1?P#Q.nF*$\" 3?`o>`'G)R\"F`]l7$$!3K!eJG?rHf'F*$\"3i$HEC^F-_*!#>7$$!35!)Qk6e(Hb'F *$\"3\")QTpZ\"\\YB(F`cl7$$!3*)4g^6;oz@iQx*G'F*$\"3&>Vhk/!*3M%!#A7$$!3 =!3L/9U=D'F*$\"3&y8`[zOJ#)*!#@7$$!3*)*)[D([C[@'F*$\"3/*\\nI3#QrYF_el7$ $!3!)z,ogppuhF*$\"3IY$[#*>=e<\"F`cl7$$!3g4Ka$\\v*RhF*$\"3;1Y@\"H-u/#F` cl7$$!3#)\\0BEm4+hF*$\"35/FI%*GzULF`cl7$$!3!**RbWR3E1'F*$\"31Y#zB]Ad%[ F`cl7$$!3p\\l6eA[DgF*$\"3Y`=efST/mF`cl7$$!31g)eweE$))fF*$\"30wVas#e8j) F`cl7$$!3'*pzrb,V]fF*$\"3Mj(3?n!3(4\"F`]l7$$!3K]Uss0E5fF*$\"3+\"o$)=R& ou8F`]l7$$!3=5-m)**GM(eF*$\"3mF`]l7$$!3yf0,Wf?)z&F*$\"3UX[L+'3jI#F`]l7$$!3w*)>H*)\\UfdF*$\"3'*)R% >tv5\"o#F`]l7$$!3&**3X%znhCdF*$\"3xoG$*zvqRIF`]l7$$!3C?JvWRQ$o&F*$\"3! zKcK%[/\"\\$F`]l7$$!3)*>Gv?A))[cF*$\"3'=!*=@6-.*QF`]l7$$!3V!f6#z&G&3cF *$\"3%zd_>qr:Q%F`]l7$$!3%*f#Qw$\\,qbF*$\"3G\">cVrpT([F`]l7$$!35g%=K%oi NbF*$\"3,79Ig8)HL&F`]l7$$!3%)z[]MQ&p\\&F*$\"3AE1!z0a'peF`]l7$$!3;!*f)f ZC!eaF*$\"3iqX'piB7V'F`]l7$$!39!p7>9!G>aF*$\"3.s,,%**R0,(F`]l7$$!3mfng %=fLQ&F*$\"3'HQ!=ln2lvF`]l7$$!35?&RR6lhM&F*$\"3M3Dd(fkh:)F`]l7$$!3M!z[ G&*)p2`F*$\"3[DS/%Q\"p%y)F`]l7$$!3M?XA`hNp_F*$\"3CP)GxQ;xU*F`]l7$$!3J5 =)3(e\"*H_F*$\"3-W)3F=N0,\"F*7$$!3Q]ls[o<&>&F*$\"3y1mB%f*\\r5F*7$$!3s \\ZpP*og:&F*$\"3Rvd+ezYT6F*7$$!3WSG#eW+o6&F*$\"3pNmsiW/87F*7$$!31]'Hw( z&*y]F*$\"3G.Sv:5<$G\"F*7$$!3B!*p8\"R$fW]F*$\"3UNEU*eWxM\"F*7$$!3/gK)R dIP+&F*$\"3*o#3a<\\_D9F*7$$!3A+&pyS9\"p\\F*$\"3=1\\lp1<#\\\"F*7$$!3UI. 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We find, however, since the 2nd and 3rd eigenenergies are so close that there have to be 2 ei genfunctions at this energy. They are the symmetric and anti-symmetric functions." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "plot([eval( rhs(sol),\{_C1=1,_C2=0\}),eval(rhs(sol),\{_C2=1E-2,_C1=0\})],x=-Pi/2.. Pi/2,color=[red,blue]);" }}{PARA 13 "" 1 "" {GLPLOT2D 882 331 331 {PLOTDATA 2 "6&-%'CURVESG6$7^q7$$!0([zEjzq:!#9$!09ZvK#f>F!#;7$$!0H&4Pu bO:F*$!0#4Oq+/(*GF-7$$!0r&RZ&=B]\"F*$!0*\\G4K4ZMF-7$$!0XUDZFDZ\"F*$!0p &o!R3$oUF-7$$!0>*o(ROFW\"F*$!0-UD6ZEY&F-7$$!0Xl#=KB49F*$!0F2;TR=L(F-7$ $!0rT)Q+tv8F*$!0\\>!3k'4&)*F-7$$!0P_$=&z#38F*$!0$Q$RzDxr\"!#:7$$!0(GTy &\\6C\"F*$!0i/vf_vs#FQ7$$!0]e3`6*y6F*$!0s-!fzUkPFQ7$$!0V@Ue*oY6F*$!0Cl @@j*\\UFQ7$$!0O%ePwY96F*$!0*y\"*\\][UYFQ7$$!0MJfk0y4\"F*$!0DL8'3P&z%FQ 7$$!0KyUlV63\"F*$!0/&*G&f(z!\\FQ7$$!0\"=XeE\"G2\"F*$!0B/*H^1[\\FQ7$$!0 IDEm\"[k5F*$!0hKJC2p(\\FQ7$$!0!))zm1:c5F*$!0JIGTxU*\\FQ7$$!0Hs4n>y/\"F *$!0Yv(\\w.+]FQ7$$!01bFR:&R5F*$!07vmFwT*\\FQ7$$!0#y`96@J5F*$!0!pB!3qn( \\FQ7$$!0e?j$o!H-\"F*$!0@'G=l%z%\\FQ7$$!0N.\"eDg95F*$!0\"R*y/>z!\\FQ7$ $!0!))o;+%*z**FQ$!0N1fFQ$!0h@W[w*e8F-7$$! /OdnB_6_F*$\"0zU6-Qi[)G0OFQ$\"0N;STB:R\"FQ7$$!/PlCl]'H$F*$\"0Ak'f?N u=FQ7$$!0l2NxBx&HFQ$\"06@eg]8a#FQ7$$!/;OA5%*=EF*$\"0vI)\\2ThLFQ7$$!0vG \"G(RjH#FQ$\"0E^En0'yUFQ7$$!/f*QVQP(>F*$\"0pzTpJLI&FQ7$$!00ik#y8P;FQ$ \"0U/\"3(*oNkFQ7$$!/#G!