Question

1 Particle in a Box with a Bump (based on B&J 4.11) Consider a particle of mass m in a 1-D double well with potential given by Vo, 05\x\<b V(x) = { 0, b<\x<c 100, [x]>c . We will study the lowest energy states, for which 0 <E<V, corresponding to tunnelling between the two wells. (a) Write down the time-independent Schödinger equation in the three regions -c<x<-b, –b< <b, and b< I< c. Write down the most general wavefunction solution in each of the regions, as well as for x >c, defining constants k, K, A, B, C, D, F, G as you need. (b) For even energy eigenfunctions * x), how must some of the coefficients be related? Use this to reduce the number of undetermined constants. Do the same for odd energy eigenfunctions yn (). In the even/odd case, write down all the boundary conditions as relations between the coefficients. (c) Use the equations between coefficients you obtained from part (b) to derive an "energy quantisation condition" in the even case. Do the same for the odd case. (Hint: You just need to derive an equation involving k,k (and of course b and c) but does not contain any of the coefficients. Since k, K uniquely determine the energy E, this will be a constraint on the possible values of even/odd energy levels E..] (d) Using L=c-b, O=kL, 0 = 1/2mVo L. KL, 0o = V 12 , rewrite the energy quantisation conditions as - tan0 = oz coth (Vez –02), for even energy eigenfunctions _87 tanh ( V 03 – 02), for odd energy eigenfunctions. -tan = (2) (e) To solve these equations for 0, we must resort to numerical or graphical means. Plot the following three functions on the same graph: - tan, ve - cocott (i vos –ou), vie - en tant (i vos –or).

Answer 1

https://www.desmos.com/calculator. Follow this link for plotting.

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