# Homework 4


Name: ______________________________

Section: _____________________________

Student ID#:__________________________

## Q1

Electrons are delocalized in the pi-orbitals of Polycyclic aromatic hydrocarbons. This becomes a particle in the box problem. Since the molecule is confined within a plane, a two-dimensional particle in a box model is a good approximation of a delocalized electron. Assume the boundary of the box is one C-H bond length beyond the carbon rings to contain the hydrogen atoms in the box. Hint: Use trigonometry for the dimensions.

Calculate the first three electronic transitions for Benzo[def]phenanthrene (bond length 1.42 Angstroms).

## Q2

Given a quantum wire made from Silicon Telluride (Si2Te3) with a 1D width of 7.5 nm,

a. What is the energy required (in electron volts) to make a transition from the state n=1 to n=2?

b. The measured value for this transition is about 1.9 eV (652.5nm). What would the mass of the electron be if this were the energy measured?

## Q3

Find the normalization constant $$B$$ in the following wavefunctions:

1. $\psi(x) = Be^{ikx}$
2. $\psi(x) = Be^{-ikx}e^{-2x^2/2a^2}$

## Q4

Show that the eigenstates to a particle in a 1D box with infinite potential satisfy the orthogonormality relationship:

$\int_{-\infty}^{\infty} \psi_n^*(x) \psi_m(x)\; dx = 1 \; \text{if } m=n$

and

$\int_{-\infty}^{\infty} \psi_n^*(x) \psi_m(x)\; dx = 0 \; \text{if } m \neq n$

## Q5

Evaluate the following commutators $$[\hat{A}, \hat{B}]$$ for the following pair of operators:

1. $$\dfrac{d}{dx}$$ and $$x$$
2. $$\dfrac{d}{dx}$$ and $$y$$
3. $$\dfrac{d^2}{dx^2}$$ and $$x$$
4. $$\dfrac{d^2}{dx^2}$$ and $$x^2$$
5. $$\dfrac{d}{dx}$$ and $$\displaystyle \int_{0}^{x} dx$$
6. $$\dfrac{1}{2}$$ and $$\dfrac{d}{dz}$$
7. $$\hat{p}_z = -i \hbar \dfrac{\partial}{\partial z}$$ and $$\hat{L}_z = -i \hbar \left( x \dfrac{\partial}{\partial y} - y \dfrac{\partial}{\partial x} \right)$$

## Q6

For the following molecules identify the number of

• degrees of freedom,
• translational degrees of freedom
• rotational degrees of freedom
• vibrational degrees of freedom
1. $$Ar$$
2. HI (not Hawaii)
3. $$C_{60}$$
4. $$H_2$$
5. $$SF_6$$
6. $$UO_2$$

## Q7

The wavefunctions for the quantum mechanical harmonic oscillator $$| \psi_v \rangle$$ in atomic units with $$\alpha = 1$$ are expressed as

$| \psi_v \rangle = N_v H_v e^{-(x-x_o)^2/2}$

with

• $$x_o$$ is the equilibrium position of the oscillator
• $$N_v$$ is a normalization factor for a specific $$v$$ value
• $$H_v$$ is the Hermite polynomial for a specific $$v$$ value (see Table M1)

What is the wavefunction (with determined normalization factor) for the harmonic oscillator in the $$v=0$$ state?

## Q8

Calculate the expectation value of $$x$$ of the quantum harmonic oscillator $$x$$ from equilibrium when the oscillator is in the $$v=0$$ and $$v=1$$ quantum states.

## Q9

Calculate the expectation value of $$x^2$$ (i.e., $$\langle x^2 \rangle$$) of the quantum harmonic oscillator $$x$$ from equilibrium when the oscillator is in the $$v=0$$ and in the $$v=1$$ quantum states.

## Q10

Use the answers from Q8 and Q9 to determined the uncertainty of position of a particle following the harmonic oscillator with $$v=0$$?

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