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In discussing molecular rotation, the quantum number \(J\) is used rather than \(l\). Using the Boltzmann distribution, calculate \(n_{J} / n_{0}\) for \(^{1} \mathrm{H}^{35} \mathrm{Cl}\) for \(J=0,5,10,\) and 20 at \(T=1025 \mathrm{K} .\) Does \(n_{J} / n_{0}\) go through a maximum as \(J\) increases? If so, what can you say about the value of \(J\) corresponding to the maximum?

The vibrational frequency for \(D_{2}\) expressed in wave numbers is \(3115 \mathrm{cm}^{-1} .\) What is the force constant associated with the bond? How much would a classical spring with this force constant be elongated if a mass of \(1.50 \mathrm{kg}\) were attached to it? Use the gravitational acceleration on Earth at sea level for this problem.

In discussing molecular rotation, the quantum number \(J\) is used rather than \(l\). Calculate \(E_{r o t} / k_{B} T\) for \(^{1} \mathrm{H}^{81}\) Br for \(J=0,5,10,\) and 20 at \(298 \mathrm{K} .\) For which of these values of \(J\) is \(E_{r o t} / k_{B} T \geq 10 . ?\)

Show by carrying out the necessary integration that the eigenfunctions of the Schrödinger equation for rotation in two dimensions, \(\frac{1}{\sqrt{2 \pi}} e^{i m_{P} \phi}\) and \(\frac{1}{\sqrt{2 \pi}} e^{i n_{l} \phi}, m_{l} \neq n_{l}\) are orthogonal

Evaluate the average of the square of the linear momentum of the quantum harmonic oscillator \(\left\langle p_{x}^{2}\right\rangle\) for the ground state \((n=0)\) and first two excited states \((n=1\) and \(n=2) .\) Use the hint about evaluating integrals in Problem \(\mathrm{P} 18.12.\)

Show by carrying out the appropriate integration that the total energy eigenfunctions for the harmonic oscilla\(\operatorname{tor} \psi_{0}(x)=(\alpha / \pi)^{1 / 4} e^{-(1 / 2) \alpha x^{2}}\) and \(\psi_{2}(x)=(\alpha / 4 \pi)^{1 / 4}$$\left(2 \alpha x^{2}-1\right) e^{-(1 / 2) \alpha x^{2}}\) are orthogonal over the interval \(-\infty

Two 3.25 g masses are attached by a spring with a force constant of \(k=450 . \mathrm{kg} \mathrm{s}^{-2} .\) Calculate the zero point energy of the system and compare it with the thermal energy \(k T\) at \(298 \mathrm{K} .\) If the zero point energy were converted to translational energy, what would be the speed of the masses?

Calculate the frequency and wavelength of the radiation absorbed when a quantum harmonic oscillator with a frequency of \(3.15 \times 10^{13} \mathrm{s}^{-1}\) makes a transition from the \(n=2\) to the \(n=3\) state.

Calculate the position of the center of mass of (a) \(^{1} \mathrm{H}^{19} \mathrm{F},\) which has a bond length of \(91.68 \mathrm{pm},\) and (b) \(\mathrm{HD}\) which has a bond length of \(74.15 \mathrm{pm}\)

The force constant for the \(^{35} \mathrm{Cl}_{2}\) molecule is \(323 \mathrm{N} \mathrm{m}^{-1}\). Calculate the vibrational zero point energy of this molecule. If this amount of energy were converted to translational energy, how fast would the molecule be moving? Compare this speed to the root mean square speed from the kinetic gas theory, \(|\mathbf{v}|_{r m s}=\sqrt{3 k_{B} T / m}\) for \(T=300 .\) K.