Calculate the natural line-width of a transition for Cu that has a lifetime of 5 ns. Estimate the contribution of Doppler broadening to the natural line-width at 2000 K and at 8,000 K.

Show that the hydrogen wave functionsφ 200 and φ21-1 are normalized. If the integrals required are not in Appendix 3, consult a table of integrals or use computer integration.

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In the canonical ensemble, we control the variables T, p, and N, and the fundamental function is the Gibbs free energy (G). But if we control T, p, and μ, then we will have a different fundamental function, Z (This is the case for cells that often regulate their temperature, pressure, and chemical potentials to maintain equilibrium). Which of the below options should the Z function equal? H - TS - μN H + TS + μN H + TS - μN G + μN F - pV - μN -H + TS + μN