>s`+8F*$\"0![QJ\\([b(FQ7$$!0]j1'yK9**F-$\"0^]@b (G)[)FQ7$$!.X5`$GBoF*$\"0ycoj,+D*FQ7$$!0+1N3Hi:&F-$\"0q.zdjQc*FQ7$$!0+ nfju\"*[$F-$\"0E*p'*fj(z*FQ7$$!0](>7ukbEF-$\"0]JLf>A))*FQ7$$!0+G%)=?@# =F-$\"0Ux43hV%**FQ7$$!0+&eY'Hf))*!#<$\"05-]j'e$)**FQ7$$!-*)3ul]:F*$\"0 )=q=ef****FQ7$$\"0+!\\gyM2rFgz$\"0Tg@v7:***FQ7$$\"0+()HJNld\"F-$\"0q\\ [J;$e**FQ7$$\"0+D*ReLUCF-$\"0dZ]2\"G+**FQ7$$\"0+joOO\"3LF-$\"0R\\R6&)y \")*FQ7$$\"0+R2UP(R]F-$\"0-Kol&*Ge*FQ7$$\".:YZQ8x'F*$\"0.OW/04E*FQ7$$ \"0]?,4mgy*F-$\"0HQ,&Q%Q_)FQ7$$\"/Ecq$z+G\"F*$\"0`jl`(Q?wFQ7$$\"0N`@@y cg\"FQ$\"0lAQ-UAa'FQ7$$\"/Tu`qFJ>F*$\"0?6PspRW&FQ7$$\"0&p(p^_wE#FQ$\"0 6(eh*4fO%FQ7$$\"/)4-)z-/EF*$\"00(Q5()*4S$FQ7$$\"0!yBYd5LHFQ$\"0LFQ7$$\"/qDJG)*)*QF*$\"0)4Gl!f!H5FQ7$$ \"/]o.*Rgg%F*$\"0eJhH68Z%F-7$$\"/`*38i8C&F*$\"0R@%G]_P8F-7$$\"/kD:Uq>f F*$!05*Qs\"3(f8F-7$$\"/^9lHQMlF*$!01fml\"yGVF-7$$\"/sq'>*Q1sF*$!0)*3$f N(*o#*F-7$$\"/p$**o%pQyF*$!0R$yS.EN;FQ7$$\"/$347O'*\\)F*$!0@!4;%>Wj#FQ 7$$\"/vxyr#e9*F*$!0!=q*3TBs$FQ7$$\"0&oja#yS[*FQ$!0qSF#)[%QUFQ7$$\"/i\\ I$HB#)*F*$!0/TScb:l%FQ7$$\"0XpXK<_)**FQ$!0'*p1()G(*z%FQ7$$\"0Fk=`5[,\" F*$!0W(pog04\\FQ7$$\"0%zcJ\\&H-\"F*$!0'oncf9[\\FQ7$$\"0gr7L*4J5F*$!0)o 89vXw\\FQ7$$\"0Ev4tV#R5F*$!0G/t0*y$*\\FQ7$$\"0#*y18)QZ5F*$!0sS_W=++&FQ 7$$\"0j(pbprb5F*$!0rqeck[*\\FQ7$$\"0M;2yXS1\"F*$!0?([Q?5y\\FQ7$$\"00Nd guB2\"F*$!0C)3^T')\\\\FQ7$$\"0w`2V.23\"F*$!0kd+7r.\"\\FQ7$$\"0=\"z!3ht 4\"F*$!0>77+;*)z%FQ7$$\"0fG3t=S6\"F*$!0;l\"*f&4ZYFQ7$$\"0G)Rj\"eq9\"F* $!0O)Go8&[C%FQ7$$\"0'z'ff(4!=\"F*$!0`'*eog`u$FQ7$$\"0fX.@<3C\"F*$!074G NSIt#FQ7$$\"0KF1u3/J\"F*$!0k>uOv)*o\"FQ7$$\"0&z\"\\B_EP\"F*$!0b;ho2=, \"FQ7$$\"061&3X$eS\"F*$!0B,5u-ib(F-7$$\"0F%4#y;!R9F*$!0B<\"pc3TcF-7$$ \"0%3)=Xw2Z\"F*$!06![^LgFVF-7$$\"0Un;7OD]\"F*$!07)>V)HBW$F-7$$\"09JUAm m`\"F*$!0?\"3!)[!f*GF-7$$\"0([zEjzq:F*F+-%'COLOURG6&%$RGBG$\"*++++\"! \")$\"\"!FejlFdjl-F$6$7ep7$F($!0@AyD'e25!#D7$F4$!0Hp/#oF6#)F-7$F>$!0h \"RvsR]=FQ7$FH$!04O1t)*ps$FQ7$$!0/(fyZ+U8F*$!0+\"Hw^lp]FQ7$FM$!0Jd\"HB ]7nFQ7$$!0i#Q[Xru7F*$!0aKaeeBj)FQ7$FS$!0M6*3J!y2\"F*7$$!0oNYbI+@\"F*$! 0R$)\\^'Q(G\"F*7$FX$!0X$ohhO%\\\"F*7$Fgn$!0q_Xgp'*o\"F*7$F\\o$!0)Qu\"Q !*z%=F*7$Fao$!0Vw_XB*4>F*7$Ffo$!0(3hDK\"e&>F*7$F[p$!09&RzwHs>F*7$F`p$! 0]'z_!*H%)>F*7$Fep$!0_2gaE<*>F*7$Fjp$!0@im(R_%*>F*7$F_q$!0'z)\\#Ro#*>F *7$Fdq$!0Q-AZPi)>F*7$Fiq$!0^S7LM_(>F*7$F^r$!0(y:h&e(f>F*7$Fcr$!0(\\\"Q !)*)e\">F*7$Fhr$!0Oy*3k$f&=F*7$F]s$!0?/9G'z*o\"F*7$Fbs$!0cNjN@B[\"F*7$ $!0vX)yeaH))FQ$!0[LimhAG\"F*7$Fgs$!0ZlJBu.3\"F*7$$!0N?v`&z*=)FQ$!0*fL! yl&\\')FQ7$F\\t$!0ztr/I)GnFQ7$$!0S4$>b!3^(FQ$!0)ea$*Qh&4&FQ7$Fat$!0f>z i]Fx$FQ7$Fft$!0fZ*oTW6?FQ7$$!0gjM_Ms@'FQ$!0pfg(F-7$F`u$!0rPkD!>\"H&F-7$Feu$!0QOHo; >$HF-7$Fju$!04qP2%z*e\"F-7$Fdv$!0%f,k$=/-\"F-7$F^w$!0\")Q>yM3,(Fgz7$Fh w$!0&Q6P>(Q?&Fgz7$Fbx$!0$3[\")H64OFgz7$F\\y$!0'*[mh8o*>Fgz7$F[[l$!0H.E b4>l%!#>7$Fi\\l$\"0NtCL?B)>Fgz7$Fc]l$\"0IT(yL3aaF-7$$\"0&e2tJ`!e&FQ$\"0==ziTZs(F- 7$Fe`l$\"0wI/#)**o4\"FQ7$$\"0v+-fVqA'FQ$\"0#[5,G3/:FQ7$Fj`l$\"0#y$Q(f! 30#FQ7$F_al$\"0a;C-?w*QFQ7$$\"00AL%>aAvFQ$\"0&\\%)y?wY^FQ7$Fdal$\"02\" fDYSkmFQ7$$\"0gAaSl\"p\")FQ$\"0K^;yr]_)FQ7$Fial$\"0ku\"p_Hh5F*7$$\"0!H %)\\;tA))FQ$\"0l$*f_iwF\"F*7$F^bl$\"0!e*)R!