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Homework 4

  • Page ID
    143067
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    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).

    Pyrene.png

    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

    Given a quantum wire made from Silicon Telluride (Si2Te3) with a 1D width of 75 nm,

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

    b. Comment on how the energy has changed as the size of the box for silicon telluride has changed.

    Q4

    Find the normalization constant \(B\) in the following wavefunctions:

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

    Q5

    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

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

    Q6

    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 \( \hat{L}_z = -i \hbar \left( x \dfrac{\partial}{\partial y} - y \dfrac{\partial}{\partial x} \right) \)
     

    Q7

    For the following molecules identify the number of

    • degrees of freedom,
    • translational degrees of freedom
    • rotational degrees of freedom
    • vibrational degrees of freedom
    1. \( Kr \)
    2. HF 
    3. \(C_{60}\)
    4. \(H_2\)
    5. \(SF_6\)
    6. \(UO_2\)

    Q8

    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?

    Q9

    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.

    Q10

    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.

    Q111

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


    Homework 4 is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts.

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