(Read pgs. 109-144)
The Dirac Bra-Ket notation is a concise and convenient
way to describe quantum states. We introduce and define the symbol

to represent a quantum state. This is called a ket, or a ket vector. It is an abstract entity, and serves to describe the "state" of the quantum system. We say that a physical system is in quantum state , where represents some physical quantity, such as momentum, spin etc, when represented by the ket .

If we have two distinct quantum states
and
,
then the following ket

where

In general, the number of linear independent kets required to express any other ket, is called the dimension of the vector space. In quantum mechanics the vector space of kets is usually non denumerable infinite. We call such a vector space Hilbert space. We assume that any physical state can be described by a ket in Hilbert space.

Dirac defined something called a bra vector, designated by . This is not a ket, and does not belong in ket space e.g. has no meaning. However, we assume for every ket , there exists a bra labeled . The bra is said to be the dual of the ket . We can ask the question: since is a ket, what is the dual (or bra vector) associated with that vector?

The answer is,

where signifies a dual correspondence. This is an anti-linear relation.

Dirac allowed the the bra's and ket's to line up back to
back, i.e.

The symbol represents a complex number that is equal to the value of the inner product of the ket with . We note, according to the above definition, that,

Dirac also defined something called an outer product,

An outer product is allowed to stand next to a ket on its left, or next to a bra on the bra's right. Lets define , then if is an arbitrary ket, one is allowed to construct

It looks like we have something like an inner product on the r.h.s of this equation. Indeed, according to

The outer product

If we take operator

where is called the

Sometimes
,
then *A* is called an hermitian
operator. Hermitian operators play a central role in quantum
theory.
Show that
,
where
is a real number, is hermitian.
Consider a hermitian operator *X*, whose eigenstates | *a*> obey the eigenvalue equation

where

With this set we can express any ket by