Homework 4

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Section: _____________________________

Student ID#:__________________________

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Q1

Use the normalization condition to find \(A\) in the following wavefunction:

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

Q2

Find the normalization constant for the wavefunction:

\[
\psi(x)=\left\{
\begin{array}{ll}
C \quad \frac{-c}{4} \leq x \leq \frac{c}{4} \\
0 \qquad \textrm{elsewhere} \\
\end{array}
\right.
\]

Q3

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

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

and

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

What is the origin of the first equation?

Q4

A particle of mass \(m\) moves in a one-dimensional box of length \(L\) with boundaries at \(x = 0\) and \(x = L\).

• For any generic wavefunction \(\psi_n(x)\) for this system, calculate the probability that the particle is found somewhere in the region 0 ≤ x ≤ L/4.
• Show how this probability depends on \(n\).
• For what value of n is there the largest probability of finding the particle in 0 ≤ x ≤ L/4?

Q5

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

1. \(\hat{C}\) and \([\hat{D},\hat{C}]\hat{E} \)
2. \(\frac{d}{dy}-y\) and \(\frac{d}{dy}+y \)
3. \(\frac{d}{dy}\) and \(\int_{0}^{y} dy \)
4. \( \frac{d^2}{dy^2}\) and \(y \)
5. \(2\) and \(\frac{d}{dy} \)

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. \(SF_6\)
2. \(CO_2\)
3. \(O_2\)
4. \(C_{60}\)
5. \(Ar\)

Q7

At what point(s) during the oscillation of a spring that obey's Hooke's law is the force on the mass the greatest?

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 mean displacement of the oscillator \(x\) from equilibrium when the oscillator is in the \(v=0\) and \(v=1\) quantum states? Explain the differences in your own words.

Q10

Calculate the mean displacement squared (\(\langle x^2 \rangle\)) of the oscillator \(x\) from equilibrium when the oscillator is in the \(v=0\) and \(v=1\) quantum states? Explain the differences in your own words.

Q11

Use the answers from Q9 and Q10 to determined the uncertainty of position of a particle following the harmonic oscillator with \(v=0\)?

Q12

For a harmonic oscillator with a mass of \(1.21 \times 10^{-25} \;kg\), the energy levels are separated by \(4.82 \times 10^{-21}\; J\). What is the force constant for the oscillator? What is the zero point energy of this oscillator?

Q13

What are two requirements for a molecule to absorb IR radiation (via its vibrations)?

Q14

Demonstrate that the probability of a vibration described by a harmonic oscillator in absorbing IR radiation form the \(v=0\) to the \(v=2\) state is forbidden. Is the \(v=1\) to \(v=0\) transition also forbidden? You will need to solve the relevant transition moment integrals for both parts of this question.

Q15

Which of the following molecule absorb in the IR?

1. \(I_2\)
2. \(HBr\)
3. \(CD_2\)
4. \(CO_2\)
5. \(CH_4\)

Q16

What do the presence of overtones in IR spectra reveal about the anharmonicity of the vibration?