HW9.nb 1. be a symmetric (parity-even) function, with a close excited state with an anti-symmetric (parity-odd) wave function.

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1 HW9.nb HW #9. Rectngulr Double-Well Potentil Becuse the potentil is infinite for» x» > + b, the wve function should vnish t x = H + bl. For < x < + b nd - - b < x < -, the potentil vnishes nd the wve function is given simply by plne wve e i k x with energy E = k ê m. For - < x <, the potentil is lrge V 0, nd the wve function dmps exponentilly, e k x with k = mhv 0 - EL = m V 0 - k. Becuse the potentil is prity-invrint VH-xL = VHxL, we expect the ground stte to be symmetric (prity-even) function, with close excited stte with n nti-symmetric (prity-odd) wve function. For the prity-even ground stte, the wve function for - < x < must be yhxl = A cosh k x. For <» x» < + b, the wve function must be yhxl = B sin khx - - bl to ensure the boundry condition t» x» = + b. For - - b < x < -, yhxl = yh-xl = - B sin khx + + bl. To mtch the logrithmic derivtive y' HL ê yhl, we y' HL yhl need ÅÅÅ = k tnh k = k cot kh-bl. The conditions for x < 0 re precisely the sme thnks to the prity. For the prity-odd excited stte, the wve function for - < x < must be yhxl = A sinh k x. For < x < + b, the wve function must be yhxl = B sin khx - - bl to ensure the boundry condition t» x» = + b. For - - b < x < -, yhxl = -yh-xl = B sin khx + + bl. To mtch the logrithmic derivtive y' HL ê yhl, we need y' HL ÅÅÅ = k coth k = k cot kh-bl. The conditions t x = - re precisely the sme thnks to the prity. yhl First we tke the limit of the inifinite potenitl brrier V 0 Ø, nd hence k Ø. Then coth k = tnh k =, nd the condition is -k cot k b = k Ø. The only wy to stisfy this eqution is by tking k b = H n + L p so tht cot k b Ø -. The lowest energy is obtined by k b = p. Nmely, to the leding order in lrge V 0, k = ÅÅÅÅ p b nd E = p ÅÅÅÅ m b for both prityeven nd odd sttes. This mkes sense becuse the wve function fits right in between two infinite potentil brriers. Now we mke the potentil brrier finite but lrge (k p ), nd two sttes must split. To see the difference between two energy levels, we note tht the only difference is between tnh k nd coth k, nd for lrge k p, the difference is exponentilly smll, tnh k = - e - k + OHe -4 k L, coth k = + e - k + OHe -4 k L. Therefore we would like to sovle k H e - k L = -k cot k b to find k > ÅÅÅÅ p for prities. We expnd k b = p - e, nd b -k cos k b -k cot k b = ÅÅÅÅÅ = ÅÅÅÅÅÅ k sin k b e + OHeL = ÅÅÅÅÅÅÅ p b e + OHe0 L, nd hence ÅÅÅÅÅÅÅ p = k H e - k L, e = ÅÅÅÅÅÅ p b e k b H e- k L. Therefore the energy eigenvlue is E = p ÅÅÅÅ H - ÅÅÅÅÅÅ m b k b H e- k LL to the leding order. The difference in the energies between the two low-lying sttes is exponentilly suppressed s expected, D E = p ÅÅÅÅÅ 8 ÅÅÅÅÅÅ m b k b e- k. To plot the wve functions, we choose = b =, m =, =, nd V 0 = 0. Much lrger V 0 mkes it impossible to find the difference between the two energy eigenvlues numericlly. We cn solve numericlly for k, k Tnh@k D == -k Cot@k bd ê. 9k ->!!!!!! m V 0 - k = ê. 8 Ø, b Ø, m Ø, Ø < ê. 8V 0 Ø 0< è!!!!!!!!!!!!!! 0 - k TnhA è!!!!!!!!!!!!!! 0 - k E ã -k Cot@kD FindRootA è!!!!!!!!!!!!!! 0 - k TnhA è!!!!!!!!!!!!!! 0 - k E == -k Cot@kD, 8k, <E 8k Ø.576<

2 HW9.nb ksol = k ê. %.576 PlotAIfAAbs@xD <, CoshA "###################### 0 - ksol xe, CoshA "###################### 0 - ksol E ÅÅÅÅÅ Å Sin@ksol HAbs@xD - LDE, 8x, -, <E Sin@ksol HAbs@D - LD Ü Grphics Ü k Coth@k D == -k Cot@k bd ê. 9k ->!!!!!! m V 0 - k = ê. 8 Ø, b Ø, m Ø, Ø < ê. 8V 0 Ø 0< è!!!!!!!!!!!!!! 0 - k CothA è!!!!!!!!!!!!!! 0 - k E ã -k Cot@kD FindRootA è!!!!!!!!!!!!!! 0 - k CothA è!!!!!!!!!!!!!! 0 - k E == -k Cot@kD, 8k, <E 8k Ø.5855< ksol = k ê. %.5855