>A\\\"F*7$Fcbl$\"0YT4$**R'p \"F*7$Fhbl$\"0)*QbLf$f=F*7$F]cl$\"01A&e#Rv\">F*7$Fbcl$\"0Ihr')*>g>F*7$ Fgcl$\"0ZS:.6`(>F*7$F\\dl$\"0\\P[Y>h)>F*7$Fadl$\"0lxDuXD*>F*7$Ffdl$\"0 ^M\"3Aa%*>F*7$F[el$\"0p?hv')>*>F*7$F`el$\"058Mp,[)>F*7$Feel$\"0/\\yKTI (>F*7$Fjel$\"0sV4g%zc>F*7$F_fl$\"0T;GCj8\">F*7$Fdfl$\"0C@$oY&)\\=F*7$F ifl$\"0)p5d7h(o\"F*7$F^gl$\"0]`*)\\-n[\"F*7$$\"0ycJSd/@\"F*$\"0*4i&*o \\%G\"F*7$Fcgl$\"0f!pt\"3+3\"F*7$$\"0Y'[vHhv7F*$\"092hldwd)FQ7$Fhgl$\" 0'Rhx+&**f'FQ7$$\"0ksx[I:M\"F*$\"0*R[tRm!4&FQ7$F]hl$\"0Yk]%4VPQFQ7$Fgh l$\"0Vb+ns7$>FQ7$Fail$\"0Ry_f%*4=)F-7$F[jl$\"0@AyD'e25F\\[m-F^jl6&F`jl FdjlFdjlFajl-%+AXESLABELSG6$Q\"x6\"Q!F\\^n-%%VIEWG6$;$!0!\\zEjzq:F*$\" 0!\\zEjzq:F*%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "Is it true that both the symmetric and the antisymmetric solution s work? Let us check the periodicity:" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "fS:=eval(rhs(sol),\{_C1 =1,_C2=0\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fSG-%)MathieuCG6%#! 0(4fmD0>>\"/++++++X!\"%,$*&\"\"$\"\"\"%\"xGF/F/" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 36 "fA:=eval(rhs(sol),\{_C1=0,_C2=1E-2\});" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fAG,$*&$\"\"\"!\"#F(-%)MathieuSG6%# !0(4fmD0>>\"/++++++X!\"%,$*&\"\"$F(%\"xGF(F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "evalf(limit(fS,x=-Pi/2));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$!09ZvK#f>F!#;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "evalf(limit(fS,x=Pi/2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$! 09ZvK#f>F!#;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "evalf(limit (diff(fS,x),x=-Pi/2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0zq$=H\\ \"o(!#D" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "evalf(limit(diff (fS,x),x=Pi/2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!0zq$=H\\\"o(!#D " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "The symmetric solution works! It is periodic to a high degree of precision." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "evalf(limit (fA,x=-Pi/2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!07LEM7]+\"!#D" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "evalf(limit(fA,x=Pi/2));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"07LEM7]+\"!#D" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "evalf(limit(diff(fA,x),x=-Pi/2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!0-4'4l5.6!#9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "evalf(limit(diff(fA,x),x=Pi/2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!0-4'4l5.6!#9" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 179 "The anti-symmetric solution also works! This is a feature that was ob served for the rotator states in the single-well problem; in fact the \+ lowest rotator states didn't display it." