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Unformatted text preview: Physical Chemistry Course Number: C362 13 Theory of Angular Momentum 1. Why do we want to study angular momentum? 2. We need to study the properties of chemical systems, how is angular momen- tum relevant? 3. Lets consider the Hamiltonian for any molecule. How does it look? (a) It includes a term thats called the nuclear kinetic energy: sum of kinetic energy of all nuclei. (Operator) (b) It includes the electronic kinetic energy: sum of kinetic energy of all electrons. (Operator) (c) Electron-nuclear electrostatic attraction. (Positive and negative charges attract each other.) (d) Electron-electron electrostatic repulsion. (e) Nuclear-nuclear repulsion. 4. So it is complicated and the full Hamiltonian has many terms. And the prob- lem of time-independent quantum chemistry is to solve for the eigenstates of this Hamiltonian, since these eigenstates determine properties of molecular systems. So we have a problem. 5. What if we consider the simplest molecular system: the hydrogen atom. Well, it has no electron-electron repulsion which turns out to be a big advantage. And it does not contain nuclear-nuclear repulsion either. 6. But even the hydrogen atom turns out to be complicated. 7. So we need a general paradigm to solve these problems. 8. What if we say: We will look for a set of operators that commute with the full Hamiltonian. 9. How does this help us? We learnt earlier that if two operators commute, they have simultaneous eigenstates. Chemistry, Indiana University 81 c circlecopyrt 2011, Srinivasan S. Iyengar (instructor) Physical Chemistry Course Number: C362 10. Hence if we look for operators that commute with the Hamiltonian, and are simpler than the Hamiltonian, we may solve for eigenstates of these simpler operators first. This way we could partition one big problem (solving for the eigenstates of the full Hamiltonian) into many small problems. 11. This is the approach we will use. OK. So what kind of simpler operators do we have in mind that may commute with the Hamiltonian. Well if we were to think classically, we might say (a) the momentum of a system is conserved in classical mechanics. So does the momentum operator commute with the Hamiltonian? Well, it turns out that this is not the case for molecular systems since the kinetic energy operator is basically the square of the momentum and it does not com- mute with the potential due to the uncertainty principle. (For crystals and other items of interest in the solid and condensed phase it is, however, possible to use the momentum operator to simplify the problem.) (b) the angular momentum is conserved in classical mechanics and it turns out that this is important. We have seen in the hydrogen atom case that the total angular momentum of a molecular system does commute with the Hamiltonian....
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- Winter '11