# Homework 4

- Page ID
- 143067

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 athe 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 (Si_{2}Te_{3}) with a 1D width of 17 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:

- \[ \psi(x) = Be^{ikx}\]
- \[ \psi(x) = Be^{-ikx}e^{-x^2/2a^2}\]

## 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:

- \(\dfrac{d}{dx}\) and \(x \)
- \(\dfrac{d^2}{dy^2}\) and \( y \)
- \(\dfrac{d^2}{dy^2}\) and \( y^2 \)
- \(\dfrac{d}{dx}\) and \(\displaystyle \int_{0}^{x} dx \)
- \(\dfrac{3}{4}\) and \(\dfrac{d}{dz} \)
- \( \hat{p}_z = -i \hbar \dfrac{\partial}{\partial z} \) 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

- \( Ne \)
- \(C_{60}\)
- \(H_2\)
- HCl
- \(UO_2\)
- \(UF_6\)

## 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 \(v=0\)?