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.Consider hypothetical one-dimensional crystal with lattice constant a. (a) Which of the following wave functions are valid Bloch states for electrons in the periodic potential of this crystal? For each

case, why do you think so? \psi_{1}(x)=A e^{i k x} \sin \left(\frac{1}{2} \frac{\pi x}{a}\right) \psi_{2}(x)=A e^{i k x} \sin \left(\frac{\pi x}{a}\right) \psi_{3}(x)=A e^{i k x} \sin \left(2 \frac{\pi x}{a}\right) Explain why wavefunctions with a form cos (pi pi /4) and sin (pi x/a) are appropriate for electrons with k = pi/a in this crystal O Explain why the two wave-functions in part (b) would be expected to have different energies.[2] d) Pt can form one-dimensional conducting chains within certain organic molecules.Use the following data for Pt to estimate the value of wave vector k at which the energies of an electron and a phonon are equal: effective electron mass m* = 13me; atomic mass 175u; spring constant between atoms K=2.9 ms atomic radius is 175 pm

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