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Unformatted text preview: 5.61 Physical Chemistry 24 Pauli Spin Matrices Page 1 Pauli Spin Matrices It is a bit awkward to picture the wavefunctions for electron spin because the electron isnt spinning in normal 3D space, but in some internal dimension that is rolled up inside the electron. We have invented abstract states and that represent the two possible orientations of the electron spin, but because there isnt a classical analog for spin we cant draw and wavefunctions. This situation comes up frequently in chemistry. We will often deal with molecules that are large and involve many atoms, each of which has a nucleus and many electrons and it will simply be impossible for us to picture the wavefunction describing all the particles at once. Visualizing one-particle has been hard enough! In these situations, it is most useful to have an abstract way of manipulating operators and wavefunctions without looking explicitly at what the wavefunction or operator looks like in real space. The wonderful tool that we use to do this is called Matrix Mechanics (as opposed to the wave mechanics we have been using so far). We will use the simple example of spin to illustrate how matrix mechanics works. The basic idea is that we can write any electron spin state as a linear combination of the two states and : c + c Note that, for now, we are ignoring the spatial part of the electron wavefunction (e.g. the angular and radial parts of ). You might ask how we can be sure that every state can be written in this fashion. To assure yourself that this is true, note that for this state, the probability of finding 2 2 the electron with spin up (down) is c ( c ). If there was a state that could not be written in this fashion, there would have to be some other spin state, , so that c + c + c 2 In this case, however, there would be a probability c of observing the electron with spin which we know experimentally is impossible, as the electron only has two observable spin states. The basic idea of matrix mechanics is then to replace the wavefunction with a vector : 1 5.61 Physical Chemistry 24 Pauli Spin Matrices Page 2 c c + c c Note that this is not a vector in physical (x,y,z) space but just a convenient way to arrange the coefficients that define . In particular, this is a nice way to put a wavefunction into a computer, as computers are very adept at dealing with vectors. Now, our goal is to translate everything that we might want to do with the wavefunction into something we can do to the vector . By going through this step-by-step, we arrive at a few rules Integrals are replaced with dot products. We note that the overlap between any two wavefunctions can be written as a modified dot product between the vectors. For example, if d + d then: 1 0 1 * * *...
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