Lecture 14.

We continued our discussion of quantum mechanics in position space,
deriving the canonical commutation relations between position and
momentum operators. We then developed Dirac notation, to discuss the
evolution of quantum states without explicit reference to their wave
functions in position space.

We noted that our derivation of quantum mechanics was a bit arbitrary,
in taking as a fundamental characteristic of a quantum state its wave
function \psi(x), or amplitude to be at any given position x. We could
have just as legitimately characterized our state by \psi(p), its
amplitude to have any fixed momentum p, or \psi(E), its amplitude to
have any allowed energy E, etc. This is because we showed that any
operators' eigenfunctions are a complete set: that is, we can write
any \psi as a sum of the eigenfunctions with appropriate coefficients,
which we know how to calculate.

To maintain this flexibility in describing the state, we adopt a more
general notation (due to Dirac). We represent the state \psi by a
"ket" |\psi >, which lives in the Hilbert space of all possible
physical states of the system. To define an inner product between
these physical states, we introduce the dual space of "bra"s <\psi|.
We noted that each ket is associated with a unique partner in the bra
space, and that the correspondence involves complex conjugation,
a|\psi> goes to <\psi | a* . We defined the inner product as a bra
acting on a ket, so that <a|b> gives the projection of |b> onto |a>.
Because of the complex conjugation, <a|b> = <b|a>* --- though the norm
of a single ket |psi>, given by <\psi|\psi>, is real.

As a basis for the ket space, we can use the eigenstates of any
operator A (that is, the states |a> for which A |a> = a |a>, so that
the value of A is definite). This means again that we can write any
state |\psi > as a sum of the eigenstates |a>, with coefficients given
by the projection <a|\psi > of |\psi> onto |a>. We obtained the
completeness relation by requiring that this decomposition in terms of
eigenstates always give us back the full state |\psi>.

We then began to discuss the special case where we choose as our basis
of eigenstates the eigenstates |x> of the position operator x. We can
insert the completeness operator, \int d^3x |x><x| ,into any
expression we have without changing it ( it just explicitly puts kets
in the |x> basis).


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KB