Question

Consider the Kronig-Penney model discussed in the lectures, where the periodic potential corresponds to an array of delta functions: However, unlike the usual model, we will take α < 0, so that we have potential wells rather than barriers. In the following, we aim to solve the time-independent Schrödinger equation 2m r (a) First consider the case where the energy E 0. Write down the general solution for ψ(z) within the interval 0 < r < a, and use the boundary conditions to obtain a set of simultancous equstions for the wave function parameters. [Please show your working Show that these equations yield the same implicit equation for the energy as was derived for the case a see E. 11.17) in the notes Plot the energy spectrum of the lowest three positive-energy bands for a Sma (b) Now consider instead the case E < 0. Write down the general solution for ψ(z) within the interval 0 < r < α and use the boundary conditions to derive an implicit equation for the energy, Show that there is only one energy band with E<0. (c) Continuing on from (b), obtain an expression for the wave function and use this to determine the probability current in the lowest band as a function of energy. Discuss why the current can be non-zero even though E <0 (which would correspond to a bound state in the case of a single potential well) Sketch the ground-state wave function across several lattice sites.

Answer 1

'The kromia.fen"สุ nnsdol dacred etechen motion in a krarg lenne rd iunt a special bde of the lenn o(r ν。6 Finite, sn this JImit , each of the maclar?der basries become detta-functio -D the potential eney Sep uol to Lero ngtol each

No-@ uwtth U'뜰 The wave Funmut cadtisrg the Clsch theprem ika 2kq ~ 79久 t Be

amd@ 2 2.f0 29A 8 ika

amd@ 2 2.f0 29A 8 ika

No toe e teo efficienta in frent of the t must mut n214a and the uouc Fun cton be comes. Bountor Condi ton c) the unuefunetn Continuo u ato
​​​
amd@ 2 2.f0 29A 8 ika

No 13 The Cembinadipoma rmuel mst of the Petad'cfetial
​​​​