HW 9: Molecular Vibrations and Harmonic Oscillator

Question-1¶

Even functions $f(-x)=f(x)$. Odd functions: $g(-x)=-g(x)$

• Use any two eigenfunctions of particle in a box and harmonic oscillator. Show wether the functions are odd or even. Show the property of the product of respective wave functions $\psi_1 \psi_2$

• Show that derivative of even function is odd and that derivative of odd function is even.

Question-2¶

• In an infrared spectrum of $H^{19}F$ there is an intense spectral line at $2000 cm^{-1}$. Calculate the force constant and vibrational frequency for this molecule.

Question-3¶

Using the table 5.1 calculate the fundamental and the first three overtones of the $^{12}C^{16}O$ molecule.

Question-4¶

Identify which integrals are zero and which are not. Explain mathematically and provide physical intution

• $\langle 0 |1\rangle$

• $\langle 1 |0\rangle$

• $\langle 1 |1\rangle$

• $\langle 0 |x |0\rangle$

• $\langle 0 |x^2 |0\rangle$

• $\langle 0 |\hat{p_x} |0\rangle$

• $\langle 0 |\hat{p^2_x} |0\rangle$

Question-5¶

Read chapters 5.11 and 5.12 and answer the following questions

• What are selection rules and what do they predict for harmonic vs unharmonic oscillators?

• Evaluate the expressions $\langle 1|x|0\rangle$ and $\langle 2|x|0\rangle$ for a harmonic oscillator?

• What are the overtones?

• Why can’t we probe vibrational spectra of $Cl_2$ using IR spectroscopy?

• How many vibrational normal modes does $CH_3CH_2CH_3$ have?