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Essentially, this posting is asking the following: Solve the Schrodinger equation with different potentials using the Fourier transform.© BrainMass Inc. brainmass.com October 16, 2018, 4:28 pm ad1c9bdddf
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The solution is attached below in two files. ...
Fourier transform is a powerful tool when used to solve ordinary and partial differential equations. This solution shows step by step how to go from the initial equation, utilizing the Fourier transform, to the inverse transform and complex contour integration to reach a solution (in an integral form). The solution contains 4 pages with full derivations and is provided in both a pdf. and Word document.
Band structure and a weak periodic potential
4. Discuss briefly the origin of the differing electronic properties of metals, semiconductors and insulators. Under what circumstances will the nearly-free electron model be useful for describing the band structure of a solid?
A weak periodic potential
W(z) = W_0 cos (2*pi*z)/a
with W_0 > 0, is imposed on a one-dimensional free-electron gas lying along the z direction. Discuss the form of the resulting wavefunctions at the first Brillouin zone boundary. Explain why a gap arises.
A simple model of a two-dimensional band structure can be obtained by summing together two one-dimentional band structures. Consider the energy band
E(k_x, k_y) = U cos (k_x * a) - U cos (k_y * a)
with U > 0, corresponding to a particular two-dimensional crystal with square lattice and period a. Give the wavevectors and energies of the highest and lowest energy states in teh band. Draw labelled constant energy contours in the first Brillouin zone near these particular states. Comment on the form of the constant-energy contours near the centre fo teh first Brillouin zone.View Full Posting Details