Lecture 18.

We continued discussion of the simple harmonic oscillator, determining
the energy spectrum and the action of creation and annihilation
operators on energy eigenstates. We then began a discussion of the
role of rotational symmetry and its generators, the angular momentum
operators, in quantum mechanics.

We had previously established that energy eigenstates of the simple
harmonic oscillator are also eigenstates | n > of the number
operator, with energy eigenvalues (n+ 1/2) \ \hbar\omega. We thus
determine the energy spectrum by finding the allowed eigenvalues n.

We found that the commutation relations of a and a\adjoint with
N determine these allowed eigenvalues.  After showing [ a,
a\adjoint ] = 1, we showed that the creation and annihilation
operators are eigenvectors under commutation with N: [N, a ] = -a,
[N, a\adjoint] = a\adjoint. Because of this, we can use a and
a\adjoint as step operators on the eigenstates of N. We showed a
|n > to be an eigenstate of N with eigenvalue n-1, and a\adjoint
|n > to be an eigenstate of N with eigenvalue n+1. This implied
that we could start with a single N eigenstate and construct a
ladder of allowed eigenstates by repeated application of a and
a\adjoint.

We then showed that nonnegative norm for any state a |n> implies
that any eigenvalue n must be greater than or equal to zero. Thus
the ladder of states must have a lowest rung, which can occur only if
a annihilates that lowest state (otherwise it would produce a state
with yet a lower eigenvalue). We showed that the eigenvalue of N of
this lowest state is n=0. This tells us that allowed eigenvalues n
are precisely the nonnegative integers, with an energy spectrum for
the simple harmonic oscillator given by E_n = (n+ 1/2) \ \hbar\omega.

We finished the simple harmonic oscillator by deriving the proper
normalization for the creation and annihilation operators' actions on
energy eigenstates.

We then began a discussion of rotations and angular momentum in
quantum mechanics. These are important both because many physical
Hamiltonians are rotationally symmetric, and because rotations are one
of the simplest examples of the kinds of symmetry groups that arise in
particle physics.

We first considered the operation of rotating a vector in 3-space,
specifically calculating how the components of that vector (taken with
respect to fixed Cartesian axes) change when we rotate it about the
z-axis. We encoded that change in a matrix R_z (\theta), which
matrix multiplies the column vector \vec{v} of initial components to
give the column vector \vec{v'} of final components. We wrote
matrices for rotations about the x and y-axes as well.

We then used Noether's theorem, which tells us that rotations about an
axis \hat{n} are generated by the angular momentum operator in the
direction \hat{n}. We considered infinitesimal rotations, Taylor
expanding both the matrices we built for these rotations, and the
exponential form given by Noether's theorem. This allowed us to
identify the angular momentum operators J_x, J_y,  and J_z.

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KB