{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 "" -1 256 "" 1 16 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 0 1 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 } {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 }} {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;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "V0:=2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "V:=x->V0*(1-cos(Pi*x/a));" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 11 "Vmax:=V(a);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "We pick an odd number of wells:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "Nw:=3;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "V mw:=piecewise(seq(op([x>a*(2*j-1) and x " 0 "" {MPLTEXT 1 0 44 "PTS:=[ seq((2*j-1)*a,j=-(Nw-1)/2..(Nw+1)/2)];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "We want to 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=500):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "A p eriodic basis allows for left- and right-travelling states with period icity 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 "ps i:=n->eval(exp(I*Pi*n*x/(Nw*a)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "int(conjugate(psi(0))*psi(2),x=PTS[1]..PTS[nops(PTS)] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "L:=int(conjugate(psi( 1))*psi(1),x=PTS[1]..PTS[nops(PTS)]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 122 "The kinetic energy for the plane-wave basis functions ca n be calculated in closed form. As an example we 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;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "4*3^2*Pi^2/(L)^2; " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 131 "Here we select the basis siz e. Note that the index labeling the plane-wave basis states ranges ove r positive and negative integers " }{TEXT 19 1 "n" }{TEXT -1 63 " (cor responding to right-travelling and left-travelling waves)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "n_max:=20; N:=2*n_max+1;" }}} {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);" }}}{EXCHG {PARA 0 "> " 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 potential we put th e strength into V(x)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "H: =Matrix(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/s qrt(L)" }{TEXT -1 105 " normalization factor in front of the plane-wav e 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);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 118 "Compared to the single-well energ y spectrum we observe that the lowest oscillator states are approximat ely replicated " }{TEXT 19 2 "Nw" }{TEXT -1 95 " times. There is a uni que ground state, and a pair of eigenvalues that appear to be degenera te." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 213 "W e could construct the eigenfunctions from the Eigenvectors command. Th e calculations with these states are tedious (slow). We prefer to conn ect with the Mathieu differential equation. This requires us to rescal e " }{TEXT 19 9 "xi=2*x*Nw" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 13 "q:=2*V0*Nw^2;" }}}{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;" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "Here " }{TEXT 19 1 "a" }{TEXT -1 39 " is the eigenvalue scaled according to " }{TEXT 19 11 "a=4*(EV-V0) " }{TEXT -1 39 ", and the potential strength 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);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "sol:=dsolve(MDE(aM(EV[1])));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "sol:=u(x) = _C1*MathieuC(-7 7049338729459/18000000000000,-4,3*x)+_C2*MathieuS(-77049338729459/1800 0000000000,-4,3*x);" }}}{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 antisymmetric 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 an tisymmetric, as the Hamiltonian is symmetric. The ground state is symm etric." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "plot([eval(rhs(s ol),\{_C1=1,_C2=0\}),eval(rhs(sol),\{_C2=1,_C1=0\})],x=-Pi/2..Pi/2,col or=[red,blue],view=[-Pi/2..Pi/2,-q/10..q/10]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 127 "The antisymmetric solution is not acceptable, it is not periodic. The ground state is unique, it has a symmetric eigenfun ction." }}{PARA 0 "" 0 "" {TEXT -1 122 "Let us plot a probability usin g the eigenenergy as a baseline, and superimpose it on the potential a s a function of theta." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 " P0:=plot([EV[1],EV[1]+(eval(rhs(sol),\{_C1=1,_C2=0,x=s/(2*Nw)\}))^2],s =-Nw*Pi..Nw*Pi,color=[black,green]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "plots[display](Pmw,P0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "sol:=dsolve(MDE(aM(EV[2])));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "sol:=u(x) = _C1*MathieuC(-191905256659097/45000 000000000,-4,3*x)+_C2*MathieuS(-191905256659097/45000000000000,-4,3*x) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 233 "We might expect the first excited state is anti-sym metric. We find, however, since the 2nd and 3rd eigenenergies are so c lose that there have to be 2 eigenfunctions at this energy. They are t he 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]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "Is it true that both the symmetric and t he antisymmetric solutions 