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 299 "At this point we should invoke ph ysics as a guide. On the one hand we might argue that there is a symme tric and an antisymmetric state. 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Let's calculate the derivative numerically:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "eta:=1E-3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$etaG$\"\"\"!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "psi_d:=x0->(evalf(eval(f_Right,x=x0+eta)-eval(f_Right ,x=x0-eta)))/2/eta;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&psi_dGj+6#%# x0G6\"6$%)operatorG%&arrowGF(,$*&#\"\"\"\"\"#F/*&-%&evalfG6#,&-%%evalG 6$%(f_RightG/%\"xG,&9$F/%$etaGF/F/-F76$F9/F;,&F=F/F>!\"\"FCF/F>FCF/F/F (F(F(6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 127 "Based on the cur rent density we define the average local velocity in QM by dividing th e probability current by the probability:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 83 "v_loc:=proc(x0) local valf,valf_d: global f_Right; \+ valf:=evalf(eval(f_Right,x=x0));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "valf_d:=psi_d(x0);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "-I/2*(conjug ate(valf)*valf_d-conjugate(valf_d)*valf)/(conjugate(valf)*valf); end: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "v_loc(0.1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0$pdbaX@=!#;$!\"!\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "plot(Re(v_loc(s/(2*Nw))),s=-Nw*Pi.. 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It describes net flow." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "In the atomic physics context: s-sta tes and generally " }{TEXT 19 3 "m=0" }{TEXT -1 85 " states are real-v alued and have no local quantum velocity associated with them. The " } {TEXT 19 3 "m>0" }{TEXT -1 77 " states have a net current flow about t he z-axis in one direction, while the " }{TEXT 19 3 "m<0" }{TEXT -1 296 " states have a net current in the opposite directions giving rise to magnetic moment interactions. In that sense these states involve c urrent loops. Likewise, in the present problem there is net flow to th e right (or to the left if the sign of the imaginary part of the wavef unction is reversed)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 11 "Exercise 1:" }}{PARA 0 "" 0 "" {TEXT -1 88 "Compile i nformation about the solutions with more wells (e.g., 5). Based on the case of " }{TEXT 19 4 "Nw=3" }{TEXT -1 102 " make predictions concern ing the nature of the states and then proceed to check out these predi ctions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 420 "We now proceed with a demonstration how energy bands emerge when \+ the potential chain is made up of more wells. We just compute the ener gy eigenvalues. The code is just a copy of the eigenvalue calculation \+ provided above. When increasing the number of wells we get more replic as of the near-degenerate ground states. Thus, we have to increase the matrix size to obtain sufficiently many (and also accurate) eigenener gies." