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\(\mathrm{P} 18.23 \quad\) The force constant for a \(^{1} \mathrm{H}^{127}\) I molecule is \(314 \mathrm{N} \mathrm{m}^{-1}\) a. Calculate the zero point vibrational energy for this molecule for a harmonic potential. b. Calculate the light frequency needed to excite this molecule from the ground state to the first excited state.

At \(300 .\) K, most molecules are not in their ground rotational state. Is this also true for their vibrational degree of freedom? Calculate \(N_{n=1} / N_{n=0}\) and \(N_{n=2} / N_{n=0}\) for the \(127 \mathrm{I}_{2}\) molecule for which the force constant is \(172 \mathrm{N} \mathrm{m}^{-1}\). At what temperature is \(N_{n=2} / N_{n=0}=0.500 ?\) Repeat the calculation for \(\mathrm{H}_{2}\) and explain the difference in the results.

Evaluate the average kinetic and potential energies, \(\left\langle E_{\text {kinetic}}\right\rangle\) and \(\left\langle E_{\text {potential}}\right\rangle,\) for the second excited state \((n=2)\) of the harmonic oscillator by carrying out the appropriate integrations.

By substituting in the Schrödinger equation for the harmonic oscillator, show that the ground-state vibrational wave function is an eigenfunction of the total energy operator. Determine the energy eigenvalue.

Calculate the reduced mass, the moment of inertia, the angular momentum, and the energy in the \(J=1\) rotational level for \(\mathrm{H}_{2}\), which has a bond length of \(74.14 \mathrm{pm}\).