Lecture 23.

We finished our study of angular momentum by explicitly constructing
the matrix representations of the angular momentum operators for j=1
and j=1/2. We also explained the relationship between the j=1
representation and the 3-dimensional ``defining'' representation that
we obtained by requiring the angular momentum operators to generate
spatial rotations. We then motivated attempts to ``relativize''
quantum mechanics.

First, we worked on writing angular momentum operators as matrices:
since rotations leave j invariant, the angular momentum operators
only mix states with a single value of j. For fixed j, we can
write the operators as 2j+1-dimensional matrices that specify their
effect on each basis state |j, m >. Explicitly, we write the
operator as a matrix O_{m' m}, whose entry in row m' and column
m gives the projection of the acted-on ket  O |j, m > onto
the basis state |j, m' >. We explicitly constructed these
matrices for the operators J^2, J_z, J_+ and J_-
--- which then determined J_x and J_y --- for the
cases j=1 and j=1/2. For j=1/2, we constructed the Pauli
matrices, whose rich structure you will explore in homework.

Because our representations |j, m > are complete, the
3-dimensional defining representation we found for J_i must be
related to the 3-dimensional j=1 representation.  We claimed the two
were related by a complex change of basis that diagonalizes the matrix
J_z, and displayed the transformation matrix.

We then began a discussion of our next challenge, to relativize
quantum mechanics. We pointed out that Schrodinger's equation comes
from enforcing a nonrelativistic dispersion relation on states, E
|\psi > = ( |p^2 | /2m + V ) |\psi >; where, to
obey the canonical commutation relations, energy and momentum become
the operators i\hbar \partial_t and -i\hbar \grad
respectively. This simple prescription for quantizing a system has
several features:

1) The nonrelativistic Hamiltonian determines a positive definite
energy spectrum. This will be important when we allow systems to
interact, because the particle can then change energy levels by
emitting a photon of the right transition energy, and arbitrarily low
energy levels mean that photons of arbitrarily large energy could be
emitted.

2) In position space, the Schrodinger equation determines a
probability density \rho = \psi^* \psi and a probability current
j = -i\hbar/ 2m\ (\psi^* \grad \psi - \psi \grad
\psi^*).  Probability is conserved locally: \partial_t \rho +
div J = 0. This conservation of probability is
essential to a probabilistic interpretation of \psi as the wave
function of a single particle.

3) Schrodinger's equation is first order in time. This means that we
only need specify an initial wave function (and no derivatives) to
fully determine the future evolution of the wave function.

We wish to ``relativize'' quantum mechanics: to enforce a relativistic
dispersion relation E^2 = p^2 + m^2, with physical objects (like the
probability current j^\mu) in manifestly Lorentz-covariant form.  In
doing this, though, we hope to maintain, as much as possible, the
structure of quantum mechanics --- particularly canonical commutation
relations and the probabilistic interpretation of the wave function.

We will begin with the obviuos approach, and its pitfalls, next time.

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KB