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "restart; Digits:=15: with(LinearAlgebra): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "a:=Pi;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG%#PiG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 " We selected a standard interval for the dynamical variable, i.e., " } {TEXT 19 8 "[-Pi,Pi)" }{TEXT -1 125 ". The re-scaled potential energy \+ is chosen such that in the oscillator limit the eigenenergies are give n as the odd integers." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "V0 :=2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#V0G\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "V:=x->V0*(1-cos(Pi*x/a));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"VGj+6#%\"xG6\"6$%)operatorG%&arrowGF(*&%#V0G\" \"\",&F.F.-%$cosG6#*(%#PiGF.9$F.%\"aG!\"\"F7F.F(F(F(6#\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Vmax:=V(a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%VmaxG\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "We pick an odd number of wells:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "Nw:=11;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#NwG\"#6 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "Vmw:=piecewise(seq(op([ x>a*(2*j-1) and x%$VmwG-%*PIECEWISEG6-7$,&\"\"#\"\"\"* &F*F+-%$cosG6#%\"xGF+!\"\"32,$*&\"#6F+%#PiGF+F1F02F0,$*&\"\"*F+F7F+F17 $F)32F9F02F0,$*&\"\"(F+F7F+F17$F)32F@F02F0,$*&\"\"&F+F7F+F17$F)32FGF02 F0,$*&\"\"$F+F7F+F17$F)32FNF02F0,$F7F17$F)32FUF02F0F77$F)32F7F02F0,$*& FPF+F7F+F+7$F)32FhnF02F0,$*&FIF+F7F+F+7$F)32F^oF02F0,$*&FBF+F7F+F+7$F) 32FdoF02F0,$*&F;F+F7F+F+7$F)32FjoF02F0,$*&F6F+F7F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "PTS:=[seq((2*j-1)*a,j=-(Nw-1)/2..(Nw+1)/2 )];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$PTSG7.,$*&\"#6\"\"\"%#PiGF)! \"\",$*&\"\"*F)F*F)F+,$*&\"\"(F)F*F)F+,$*&\"\"&F)F*F)F+,$*&\"\"$F)F*F) F+,$F*F+F*,$*&F7F)F*F)F),$*&F4F)F*F)F),$*&F1F)F*F)F),$*&F.F)F*F)F),$*& F(F)F*F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "Pmw:=plot(Vmw ,x=PTS[1]..PTS[nops(PTS)],numpoints=500):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 35 "psi:=n->eval(exp(I*Pi*n*x/(Nw*a)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$psiGj+6#%\"nG6\"6$%)operatorG%&arrowGF(-%%evalG 6#-%$expG6#*.%#PiG\"\"\"9$F4%\"xGF4%#NwG!\"\"%\"aGF8^#F4F4F(F(F(6#\"\" !" