Chapter 11: Problem 13
It is said that a laser is not a source of energy but a converter of energy. Explain.
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Chapter 11: Problem 13
It is said that a laser is not a source of energy but a converter of energy. Explain.
These are the key concepts you need to understand to accurately answer the question.
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Consider the Fermi distribution of (11-24), \(n(\mathscr{E})=1 /\left[e^{\left(\mathscr{E}-\mathscr{E}_{F}\right) / k T}+1\right]\). (a) Show that \(n(\mathscr{E})=1-n\left(2 \mathscr{E}_{F}-\mathscr{E}\right) ;\) that is, with \(\mathscr{E}-\mathscr{E}_{F}=\delta\), show that \(n\left(\mathscr{E}_{F}+\delta\right)=1-n\left(\mathscr{E}_{F}-\delta\right)\). This proves that the distribution has a symmetry about \(n\left(\mathscr{E}_{F}\right)=1 / 2 .\) (b) Find \(n(\mathscr{E})\) for \(\delta=\) \(\mathscr{E}-\mathscr{E}_{F}=k T\), or \(2 k T\), or \(4 k T\), or \(10 k T\). Make a rough sketch of \(n(\mathscr{E})\) versus \(\mathscr{E}\) for any \(T>0\). (c) What percent error is made by approximating the Fermi distribution by the Boltzmann distribution when \(\delta / k T=1,2,4,10 ?\)
In writing about experiments on the scattering of \(\alpha\) particles in helium Rutherford said, "On account of the impossibility of distinguishing between the scattered alpha particles and the projected He nuclei, the results are subject to a certain ambiguity." Explain how an awareness of quantum statistics could have removed the ambiguity. What determines whether a gas obeys Bose or Fermi distributions?
In Debye's model of a solid, the maximum frequency \(v_{m}\) corresponds to a minimum wavelength. Because of the discrete nature of a solid this minimum wavelength corresponds to a vibration in which adjacent atoms move \(180^{\circ}\) out of phase with one another; that is, the interatomic spacing is half a wavelength. Is this plausible? Explain.
In a one-dimensional system the number of energy states per unit energy is \((l / h) \sqrt{2 m / \mathscr{E}}\), where \(l\) is the length of the sample and \(m\) is the mass of the electron. There are \(\mathcal{N}\) electrons in the sample and each state can be occupied by two electrons. (a) Determine the Fermi energy at \(0^{\circ} \mathrm{K}\). (b) Find the average energy per electron at \(0^{\circ} \mathrm{K}\).
Justify the assumption that conduction electrons behave approximately as a system of free noninteracting particles.
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