Lecture 16.
Handouts: 
10. Homework 3 Solutions
11. Homework 4

We reviewed the representation of quantum states and operators as
vectors and matrices, given a specific choice of basis eigenstates. We
also reviewed the distinction between passive transformations, where
we transform the basis eigenstates; and active transformations, where
we actually transform the physical states. The two cases correspond in
mechanics to using a rotated set of coordinate axes, versus physically
rotating a particle position. In the first case, only our description
of the system, but none of its physical characteristics, change; while
in the second, the physical state itself changes. We reviewed that in
the first case, kets are left-multiplied by a unitary matrix U; bras
are right-multiplied by U\adjoint; and operators undergo the
similarity transformation O goes to U O U\adjoint, so that no
physical inner products or expectation values change. For the second
case, we are only transforming the states, with the operators
unchanged, so that expectation values typically change.

As a second example of such a transformation operator U (we've already
discussed time translation), we considered spatial translation.  We
calculated the commutator between the position operator and the
spatial translation operator which shifts x-eigenstates position by an
infinitesimal vector \epsilon. Using the canonical commutation
relations between position and momentum, we found that the unitary
operator with the desired commutation relation was given by the
identity minus i \epsilon \dot p /\hbar. We explained why this made
sense in the position space basis, where i p/ \hbar is just the
gradient operator, so that our spatial translation operator just takes
\psi(x) to the first order Taylor expansion of \psi (x + \epsilon).

We discussed building up a finite spatial translation from the
infinite repetition of infinitesimal ones (in the careful limit where
the total distance a = n \epsilon, where n is the number of
repetitions, is finite). This gives as the spatial translation
operator the exponential of -i \epsilon \dot p/\hbar, which is
manifestly unitary. We noted that this form is standard in quantum
mechanics: the operator \exp {-iaQ/\hbar} performs a transformation by
an amount a, where Q is the operator for the charge associated with
the transformation by Noether's theorem. We wrote out the examples of
time translation, spatial tranlation, and spatial rotation explicitly.

---
KB