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "L:=int(conjugate(psi(1 ))*psi(1),x=PTS[1]..PTS[nops(PTS)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"LG,$*&\"#A\"\"\"%#PiGF(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Here we pick the matrix size:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "n_max:=50; N:=2*n_max+1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&n_maxG\"#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG \"$,\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "VME:=unapply(int( V(x)*exp(Pi*I*dn*x/(Nw*a)),x=x1..x2),x1,x2,dn);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$VMEGj+6%%#x1G%#x2G%#dnG6\"6$%)operatorG%&arrowGF*,$* *\"#A\"\"\",2*(^#!\"\"F1-%$expG6#*(^##F1\"#6F19&F19$F1F1)F=\"\"#F1F1*& ^#\"$@\"F1F6F1F1**F?F1F6F1-%$cosG6#F>F1^#F1F1F1**F " 0 "" {MPLTEXT 1 0 51 "H:=Matrix(N,N,shap e=hermitian,datatype=complex[8]):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "We should have a " }{TEXT 19 23 "1/sqrt(2*a) = 1/sqrt(L)" }{TEXT -1 65 " normalization factor! We can fix this by dividing the matrix b y " }{TEXT 19 7 "L=Nw*2a" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "for i1 from 1 to N do: n1:=-n_max+i1-1: for i2 from \+ 1 to i1 do: n2:=-n_max+i2-1: if i2=i1 or n1-n2=Nw then" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 203 "H[i1,i2]:=evalf(1/L*add(limit(VME(PTS[i0],PTS [i0+1],s),s=n1-n2),i0=1..Nw)): else H[i1,i2]:=evalf(1/L*add(VME(PTS[i0 ],PTS[i0+1],n1-n2),i0=1..Nw)): fi: od: H[i1,i1]:=H[i1,i1]+evalf(4*n1^2 *Pi^2/(L)^2): od: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Eigenvalues(H ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "EV:=convert(%,list);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%#EVG7aq$\"0E6c9.()H*!#:$\"0,e.>%*GI* F($\"0'*e.>%*GI*F($\"0V]znsTJ*F($\"0i]znsTJ*F($\"0fulXC!H$*F($\"0#fdcW -H$*F($\"0q#p-VvU$*F($\"06$p-VvU$*F($\"02p'[1&4N*F($\"0Hr'[1&4N*F($\"0 0pR%o #QF?$\"0ex$>R%o#QF?$\"0Ti)3U^gRF?$\"0_i)3U^gRF?$\"0,iJ+!>CTF?$\"09iJ+! >CTF?$\"09wI\"HG#G%F?F`o$\"0b\\SF\"32ZF?$\"0k\\SF\"32ZF?$\"0Yq$*yZc\" \\F?$\"0bq$*yZc\"\\F?$\"0IC%)H**G=&F?$\"0MC%)H**G=&F?$\"0\\;\"fB/zaF?$ \"0g;\"fB/zaF?$\"08l4-`vz&F?$\"0Kl4-`vz&F?$\"0_RCq@I6'F?$\"0j/[VnC;'F? $\"0'y%p%3<(\\'F?$\"0'z%p%3<(\\'F?$\"08yR,[Q(oF?$\"0MyR,[Q(oF?$\"0v/&[ X$*osF?$\"0&\\][X$*osF?$\"0;dx+**=o(F?$\"0ddx+**=o(F?$\"0@m'ROY7\")F?$ \"0\\m'ROY7\")F?$\"0ZC4\"pXg&)F?$\"0kC4\"pXg&)F?$\"0R$40.QF!*F?$\"0d$4 0.QF!*F?$\"0q*[&4c'4&*F?$\"03!\\&4c'4&*F?$\"0$[:R4!4+\"!#8$\"0&[:R4!4+ \"F\\s$\"0hiwXOD0\"F\\s$\"0kiwXOD0\"F\\s$\"08GmF\\ s$\"0ccR\"pBi>F\\s$\"0DcU#fxQ?F\\s$\"0PcU#fxQ?F\\s$\"0[A>m#)p6#F\\s$\" 0]A>m#)p6#F\\s$\"0:,>7co>#F\\s$\"0D,>7co>#F\\s$\"0PpzQ&RyAF\\s$\"0YpzQ &RyAF\\s" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "PEV:=plot([seq( EV[i],i=1..N)],x=PTS[1]..PTS[nops(PTS)],color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "plots[display](Pmw,PEV, title=\"periodic \+ stitching at the ends\",view=[-L/2..L/2,0..1.2*Vmax]);" }}{PARA 13 "" 1 "" {GLPLOT2D 1037 356 356 {PLOTDATA 2 "6eq-%'CURVESG6$7]^m7$$!3;.([* =>vbM!