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) Electronnuclear electrostatic attraction. (Positive and negative charges
attract each other.)
(d) Electronelectron electrostatic repulsion.
(e) Nuclearnuclear repulsion.
4. So it is complicated and the full Hamiltonian has many terms. And the prob
lem of timeindependent 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 electronelectron repulsion which turns out to be a big advantage.
And it does not contain nuclearnuclear 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
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2011, Srinivasan S. Iyengar (instructor)
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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
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 Angular Momentum, Srinivasan S. Iyengar

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