237module10

237module10 - Consider the hydrogen atom (ignoring the...

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Quantum Mechanics I - Module 10 Hydrogen Atom Practice Consider the hydrogen atom and its eigenstates, omitting effects due to fine structure. 1. For the hydrogen eigenstate | 210 i , give the (a) expectation value of the energy (b) expectation value of the total orbital angular momentum (c) expectation value of the z component of orbital angular momentum (d) expectation value of the x component of orbital angular momentum (e) possible values of total angular momentum 2. Now replace the electron with a muon, which has a mass of m μ 200 m e . (a) What is the ratio of the ground state energy of the muonic atom to that of the hydrogen atom? (b) What is the ratio of the wavelength of a transition from the n = 2 to the n = 1 state of the muonic atom to that of hydrogen? More Hydrogen Atom Practice
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Unformatted text preview: Consider the hydrogen atom (ignoring the spin-orbit interaction), with eigen-functions given by the chart on page 243 of your textbook. 1. For a given eigenfunction nlm , what is the energy? 2. Consider the r dependence: (a) Find the expectation value of r for the n = 1 , l = 0 state by explicit calculation. Check units. (b) Do you expect the expectation value of r for, say, the n = 2 , l = 0 state to be bigger or smaller? Why? 3. Suppose we had singly ionized helium (2 protons and 2 neutrons in the nucleus, one electron) instead of hydrogen. (a) By what factor would the energy levels E n change with respect to hydrogen? (b) By what factor would the fractional energy spacing E n +1-E n E n change with respect to hydrogen? 1...
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