#;$\"\"%\"\"!7$$!3I?'f*G/ETMF*$\"3'\\h@sRO!zR!#<7$$!3z*o05Y^'GMF *$\"3yUTP9\\+FRF37$$!32?2\"yNrWT$F*$\"3(H$GwVx*>$QF37$$!3\")>rofs>+MF* $\"3A$p9+eC#*p$F37$$!31S-'Q+\"*fQ$F*$\"3+'y]UXgF`$F37$$!3!)*R\"=')*>GP $F*$\"3mS5J'yb2N$F37$$!3sz+Fg@=fLF*$\"3P(o*puloPJF37$$!31![.7!z2XLF*$ \"3aDo^Nz:&*GF37$$!39!euY()=5L$F*$\"3$*yW<4#4dj#F37$$!3D!f:yVdlJ$F*$\" 3=I'pE0*zbBF37$$!3uf'ejz>QI$F*$\"3qDb_VZ!H5#F37$$!3=I.)*G-[*G$F*$\"3l; _%ejy\"3vKF*$\"3G-b<&)QbK:F37$$!37+)*G5i?hKF*$\"3y:5; :j3o7F37$$!35g\"4>'eg[KF*$\"3Fvm?xK)*R5F37$$!3B!z_*GGiLKF*$\"39mgAK*)R ))y!#=7$$!3.q5rM-$4A$F*$\"3Ym@bxT8rfFip7$$!3#*z$4-tlh?$F*$\"3dmEiPczES Fip7$$!3#*pj(>k'4$>$F*$\"3'*='e7$$!3)*\\B(Ga!)z:$F*$\"3;*Qsh-C'zE Fcr7$$!3%)z1a\"Hd3:$F*$\"3i\"3P')*Q@x&)!#?7$$!3\"**zzJ#fJWJF*$\"3FCO+Q *QdT(!#@7$$!3$*H*=[bux8$F*$\"3)Q\\nbt5wX\"F^s7$$!34gNL0)=28$F*$\"3sA9h [6C\"=\"Fcr7$$!3%*z\"[e0jO7$F*$\"3*pwxe.-h?$Fcr7$$!3-+@Qu^+4JF*$\"3C_I \"*)HzD0\"Fip7$$!3++kZXaC'4$F*$\"3[7Pnc6P@?Fip7$$!3))RSp)ekC3$F*$\"3=z +K0,X&R$Fip7$$!3/g+=:wAoIF*$\"3u$>Kyu(HX^Fip7$$!3))\\499&*HaIF*$\"3I-+ 2%RN$[rFip7$$!3))4g@gL#3/$F*$\"3)Hir@(*RPK*Fip7$$!335P!*)Qge-$F*$\"3=) yT#)f9k>\"F37$$!30qXn$\\:C,$F*$\"3#\\mPqaj\"\\9F37$$!3%*R?E)>g!)*HF*$ \"3g>I'fW%))HF37$$!3/+K'H&4$3(HF*$ \"3=bXH,%)ysAF37$$!3)*Rz$))*)\\u&HF*$\"3*yuX8ZzY`#F37$$!3&*\\:qTGYVHF* $\"3cvm]j*Q\")z#F37$$!3%*R%\\n$zyHHF*$\"3^-zZjIoSIF37$$!3-?))ey:Z:HF*$ \"3-q%3'*\\8PF$F37$$!3!*>19CKo,HF*$\"3i8u\"\\GjNZ$F37$$!3/q6ASEe()GF*$ \"3)f/b;\"R)*[OF37$$!3;+eD;))ftGF*$\"3S%*\\P9Pj!z$F37$$!3')4aLM\"\\2'G F*$\"3CX)fB&*G+*QF37$$!3;!yo([>-YGF*$\"3)H$*4B#falRF37$$!3,]8gjeVRGF*$ \"3/zzXW8h&)RF37$$!3&)>RVy(\\G$GF*$\"3Orn()))p1(*RF37$$!3%)z6(zjFe#GF* $\"3'))e]@;U(**RF37$$!3))H%3v\\0)=GF*$\"3!psXocgD*RF37$$!3)*\\138W37GF *$\"3'zG\"\\7r[wRF37$$!33qGlGLO0GF*$\"3i.kvlFvV!z#F*$ \"3?egAFYokQF37$$!3'*fKr!)G^xFF*$\"3O1bn$yEfv$F37$$!3/gs#3HVGw#F*$\"3m %H$*pb:rf$F37$$!3')*Hx@bt%\\FF*$\"3'[-o3*=RAMF37$$!3%)zEBf\"\\[t#F*$\" 3Fc[TSQ\"yIAs#F*$\"3Ci9*z8D;*HF37$$!3#*fyq#F*$\"3' \\\\?S\")fEt#F37$$!3-]s:\"QyRp#F*$\"3o`v`7'4\"oCF37$$!3!**ef)3f5!o#F*$ \"3ijWFbSt%>#F37$$!3!*fY)3Z%GmEF*$\"3QEu+mnj=>F37$$!3=q/&fR1Il#F*$\"33 x/M5.za;F37$$!3&)zKIXClQEF*$\"3]O*)e9-aw8F37$$!3%*>Ja*3s\\i#F*$\"3985` $)=?B6F37$$!3**zLHfTd5EF*$\"3fuSW*Q$pDF*$\"39z'y-!yy hIFip7$$!3'*>y@nadbDF*$\"3%odlSLzGw\"Fip7$$!3+I![>Y*>TDF*$\"3!34.t(**p ZxFcr7$$!37S)zijez_#F*$\"3WZdobsZ_@Fcr7$$!3&*HM&zUy6_#F*$\"3cq*e&Q;bWi F^s7$$!38?qi>#)R9DF*$\"3'=70&4-ej7Fds7$$!39Sl*R17p]#F*$\"3'paIfD9i/%F^ s7$$!3')\\gO3fU*\\#F*$\"3\")o`pmwm9>Fcr7$$!33S&**oqpe[#F*$\"3UxSZ7t8ju Fcr7$$!3')\\To^f+sCF*$\"3!=w\\3eG!z;Fip7$$!3++ctpo!zX#F*$\"3'y3H8;;!)) 