hw05sol - Introduction to Quantum and Statistical Mechanics...

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Introduction to Quantum and Statistical Mechanics Homework 5 Solution Prob. 5.1 A quantum well laser is operated by electrons relaxing from the first excited state E 2 to the ground state E 1 in a finite-potential well, which is formed by a superlattice of semiconductors with different bandgap. Assume the electron has an effective mass m * of 0.067 m 0 where m 0 is the rest mass of electrons in vacuum. Given the well width as L and well depth as V o , (a) To avoid transition in other levels, it is desirable to have exactly two levels in the quantum well. Express the relation of V o with L and with 2 * 2 2 2 L m E L π = (8 pts) E L ≤ V o < 4E L . (b) Will E 2 = 4E 1 ? Will this be a reasonable approximation? (6 pts) From the plot of the transcendental equation, E 2 will not be exactly four times of E 1 , but however, the closer V o is to 4E L , the better the approximation. For example, when V o = 3E L , E 1 = 0.53E L and E 2 = 1.96E L which are pretty close to 4 times. (c) If the emitted wavelength is at 10.6µm (middle infrared), use E 2 and E 1 close to the infinite well levels to obtain an estimate of the quantum well width in nm. (8 pts) 2 * 2 2 1 2 2 3 L m E E c h λ = - = ; L = 6.9nm. (d) Graphically justify that the approximation of using E 2 and E 1 from the infinite potential well is acceptable from the transcendental equation plot. Notice that this approximation makes the dependence on V o negligible. ( 8 pts ) x=0 x=L/2 V o x=-L/2 E 2 E 1 1
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Prob. 5.2
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hw05sol - Introduction to Quantum and Statistical Mechanics...

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