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Chemistry LibreTexts

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).

    Pyrene.png

    Q2

    Given a quantum wire made from Silicon Telluride (Si2Te3) 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:

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

    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:

    1. \(\dfrac{d}{dx}\) and \(x \)
    2. \(\dfrac{d^2}{dy^2}\) and \( y \)
    3. \(\dfrac{d^2}{dy^2}\) and \( y^2 \)
    4. \(\dfrac{d}{dx}\) and \(\displaystyle \int_{0}^{x} dx \)
    5. \(\dfrac{3}{4}\) and \(\dfrac{d}{dz} \)
    6. \( \hat{p}_z = -i \hbar \dfrac{\partial}{\partial z} \) 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. \( Ne \)
    2. \(C_{60}\)
    3. \(H_2\)
    4. HCl
    5. \(UO_2\)
    6. \(UF_6\)

    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 \(v=0\)?