3502_16_octo16_LG - Chem 3502/5502 Physical Chemistry...

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Chem 3502/5502 Physical Chemistry II (Quantum Mechanics) 3 Credits Fall Semester 2009 Laura Gagliardi Lecture 16, October 16, 2009 Solved Homework (Homework for grading is also due today) Given that for a hydrogenic atom H = T + V = 1 2 " 2 ! ! 1 where we have explicitly written the proper kinetic and potential energy operators. We are reminded/given that H = ! Z 2 2 n 2 and r ! 1 = Z n 2 so it is trivial to determine T = Z 2 2 n 2 Thus, the expectation value of the kinetic energy is 1/2 times the expectation value of the potential energy. This relationship of < T > = (1/2)< V > is a manifestation of what is known as the quantum mechanical virial theorem, and it holds true for all wave functions where the potential energy term in the Hamiltonian operator depends only on r –1 to one or more nuclei. Atomic Spectroscopy The hydrogenic (one-electron) atom has 3 quantum numbers associated with each wave function. Two of these are from the spherical harmonics, and we already know the selection rules on the spherical harmonics: ! l = ± 1 and ! m l = 0, ± 1 (16-1) without derivation we will simply accept that the selection rule for Δ n is that absorption/emission is allowed for any change in n (note that n must change from one
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16-2 value to another, or else E , which depends only on n , fails to change, and then there is no opportunity to absorb or emit a photon in the first place!) The figure on the next page illustrates allowed transitions in the spectrum of a hydrogenic atom. Note that, although the figure is terribly complicated in the sense that many, many transitions are allowed, an actual measured spectrum would be relatively simple, because of the very small number of different Δ E values. A single photon frequency is associated with each Δ E , and thus there is a single ν for every n = 1 n = 2 transition, and a single ν for every n = 2 n = 3 transition, irrespective of the actual orbitals involved at the particular principal quantum number level. Truth is, of course, life is not quite that simple. It's only that simple if you use a low-resolution spectrometer. If you look more carefully, or you make the experiment a bit more complicated, suddenly you find a lot of new lines in the spectrum (different photon frequencies). Let's start with the most profound complication. Electron Spin In 1922, Stern and Gerlach did the following experiment. 1) Heat a block of silver until it vaporizes (whoa. ..) 2) Arrange the pressure in the experimental system such that the gas of silver atoms collimates into a "beam" that passes through the poles of a magnet. 3) Observe where the silver atoms strike a target behind the magnet. Here's what Stern and Gerlach expected. The silver atom can be thought of as being like a very big hydrogen atom. That's because all of the electrons but one completely fill principal quantum number levels 1 to 3 and the 4s, 4p, and 4d levels. So, for the one remaining electron in the 5s orbital, it’s a little bit like being around a nucleus of unit positive charge since “underneath” it, it sees 47 protons and 46 electrons that are spherically symmetric. With that picture, it should be clear that the total angular
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3502_16_octo16_LG - Chem 3502/5502 Physical Chemistry...

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