3 HW9.nb <, SinhA "###################### 0 - ksol xe, SinhA "###################### 0 - ksol E Sign@xD ÅÅÅÅÅ Å Sin@ksol HAbs@xD - LDE, 8x, -, <E Sin@ksol HAbs@D - LD Ü Grphics Ü. Periodic delt-function Potentil () Using the form of the wve function given in the problem, yh-el = A + B yh+el = e i k HA e -i k + B e i k L y' H-eL = i kha + BL y' H+eL = i k e i k HA e -i k - B e i k L The wve function is continuous yh-el = yh+el, but its derivtive is discontinuous becuse of the delt-function potentil, yh+el - yh-el = ÅÅÅ m l yh0l. If you wnt to solve for "k" by hnd the most expedient method is to write the two equtions in the form of mtrix multiplying the column vector (A, B): eqn = Collect@A + B - E I k HA E -I k + B E I k L, 8A, B<D eqn = CollectAI k E I k HA E -I k A H - Â k-â k L + B H - Â k+â k L - B E I k L - I k HA - BL - m l ÅÅÅ HA + BL, 8A, B<E A J-Â k + Â Â k-â k k - ÅÅÅ m l N + B JÂ k - Â Â k+â k k - ÅÅÅ m l N

4 HW9.nb 4 mymtrix = 88Coefficient@eqn, AD, Coefficient@eqn, BD<, 8Coefficient@eqn, AD, Coefficient@eqn, BD<<; mymtrix êê MtrixForm i j k -  k-â k -  k+â k y  k -   k+â k k - m l z - k +   k-â k k - m l The eqution we wnt to solve is thus: mymtrix.8a, B< ã 80, 0< 9A H -  k-â k L + B H -  k+â k L, A J- k +   k-â k k - ÅÅÅ m l N + B J k -   k+â k k - ÅÅÅ m l N= ã 80, 0< For there to be nontrivil solution, the determinnt of the coefficient mtrix must be equl to zero. Mthemtic is not convenient for doing this mnipultion, but it isn't hrd to clculte the x determinnt by hnd nd set it equl to zero. It yields qudrtic eqution for e i k which cn be esily solved using the qudrtic eqution to give the required result. If you wnt to use Mthemtic to rrive t the solution, here is n exmple method: SolveA 9A + B ã E I k HA E -I k + B E I k L, I k E I k HA E -I k - B E I k L - I k HA - BL ã m l ÅÅÅ HA + BL=, 8B, k<e Solve::ifun : Inverse functions re being used by Solve, so some solutions my not be found; use Reduce for complete solution informtion. More 99B Ø ÅÅÅ m l J-A m l - A -  k m l -  A k +  A -  k k -  A -  k "####################################################################################################### ########## -4  k k 4 + H m l -   k m l + k +  k k L N, k Ø - ÅÅÅÅ J LogA- ÅÅÅÅÅ k J - k J- m l +   k m l - k -  k k + "####################################################################################################### ########## -4  k k 4 + H m l -   k m l + k +  k k L NNEN=, 9B Ø ÅÅÅ m l J-A m l - A -  k m l -  A k +  A -  k k +  A -  k "####################################################################################################### ########## -4  k k 4 + H m l -   k m l + k +  k k L N, k Ø - ÅÅÅÅ J LogA- ÅÅÅÅÅ k J - k J- m l +   k m l - k -  k k - "####################################################################################################### ########## -4  k k 4 + H m l -   k m l + k +  k k L NNEN== T h e r e f o r e, e i k = - ÅÅÅÅ k e-i k i j-i m lh - e i k L - kh + e i k L "################################ -4 4 k e i k + ################################ Hi m lh - e i k L ################ + kh ################ + e i k ####### LL y z k = - ÅÅÅÅ i j-m l sin k - k cos k "################################ -4 ############################################# k 4 k + H m l sin k + k cos k L y z k = cos k + m l sin k i - Hcos k + m %%%%%%%%%%%%%%%%%%%% l sin k k k L.