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\});" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 36 "fA:=eval(rhs(sol),\{_C1=0,_C2=1E-2\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "evalf(limit(fS,x=-Pi/2));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "evalf(limit(fS,x=Pi/2));" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "evalf(limit(diff(fS,x),x=-P i/2));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "evalf(limit(diff( fS,x),x=Pi/2));" }}}{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));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "evalf(limit(fA,x=Pi/2));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "evalf(limit(diff(fA,x),x=-Pi/2));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "evalf(limit(diff(fA,x),x=Pi/2));" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 179 "The anti-symmetric solution also \+ works! This is a feature that was observed for the rotator states in t he single-well problem; in fact the lowest rotator states didn't displ ay it." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 299 "At this point we should invoke physics as a guide. On the one han d we might argue that there is a symmetric and an antisymmetric state. The antisymmetric state, e.g., violates the expectation that the prob ability density is the same in all three wells, as the middle one come s out practically empty:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "P1:=plot([EV[2],EV[2]+(eval(rhs(sol),\{_C1=0,_C2=1E-2,x=s/(2*Nw) \}))^2],s=-Nw*Pi..Nw*Pi,color=[black,magenta]):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 26 "plots[display](Pmw,P0,P1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 194 "We can also argue that since the energy levels a re degenerate, any linear combination of the two eigenfunctions found \+ (and associated with this energy value) is also an acceptable eigenfun ction." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 "In particular, we can construct a left-travelling and a right-travell ing solution by using " }{TEXT 19 2 "+I" }{TEXT -1 5 " and " }{TEXT 19 2 "-I" }{TEXT -1 206 " in front of the anti-symmetric state, and a \+ real constant in front of the symmetric one. We apply a trial-and-erro r strategy to ensure an equal peak height over the three wells for the probability density." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "ev al(fS,x=0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "f_Right:=fS+ I*0.435*fA;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "P1:=plot([E V[2],EV[2]+(eval(abs(f_Right)^2,\{x=s/(2*Nw)\}))^2],s=-Nw*Pi..Nw*Pi,co lor=[black,magenta]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "pl ots[display](Pmw,P0,P1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 176 "To p rove that these states represent travelling electrons (around the ring made up of three sites) we have to calculate the probability current \+ density (we use units with mass " }{TEXT 19 3 "m=1" }{TEXT -1 2 ")." } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "jPC:=psi->-I/2*(conjugate( psi)*diff(psi,x)-psi*diff(conjugate(psi),x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "#evalf(limit(jPC(f_Right)/f_Right,x=0.1));" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "Maple is having a hard time. Let's calculate the derivative numerically:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "eta:=1E-3;" }}}{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;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 127 "Based on the current d ensity we define the average local velocity in QM by dividing the prob ability current by the probability:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "v_loc:=proc(x0) local valf,valf_d: global f_Right; va lf:=evalf(eval(f_Right,x=x0));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "v alf_d:=psi_d(x0);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "-I/2*(conjugat e(valf)*valf_d-conjugate(valf_d)*valf)/(conjugate(valf)*valf); end:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "v_loc(0.1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "plot(Re(v_loc(s/(2*Nw))),s=-Nw*Pi.. Nw*Pi);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 137 "The local velocity is very high when the particle is tunneling through one of the barriers! What is it near the minimum in the potential?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "plot(Re(v_l oc(s/(2*Nw))),s=-Nw*Pi..Nw*Pi,view=[-Nw*Pi..Nw*Pi,0..0.1],numpoints=50 0);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 137 "This may seem strange: th e quantum local velocity is lowest near the potential minima where a c lassical particle would be moving fastest." }}{PARA 0 "" 0 "" {TEXT 259 9 "Question:" }{TEXT -1 105 " does the high local velocity inside \+ the potential barrier have anything to say about the tunneling time?" }}{PARA 0 "" 0 "" {TEXT -1 163 "Concerning the interpretation of the l ocal quantum velocity keep in mind that it is zero for oscillators (th e eigenfunction is real-valued). It describes net flow." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "In the atomic physic s context: s-states and generally " }{TEXT 19 3 "m=0" }{TEXT -1 85 " s tates are real-valued and have no local quantum velocity associated wi th them. The " }{TEXT 19 3 "m>0" }{TEXT -1 77 " states have a net curr ent flow about the z-axis in one direction, while the " }{TEXT 19 3 "m <0" }{TEXT -1 296 " states have a net current in the opposite directio ns giving rise to magnetic moment interactions. In that sense these st ates involve current loops. Likewise, in the present problem there is \+ net flow to the right (or to the left if the sign of the imaginary par t of the wavefunction is reversed)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT 258 11 "Exercise 1:" }}{PARA 0 "" 0 "" {TEXT -1 88 "Compile information about the solutions with more wells (e.g., \+ 5). Based on the case of " }{TEXT 19 4 "Nw=3" }{TEXT -1 102 " make pre dictions concerning the nature of the states and then proceed to check out these predictions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 420 "We now proceed with a demonstration how energy band s emerge when the potential chain is made up of more wells. We just co mpute the energy eigenvalues. The code is just a copy of the eigenvalu e calculation provided above. When increasing the number of wells we g et more replicas of the near-degenerate ground states. Thus, we have t o increase the matrix size to obtain sufficiently many (and also accur ate) eigenenergies." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "restart; Digits:=15: with(Li nearAlgebra):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "a:=Pi;" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "We selected a standard interval fo r the dynamical variable, i.e., " }{TEXT 19 8 "[-Pi,Pi)" }{TEXT -1 125 ". The re-scaled potential energy is chosen such that in the oscil lator limit the eigenenergies are given as the odd integers." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "V0:=2;" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 25 "V:=x->V0*(1-cos(Pi*x/a));" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 11 "Vmax:=V(a);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "We pick an odd number of wells:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "Nw:=11;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "Vmw:=piecewise(seq(op([x>a*(2*j-1) and x " 0 "" {MPLTEXT 1 0 44 "PTS:=[seq((2*j-1)*a,j=-(Nw-1)/2..(Nw+1)/2)];" }}}{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)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "L:=in t(conjugate(psi(1))*psi(1),x=PTS[1]..PTS[nops(PTS)]);" }}}{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;" }}} {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);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "H:=Matrix(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/sqrt(L)" }{TEXT -1 65 " normalization factor! We c an 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 t o N do: n1:=-n_max+i1-1: for i2 from 1 to i1 do: n2:=-n_max+i2-1: if i 2=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)): els e 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);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "PEV:=plot([seq(EV[i],i=1..N)],x=PTS[1]..PTS[nops(PTS)],color=blu e):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "plots[display](Pmw,P EV, title=\"periodic stitching at the ends\",view=[-L/2..L/2,0..1.2*Vm ax]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 158 "We observe that for the energy levels that correspond to excitations in the single-well probl em the spread of eigenvalues is larger than for the ground state." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "An invest igation 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 stat e 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 wavefuncti ons are delocalized. One can say that tunneling plays an important rol e all the time." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "Even at energies above the potential barriers the energy \+ spectrum displays a band structure with " }{TEXT 19 2 "Nw" }{TEXT -1 73 " states per band. For a (one-dimensional) macroscopic lattice the \+ number " }{TEXT 19 2 "Nw" }{TEXT -1 173 " is huge and the energy bands are practically continuous. The problem of conductivity (metals, insu lators, semiconductors) can be understood in terms of this band struct ure." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 11 " Exercise 2:" }}{PARA 0 "" 0 "" {TEXT -1 130 "Explore the band structur e by increasing the number of wells. Will the gaps fill 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 yield enough eigen values 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 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 8 "Summary:" }}{PARA 0 "" 0 "" {TEXT -1 139 "The Bloch theor em strictly speaking is applicable for an ifinite sequence of potentia l wells, in which case continuous energy bands emerge." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 291 "For a periodic chai n of potential wells we can construct eigenstates that resemble the Bl och states in the sense that the degenerate energy levels allow eigens tates which have identical density over each well, and which correspon d 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 peri odic boundary condition through which the lattice is stitched to form \+ a ring conflicts with the Bloch condition. This is (probably) the reas on for the possibility of eigenstates that don't agree with the Bloch \+ theorem with densities that aren't necessarily periodic. For large " } {TEXT 19 2 "Nw" }{TEXT -1 56 " the violation of the Bloch condition sh ould not matter." }}}}{MARK "105" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }