Name: ______________________________ Section: _____________________________ Student ID#:__________________________ Template:HideTOC 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 Find the normalization constant \(B\) in the following wavefunctions: \[ \psi(x) = Be^{ikx}\] \[ \psi(x) = Be^{-ikx}e^{-2x^2/2a^2}\] Q3 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 \[ \int_{-\infty}^{\infty} \psi_n^*(x) \psi_m(x)\; dx = 0 \; \text{if } m \neq n\] Q4 Evaluate the following commutators \( [\hat{A}, \hat{B}] \) for the following pair of operators: \(\dfrac{d}{dx}\) and \(x \) \(\dfrac{d}{dx}\) and \(y \) \(\dfrac{d^2}{dx^2}\) and \( x \) \(\dfrac{d^2}{dx^2}\) and \( x^2 \) \(\dfrac{d}{dx}\) and \(\displaystyle \int_{0}^{x} dx \) \(\dfrac{1}{2}\) and \(\dfrac{d}{dz} \) \( \hat{p}_z = -i \hbar \dfrac{\partial}{\partial z} \) and \( \hat{L}_z = -i \hbar \left( x \dfrac{\partial}{\partial y} - y \dfrac{\partial}{\partial x} \right) \) Q5 For the following molecules identify the number of degrees of freedom, translational degrees of freedom rotational degrees of freedom vibrational degrees of freedom \( Ar \) HI \(C_{60}\) \(H_2\) \(SF_6\) \(UO_2\) Q6 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? Q7 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. Q8 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. Q9 Use the answers from Q7 and Q8 to determined the uncertainty of position of a particle following the harmonic oscillator with \(v=0\)?