5 HW9.nb 5 As suggested in the problem, we define d = ÅÅÅÅÅÅÅ, nd the expression simplifies to e i k = cos k + ÅÅÅÅÅÅ k d sin k i - Hcos k + ÅÅÅÅÅÅ %%%%%%%%%%%%%%%%% k d sin k L. (b) m l In the limit d Ø, which is nothing but free prticle without potentil, we hve e i k = cos k i - cos!!!!!!! k = e i k, nd hence k = Hk + ÅÅ p n L. Or equivlently, k is the momentum modulo ÅÅ p n. Therefore, k nd hence the energy grows continuously s function of k. This cn be seen with lrge enough d numericlly: PlotA-I ê * LogACos@k D + Sin@k D + I * - i jcos@k D + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% k d k k d Sin@k Dy z E ê. 8 Ø, d Ø 00< ê. 8k Ø p x<, 8x, 0, 6<, PlotRnge Ø 8-Pi, Pi<E PlotA-I ê * LogACos@k D + Sin@k D - I * - i jcos@k D + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% k d k k d Sin@k Dy z E ê. 8 Ø, d Ø 00< ê. 8k Ø p x<, 8x, 0, 6<, PlotRnge Ø 8-Pi, Pi<E  LogACos@k D + Power@áàD áà +  Plus@áàD!!!!!!!!!!!!! E ê. 8 Ø, d Ø 00< ê. 8k Ø p x< is not mchine-size rel number t x =.5`*^-7. More  LogACos@k D + Power@áàD áà +  Plus@áàD!!!!!!!!!!!!! E ê. 8 Ø, d Ø 00< ê. 8k Ø p x< is not mchine-size rel number t x = `. More  LogACos@k D + Power@áàD áà +  Plus@áàD!!!!!!!!!!!!! E ê. 8 Ø, d Ø 00< ê. 8k Ø p x< is not mchine-size rel number t x = `. More Generl::stop : Further output of Plot::plnr will be suppressed during this clcultion. \! \HMore \L Ü Grphics Ü Â Log@Cos@k D + Power@áàD áà -  Power@áàDD ê. 8 Ø, d Ø 00< ê. 8k Ø p x< is not mchine-size rel number t x =.5`*^-7. More  Log@Cos@k D + Power@áàD áà -  Power@áàDD ê. 8 Ø, d Ø 00< ê. 8k Ø p x< is not mchine-size rel number t x = `. More  Log@Cos@k D + Power@áàD áà -  Power@áàDD ê. 8 Ø, d Ø 00< ê. 8k Ø p x< is not mchine-size rel number t x = `. More

6 HW9.nb 6 Generl::stop : Further output of Plot::plnr will be suppressed during this clcultion. \! \HMore \L Ü Grphics Ü Show@%, %%D (c) - Ü Grphics Ü Looking t the eqution e i k = cos k + ÅÅÅÅÅÅ k d sin k i - Hcos k + ÅÅÅÅÅÅ %%%%%%%%%%%%%%%%% k d sin k L, if the rgument of the squre root is negtive, the l.h.s. becomes pure rel nd cnnot stisfy the eqution for rel k. Therefore there is no solution when» cos k + ÅÅÅÅÅÅ sin k» >. When d is finite but lrge, the combintion exceeds unity for k = n p + e (e > 0). This cn be k d seen by expnding it in terms of e, cos Hn p + el = H-L n I - ÅÅÅÅÅ e + OHe4 LM, sinhn p + el = H-L n He + OHe LL, nd hence cos k + ÅÅÅÅÅÅ k d sin k = H-Ln I + ÅÅÅÅÅÅ k d e - ÅÅÅÅÅ e + OHe LM, nd the mgnitude exceeds unity for 0 < e < ÅÅ k d > ÅÅÅÅÅ. The gp n p d must exist just bove k = ÅÅÅÅÅÅÅ n p for ny n, while the gp becomes smller for lrge n. (d) First for wek potentil d =,

