Make a plot of E vs. q for the band structure in Section 5.3.2. Use α = 1 (in units where m = h = a = 1 ). Hint: In Mathematica, ContourPlot will graph E(q) as defined implicitly by Equation 5.71. On other platforms the plot can be obtained as follows: Choose a large number (say 30,000) of equally

Question

Make a plot of E vs. q for the band structure in Section 5.3.2. Use α = 1 (in units where [latex] m=\hbar=a=1 [/latex] ). Hint: In Mathematica, ContourPlot will graph E(q) as defined implicitly by Equation 5.71. On other platforms the plot can be obtained as follows:

Choose a large number (say 30,000) of equally-spaced values for the energy in the range E = 0 and E = 30.

For each value of E, compute the right-hand side of Equation 5.71. If the result is between -1 and 1 , solve for q from Equation 5.71 and record the pair of values [latex] \{q, E\} ext { and }\{-q, E\} [/latex] (there are two solutions for each energy).

You will then have a list of pairs [latex] \{\{q 1, E 1\},\{q 2, E 2\}, \ldots\} [/latex] which you can plot.

[latex] \cos (q a)=\cos (k a)+\frac{m \alpha}{\hbar^{2} k} \sin (k a) [/latex] (5.71).

Accepted Answer

[latex] ext { ContourPlot }[\operatorname{Cos}[x]=\operatorname{Cos}[\sqrt{2 y}]+(\operatorname{Sin}[\sqrt{2 y}]) / \sqrt{2 y},\{x,-\pi, \pi\},\{y, 0,50\}] [/latex].

Question 5.28: Make a plot of E vs. q for the band structure in Section 5.3...

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