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MSE300 Final Assignment Fall 2014

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There are four questions that are each worth 25 points. To receive credit you must show all of your work. You may look up formulas, values for constants and other information/data, and you can use tools like Maple and MATLAB, but this assignment must be completed on your own without the help of others. Exams must be turned in at my office before Friday, December 12 at 4 pm. I attest that I completed this assignment without any help from classmates or other individuals.

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1. The hexatriene molecule is sketched below. Each carbon atom has four valence electrons, three of which are tied up in covalent (sigma) bonds to neighboring atoms, leaving six electrons that can move between the ends of the molecule. Assuming a one-dimensional particle-in-a-box containing the 6 electrons that is 0.73 nanometers long, calculate the minimum energy in eV needed to excite an electron from the highest filled energy state to the lowest unfilled state, and give the wavelength and approximate color of light that corresponds to this adsorption energy. How close is this wavelength to the experimental value of 2580 Angstroms? Remember that there are 2 electrons per allowed energy state.

MSE300 Final Assignment Fall 2014

2. The Kronig‐Penney (K‐P) Model: This is one of the first models to use quantum mechanics to show there can be disallowed energies when a particle travels through a periodic potential. The potential energy in the K‐P model is composed of a periodic array of square wells that can be illustrated by the solid black lines below:

In this model we take the potential energy to be zero in the regions corresponding to a2, while U is the potential energy in regions corresponding to a1. To model a material, we can envision the traveling particle as an electron and the wells in the periodic potential coming from the positive nuclei. The disallowed energies are then the band gaps. The Bloch theorem of quantum mechanics (which we didn’t cover in class) states that the wave function in a periodic potential can be expressed as

Eq.(1) where is a wave function appropriate for one cell (which in our case has length a = a1 + a2) and k is the wave vector for the particle given by where is the particle wave length. The value of the wave vector can go from 0 (infinite wavelength) up to the first Brillioun zone. We call k a wave vector because in a three dimensional crystal an electron can travel in different directions, with a different wave length in different directions. We are going to use two functions for , one for the regions denoted by a1 in the figure above where the potential energy is equal to U, and another for the regions denoted a2 where the potential energy is zero. Both of the functions have two parameters that will be determined by requiring that the total wave function and its derivative is continuous throughout the cell. For the case where E>U (the total energy is greater than the potential energy), in the region denoted a1 this function is

Eq.(2)

where

. Eq.(3)

and in the region denoted a2 this function is

e‐

Z+ Z+ Z+ Z+ Z+

a1 a2

a U

MSE300 Final Assignment Fall 2014

Eq.(4)

where

Eq.(5)

The coefficients A, B, C, and D in Equations (2) and (4) must be determined so that the wave function and its first derivative are continuous at all values of x. Without going through the details (which are tedious but not difficult), this leads to the Eq.(6) that relates the wave vector k of the particle to the properties of the periodic potential (i.e. the values of a1, a2,  and 

cos cos sin sin Eq.(6)

For the case where E<U (the total energy is less than the potential energy), the equation for  is

Eq.(7)

and an equation equivalent to Eq.(6) is

cosh cos sinh sin . Eq.(8)

Part (a) Show mathematically that when E=U, Eq.(6) is equal to Eq.(8)

For this condition =0, which leads to sin(0)/0 and sinh(0)/0. You will have to find values for these two expressions to complete this derivation.

Part (b) Assume a model for an electron in a crystal where a1=1.75 Angstroms, a2=0.25 Angstroms (so that a = a1 + a2 = 2 Angstroms), and U=10 eV. Using this model plot the energy (in eV) of the electron as a function of +/‐ wave vector k (e.g. the particle’s momentum) in units of 1/Angstroms for several “branches” of the dispersion relation (you should obtain a plot similar to that shown to the right). You should try electron total energies that are both greater than and less than U. The band gaps should occur at

and at k=0. From these results give the lowest allowed energy (the bottom of the first band at k=0), and the 1st and the 2nd band gaps in eV.

Hint1: Rather than choosing k and solving for E, it would be easier to try different values of E, and use Eq.(6) or Eq.(8) to solve for the corresponding value for k. There are values for E for which Eq.(6) or Eq.(8) cannot be solved; these are the disallowed energy states.

Hint2: You may want to convert everything to the MKS system of units, and then convert your final answers to eV and 1/Angstroms.

MSE300 Final Assignment Fall 2014

3. Quantum Harmonic Oscillator: The potential energy U of a harmonic oscillator is given by

Eq.(9)

where is the frequency, x is the diplacement from the minimum energy value (it can be

negative or positive), and k and m are the force constant and mass of the spring, respectively.

Part (a): Using Eq.(9), write down Schroedinger’s equation for the harmonic oscillator.

Part (b): Show that the equation

Eq.(10)

is a solution to Schroedinger’s equation for the harmonic oscillator, and that this solution leads to a total energy of . To do this you can substiute Eq.(10) into your Schroedinger’s equation, take the second derivative, and collect terms. Because the total energy is a constant and doesn’t depend on position, x should cancel out of the equation.

Part (c): The probability of finding a particle, or in the case of the harmonic oscillator the probabiity of finding a particle displacement x, is proportional to the square of the wave function. Because a particle has to be some place (or there has to be some displacement of a quantum oscillator) a wave function should be normalized so that the total probability integrated over all space is 1. In one dimension this is given as

∗ 1 Eq.(11)

Determine whether Eq.(10) is properly normalized, e.g. that it satisfies Eq.(11).

MSE300 Final Assignment Fall 2014

4. Part (a). Briefly describe the contributions to the development of quantum mechanics made by each of the following people: Plank, Einstein, Rutherford, Bohr, DeBroglie, Davisson and Germer, Schrödinger.

Part (b). Calculate the wavelength of a baseball traveling at 90 miles per hour, and explain why batters don’t have to worry about a baseball showing quantum effects such as interference with the bat.