7 HW9.nb 7 PlotA-I ê * LogACos@k D + k d 8k Ø p x<, 8x, 0, 4<, PlotRnge Ø 8-Pi, Pi<E Sin@k D + I * - i jcos@k D + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% k k d Sin@k Dy z E ê. 8 Ø, d Ø < ê. PlotA-I ê * LogACos@k D + Sin@k D - I * - i jcos@k D + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% k d k k d Sin@k Dy z E ê. 8 Ø, d Ø < ê. 8k Ø p x<, 8x, 0, 4<, PlotRnge Ø 8-Pi, Pi<E  LogACos@k D + Power@áàD áà +  Plus@áàD!!!!!!!!!!!!! E ê. 8 Ø, d Ø < ê. 8k Ø p x< is not mchine-size rel number t x = `*^-7. More  LogACos@k D + Power@áàD áà +  Plus@áàD!!!!!!!!!!!!! E ê. 8 Ø, d Ø < ê. 8k Ø p x< is not mchine-size rel number t x = `. More  LogACos@k D + Power@áàD áà +  Plus@áàD!!!!!!!!!!!!! E ê. 8 Ø, d Ø < ê. 8k Ø p x< is not mchine-size rel number t x = `. More Generl::stop : Further output of Plot::plnr will be suppressed during this clcultion. \! \HMore \L Ü Grphics Ü Â Log@Cos@k D + Power@áàD áà -  Power@áàDD ê. 8 Ø, d Ø < ê. 8k Ø p x< is not mchine-size rel number t x = `*^-7. More  Log@Cos@k D + Power@áàD áà -  Power@áàDD ê. 8 Ø, d Ø < ê. 8k Ø p x< is not mchine-size rel number t x = `. More  Log@Cos@k D + Power@áàD áà -  Power@áàDD ê. 8 Ø, d Ø < ê. 8k Ø p x< is not mchine-size rel number t x = `. More Generl::stop : Further output of Plot::plnr will be suppressed during this clcultion. \! \HMore \L

8 HW9.nb Ü Grphics Ü Show@%, %%D Ü Grphics Ü There re gps just bove k = ÅÅÅÅÅÅÅ n p the prt (c)., nd the gps become smller for higher n s expected from the nlytic considertions in Now for strong potentil d = ÅÅÅÅ, PlotA-I ê * LogACos@k D + k d Sin@k D + I * - i jcos@k D + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% k k d Sin@k Dy z E ê. 8 Ø, d Ø ê < ê. 8k Ø p x<, 8x, 0, 4<, PlotRnge Ø 8-Pi, Pi<E PlotA-I ê * LogACos@k D + Sin@k D - I * - i jcos@k D + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% k d k k d Sin@k Dy z E ê. 8 Ø, d Ø ê < ê. 8k Ø p x<, 8x, 0, 4<, PlotRnge Ø 8-Pi, Pi<E  LogACos@k D + Power@áàD áà +  Plus@áàD!!!!!!!!!!!!! E ê. 9 Ø, d Ø ÅÅÅÅ = ê. 8k Ø p x< is not mchine-size rel number t x = `*^-7. More  LogACos@k D + Power@áàD áà +  Plus@áàD!!!!!!!!!!!!! E ê. 9 Ø, d Ø ÅÅÅÅ = ê. 8k Ø p x< is not mchine-size rel number t x = `. More  LogACos@k D + Power@áàD áà +  Plus@áàD!!!!!!!!!!!!! E ê. 9 Ø, d Ø ÅÅÅÅ = ê. 8k Ø p x< is not mchine-size rel number t x = `. More

9 HW9.nb 9 Generl::stop : Further output of Plot::plnr will be suppressed during this clcultion. \! \HMore \L Ü Grphics Ü Â Log@Cos@k D + Power@áàD áà -  Power@áàDD ê. 9 Ø, d Ø ÅÅÅÅ = ê. 8k Ø p x< is not mchine-size rel number t x = `*^-7. More  Log@Cos@k D + Power@áàD áà -  Power@áàDD ê. 9 Ø, d Ø ÅÅÅÅ = ê. 8k Ø p x< is not mchine-size rel number t x = `. More  Log@Cos@k D + Power@áàD áà -  Power@áàDD ê. 9 Ø, d Ø ÅÅÅÅ = ê. 8k Ø p x< is not mchine-size rel number t x = `. More Generl::stop : Further output of Plot::plnr will be suppressed during this clcultion. \! \HMore \L Ü Grphics Ü

10 HW9.nb 0 Show@%, %%D Ü Grphics Ü The result is highly distorted from the free-prticle cse. Nonetheless the bnd structure is clerly seen. (e) It is prity, tht chnges the overll sign of k. This cn be seen from the explicit form of the wve function, yhxl = A e i k x + B e -i k x for H- < x < 0L yhxl = e i k HA e i khx-l + B e -i khx-l L for 0 < x <. The prity trnsforms it to yhxl = e i k HA e i kh-x-l + B e -i kh-x-l L = B e i Hk+kL e i k x + A e i Hk-kL e -i k x = A' e i k x + B' e -i k x for H- < x < 0L yhxl = B e i k x + A e -i k x = e -i k HB e i Hk+kL e i k Hx-L + A e i Hk-kL e -i k Hx-L L = e -i k HA' e i khx-l + B' e -i khx-l L 0 < x <. The two wve functions re relted by the chnge A Ø A' = B e i Hk+kL, B Ø A e i Hk-kL, e i k Ø e -i k. f o r