ECE 663 Fall 2002



ECE 6163-1 Fall 2009

Homework Assignment #2 QUANTUM MECHANICS Due 9/9/09

Please show all relevant steps to get full credit for your solution. You are not allowed to use any unauthorized help (e.g. solution manuals, solutions from other years, etc). Any violations will automatically be reported to the honor committee. You are allowed to discuss your solutions in a group, but make sure your solution is your own.

1. The exciton is a hydrogen atom-like entity encountered in advanced semiconductor device work (particularly in III-V optoelectronic devices). It consists of an electron bound to a positively charged particle (a hole) of approximately the same mass and same but opposite charge. The Bohr atom results can be used to compute the allowed energy states etc of the exciton provided that the reduced mass is used mr=m+m-/(m++m-)(m0/2 instead of the electron mass. The (0 used in the Bohr formula must also be replaced by Ks(0 where Ks is the semiconductor dielectric constant. (use Ks=12). Determine the ground state exciton energy (n=1) [10 points]

2. The simple harmonic oscillator has equally spaced eigenvalues that look like (n+1/2) ħω where ω is the angular frequency of the oscillator. Consider now the half-oscillator shown below, whose potential equals a regular oscillator for x > 0 and equals infinity (hard wall) for x < 0. The hard wall imposes additional boundary conditions on the regular oscillator solutions. From this constraint alone, and the above information, draw the eigenvectors (‘modes’) of the half-oscillator and find its eigenvalues (you could run a numerical solution in addition, just for confirmation). Explain the reasoning behind your analysis [10 points]

3. At a time t=0 a particle is represented by the wave function:

[pic]

a) Normalize ( (that is, find C in terms of d and L)

b) Sketch ((x,0) as a function of x.

c) Where is the particle most likely to be found?

d) What is the probability of finding the particle to the left of d? Check your result in the limiting cases of L=d and L=2d

e) What is the expectation value of x? [5 x 2 = 10 points]

4. (a) Confirm, as pointed out in the text that (px(=0 for all energy states of a particle in a 1-D box, keeping in mind that the states are normalized. Let’s assume that the length of the box is L.

(b) Determine (x( for all energy states of a particle in a 1-D box.

(c) Determine (x2( for all energy states of a particle in a 1-D box.

(d) Determine the variance σx2 = (x2(-(x(2

(e) Determine the variance σp2 = ( px 2(-( px (2

(f) Show that the fluctuations are correlated, in other words, that σx.σp ≥ ħ/2.

This is the celebrated uncertainty principle that states that reducing the accuracy of one measurement (e.g. reducing σx) increases the inaccuracy of the other (increasees σp).

[3 + 3 + 3 + 3 + 3 + 5 = 20 points]

5. A particle with mass m and kinetic energy E (E>0) approaches an abrupt potential drop of height V0 as depicted in the figure below. What is the probability that it will “reflect” back if E=V0/4? [10 points]

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V(x)

x

-V0

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