Next: Projection operators Up: Foundations Previous: Expansion in terms delta

Bra-ket notation

(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

(1.24)

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

(1.25)

where ci is a complex number, is also a possible
state for the system.

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,

(1.26)

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.

(1.27)

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,

(1.28)

Dirac also defined something called an outer product,

(1.29)

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

(1.30)

It looks like we have something like an inner product on the r.h.s of
this equation. Indeed, according to the associative axiom of
multiplication, we are allowed to put parenthesis around the quantity and equate it to the value of the inner product . Or

(1.31)

The outer product X is an operator in Hilbert space. It acts on
ket from the left and turns it into another ket . Be careful! for
has no meaning, however does.

(1.32)

If we take operator A and operate on a ket , is dual to it? In general it is not, however the
dual of is

(1.33)

where is called the hermitian conjugate of
operator A. Show that the hermitian conjugate of is
.

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

(1.34)

where a is an eigenvalue. Suppose these eigenvalues are distinct,
then set are mutually orthonormal and form a set of
basis kets in Hilbert space, provided that

(1.35)

With this set we can express any ket by

(1.36)

Projection operators

Matrix representation of operators

Unitary operators

Next: Projection operators Up: Foundations Previous: Expansion in terms delta
Bernard Zygelman
1999-09-21