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Chapter 37: Problem 32

A molecule with rotational inertia \(I\) undergoes a transition from the lth rotational level to the \((l-1)\) th level. Show that the wavelength of the emitted photon is \(\lambda=4 \pi^{2} I c / h l\)

Short Answer

Expert verified
The wavelength of the emitted photon is \(\lambda = 4*\pi^{2}*I*c/(h*l)\)

Step by step solution

01

Understand the Rotational Energy Level

Rotational energy of a molecule in a level \(l\) is described by the equation \(E = l*(l+1)*h^{2}/(8*\pi^{2}*I)\), where \(h\) is the Planck constant, \(I\) is the rotational inertia and \(l\) is the rotational quantum number.
02

Calculate Energy Difference Between Two Levels

Calculate the energy difference, \(\Delta E\), between the two levels \(l\) and \(l-1\). That is, \(\Delta E = E(l) - E(l-1) = [(l*(l+1))-(l*(l-1))]*h^{2}/(8*\pi^{2}*I) = 2l*h^{2}/(8*\pi^{2}*I)\).
03

Use Planck's Law to Find Wavelength

Since, from the Planck's law, energy \(\Delta E\) of a photon is given by \(h*c/\lambda\), this equation can be rearranged to find wavelength \(\lambda\). Therefore, \(\lambda = h*c/\Delta E\).
04

Substitute \(\Delta E\) into the formula for wavelength

Plug \(\Delta E\) from step 2 into the wavelength formula. So, \(\lambda = h*c/\Delta E = h*c*(8*\pi^{2}*I)/(2l*h^{2}) = 4*\pi^{2}*I*c/(h*l)\).

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Most popular questions from this chapter

You're trying to explain to your classmates how classical and quantum descriptions of electrical conduction in metals differ. Using copper's Fermi energy (7.0 eV), you calculate the associated electron speed, then compare your result with the classical thermal speed for an electron at room temperature \((300 \mathrm{K}) .\) What do you find, and how does this help with your explanation?

Integrating Equation 37.5 over all energies gives the total number of states per unit volume in a metal. Therefore, integrating from \(E=0\) to \(E=E_{\mathrm{F}}\) - that is, over the occupied states only-gives the number of conduction electrons per unit volume. Carry out this integration to show that the electron number density is given by $$n=\left(\frac{2^{9 / 2} \pi m^{3 / 2}}{3 h^{2}}\right) E_{\mathrm{F}}^{3 / 2}$$

The critical field in a niobium-titanium superconductor is \(15 \mathrm{T}\) What current in a 5000 -turn solenoid \(75 \mathrm{cm}\) long will produce a field of this strength?

A molecule absorbs a photon and jumps to the next higher rotational state. If the photon energy is three times what would be needed for a transition from the rotational ground state to the first rotational excited state, between what two levels is the transition?

The Fermi energy in metals is much higher than the thermal energy at typical temperatures. Why does this make the mean speed of conduction electrons nearly independent of temperature?
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