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64" "Curve 65" "Curve 66" "Curve 67" "Curve 68" "Curve 69" "Curve 70" "Curve 71" "Curve 72" "Curve 73" "Curve 74" "Curve 75 " "Curve 76" "Curve 77" "Curve 78" "Curve 79" "Curve 80" "Curve 81" "C urve 82" "Curve 83" "Curve 84" "Curve 85" "Curve 86" "Curve 87" "Curve 88" "Curve 89" "Curve 90" "Curve 91" "Curve 92" "Curve 93" "Curve 94 " "Curve 95" "Curve 96" "Curve 97" "Curve 98" "Curve 99" "Curve 100" " Curve 101" "Curve 102" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 158 "We obs erve that for the energy levels that correspond to excitations in the \+ single-well problem the spread of eigenvalues is larger than for the g round state." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "An investigation of the energy levels around " }{TEXT 19 5 "E=2 .6" }{TEXT -1 22 " shows that there are " }{TEXT 19 5 "Nw=11" }{TEXT -1 120 " states which come as 5 degenerate pairs plus one extra state. The extra state is at the top of the 'band' in this case." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 115 "For each energy \+ the wavefunctions are delocalized. One can say that tunneling plays an important role all the time." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 97 "Even at energies above the potential barr iers the energy spectrum displays a band structure with " }{TEXT 19 2 "Nw" }{TEXT -1 73 " states per band. For a (one-dimensional) macroscop ic lattice the number " }{TEXT 19 2 "Nw" }{TEXT -1 173 " is huge and t he energy bands are practically continuous. The problem of conductivit y (metals, insulators, semiconductors) can be understood in terms of t his band structure." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 258 11 "Exercise 2:" }}{PARA 0 "" 0 "" {TEXT -1 130 "Explore t he band structure by increasing the number of wells. Will the gaps fil l in as a result of increasing the number of wells " }{TEXT 19 2 "Nw" }{TEXT -1 107 "? Make sure the matrix size is sufficiently large to yi eld enough eigenvalues and to ensure their accuracy." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 261 8 "Summary:" }}{PARA 0 "" 0 "" {TEXT -1 139 "The Bloch theorem strictly speaking is applicable f or an ifinite sequence of potential wells, in which case continuous en ergy bands emerge." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 291 "For a periodic chain of potential wells we can construct eigenstates that resemble the Bloch states in the sense that the dege nerate energy levels allow eigenstates which have identical density ov er each well, and which correspond to net flow of particle density to \+ the right or to the left." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 285 "The periodic boundary condition through which the lattice is stitched to form a ring conflicts with the Bloch condition . This is (probably) the reason for the possibility of eigenstates tha t don't agree with the Bloch theorem with densities that aren't necess arily periodic. For large " }{TEXT 19 2 "Nw" }{TEXT -1 56 " the violat ion of the Bloch condition should not matter." }}}}{MARK "0 0 0" 32 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }