Lecture 15.

We continued our discussion of Dirac notation, looking at implications
in the familiar basis of position eigenstates; discussing treatment in
an arbitrary basis where operators and physical states appear as
matrices and vectors; considering changes of basis; and considering
changes of physical state.

First we used the completeness relation for a basis of position
eigenstates to note the following: <x|\psi >, the projection of
\psi onto an x-eigenstate, is just the ordinary wave function
\psi (x) --- the amplitude for the state |\psi> to be at position
x. Thus our wave function, which we had earlier considered
fundamental, is just an explicit projection of the state |\psi> onto
a position eigenstate basis. We found the norm <\psi |\psi > was
just the integrated probability associated with the wave function
\psi (x); hence physical states have norm 1. Finally we found that
the inner product <\phi |\psi > took the familiar form \intd^3 x
\phi^*(x) \psi(x), which arose continually as the coefficient term
in eigenfunction expansions of \psi(x). We saw that using the
completeness relation on <x|\psi >, for eigenstates of a desired
operator A, automatically produced the eigenfunction expansion of
\psi (x) in terms of eigenfunctions \phi_a (x).

We claimed that quantum mechanics looked simpler in this
basis-independent formalism. Physical states are kets of norm 1; they
evolve according to the Schrodinger equation; and observables
correspond to operators A with eigenstates |a> having eigenvalues
a: A |a> = a |a>. We measure eigenvalue a in state \psi with
probability amplitude <a|\psi>, determining the expectation value
<\psi | A |\psi > (where, since A is Hermitian, we can take this
to be the projection of the ket A |\psi> onto the bra <\psi |, or
of the ket |\psi> onto the bra <\psi | A, equivalently).

Of course, to actually do any calculations, we have to work in some
basis. We can choose a basis convenient to each specific calculation.
We reviewed how, in a given basis, we may represent operators as
matrices, kets as column vectors of basis coefficients, and bras as
the transpose of those column vectors. Action of an operator A on a
ket v_i (where v_i are the coefficients of basis kets |i>) then
corresponds to matrix multiplication of the column vector v by the
matrix A, which we showed how to compute in class. 

We then discussed change of basis --- a change in the basis
eigenstates we choose to expand our system in, and not a change of any
physical state. We showed that a transformation of our basis
eigenstate |i> to |i>' = \sum_j A_{ji} |j>, could be written
equivalently as matrix multiplication of the column vector of basis
coefficients, v \rightarrow A v, in terms of the old basis
eigenstates.  Similarly, a bra with row vector v* went to v*
A\adjoint . This meant that a state's norm would change, unless A is
unitary: A\adjoint  A = 1. Since a physical state's norm is always 1,
changes of basis must be implemented by unitary matrices. We further
showed that expectation values would change under our change of basis
unless the operators transform by similarity transformation: O
\rightarrow A O A\adjoint .

Finally, we discussed physical changes of state, when we consider
actually examining two distinct physical states --- say rotating a
position eigenstate by 30 degrees. Transformations relating such
states must still be unitary, to preserve the norm 1 of the physical
state. However, such transformations do not affect our definitions of
the operators corresponding to observables. Thus expectation values of
an operator in the transformed state will generally differ from those
of the original state, unless that operator commutes with the unitary
transformation matrix.

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KB