Physics 251 Spring 1999
Modern Physics
Homework Assignment # 9, Due (in class) Monday April 12

Reading Assignment: Krane, Chapter 9 sections 5,6,7 NOTE: On p. 293, Krane states that the selection rules for vibrations and rotations are $\vert\Delta N\vert=1$ and $\vert\Delta L\vert=1$.He further states ``all transitions must simultaneously satisfy both selection rules." This is not quite right. Pure rotational transitions are allowed. They obey $\vert\Delta N\vert=0$ and $\vert\Delta L\vert=1$.Problem Assignment:

1.

Krane, Chapter 9, p.298, #2 (Assume the K atom is a K⁺-ion and the Br atom is a Br^--ion.)

2.

Krane, Chapter 9, p.299, #12. 2 points. The amplitude A_n of the n-th harmonic oscillator energy eigenstate can be found by setting the energy $E_n=(n+1/2)\hbar\omega$ equal to the potential energy $m\omega^2 A_n^2 /2$ when the corresponding classical oscillator reaches its turning point. (The same formula can also be derived from the quantum wavefunction using as a definition $A_n^2 = \int dx \vert\psi_n(x)\vert^2 x^2$.)

3.

Krane, Chapter 9, p.300, #13. The vibrational frequency of the H₂ molecule is 1.32$\times 10^{14}$Hz, not 10¹² as given.

4.

Figure 9.31 shows the molecular absorption spectrum of HCl molecules near 0.36eV (in the infrared.) These transitions correspond to a vibrational excitation from N=0 to N=1 with simultaneous changes of the rotational quantum number. (The initial state is N=0, the vibrational ground state, if the vapor is not too hot.) The Cl atom has two isotopes (see Appendix B, p556) with masses 35u and 37u. Using the measured energy 0.358eV (from fig. 9.31) as the vibrational excitation energy, calculate the expected splitting of the vibrational energy levels of the two isotopes.

5.

From the same figure it is clear that the rotational energy constant $\hbar^2/I \approx 0.0026eV$. Calculate the expected isotope splitting of the rotational levels. Explain the fine structure in the measured spectrum of fig. 9.31.

6.

Krane, Chapter 9, p.300, #21. The equilibrium separation in the HBr molecule is 1.4144$\times 10^{-10}$m.