Choosing a representation means choosing
an orthonormal basis in the state space E. Assume the set {|u_{i}>} is a
discretely labeled basis and the set {|ω_{a}>}
is a continuously labeled basis for E.

<u_{i}|u_{i}> = δ_{ij} , <ω_{a}|ω_{a'}>
= δ(a - a’).

Any ket |ψ> may be written as |ψ> = ∑_{i}c_{i}|u_{i}>,
where c_{i} = <u_{i}|ψ>.

|ψ> = ∑_{i}<u_{i}|ψ>|u_{i}> = ∑_{i}|u_{i}><u_{i}|ψ>
= I|ψ> = |ψ>.

∑_{i}|u_{i}><u_{i} is the projector into the subspace
spanned by {|u_{i}>}. If {|u_{i}>} is a basis then ∑_{i}|u_{i}><u_{i}|
= I is the identity operator.
This is called
the closure relation.

Similarly, any ket |ψ> may be written as

|ψ> = ∫da c(a)|ω_{a}>, where c(a) = <ω_{a}|ψ>.

|ψ> = ∫da <ω_{a}|ψ>|ω_{a}> = ∫da |ω_{a}><ω_{a}|ψ>
= I|ψ> = |ψ>.

∫da |ω_{a}><ω_{a}| is the
projector into the subspace spanned by {|ω_{a}>}.
If {|ω_{a}>} is a basis then ∫da |ω_{a}><ω_{a}| = I is the identity operator.
This is called
the closure relation for a continuously
labeled basis.

In the basis {|u_{i}>} the ket |ψ>
is completely specified by its components c_{i} = <u_{i}|ψ>.
The corresponding bra is completely specified by its
components c_{i}* = <ψ|u_{i}> in the
basis {<u_{i}|}.

By convention, we represent a ket as a one column matrix of its components in a given
basis.

Representation of |ψ>:

We represent a bra as one row matrix of its components in a given basis.

Representation of <ψ|: (<ψ|u_{1}> <ψ|u_{2}>
<ψ|u_{3}> ... ) = (c_{1}* c_{2}* c_{3}*
... )

Let Ω be a linear operator and let {|u

Then |ψ> = ∑

We find b

We can write Ω|ψ> = |Φ> as a matrix equation.

The column matrices are representations of the kets
|Φ>
and |ψ> and the square matrix is a representation of
the operator Ω in the {|u_{i}>} basis.
The Ω_{ij} = <u_{i}|Ω|
u_{j}> are the matrix elements
of the operator Ω. The matrix elements are scalars.

By knowing the components of |ψ> in the basis {|u_{i}>} and the actions
of the operator Ω on the basis vectors, we can calculate the components of |Φ> =
Ω|ψ> in the same basis.

The matrix elements of Ω are Ω_{ij} = <u_{i}|Ω|u_{j}>.

The matrix elements of Ω^{†} are Ω^{†}_{ij} = <u_{i}|Ω^{†}|u_{j}>
= <u_{j}|Ω|u_{i}>^{*} = Ω_{ji}^{*}.

If Ω is Hermitian, Ω = Ω^{†}, then
Ω_{ij} = Ω_{ji}^{*}.
In particular, Ω_{ii} = Ω_{ii}^{*}.
The diagonal matrix elements of a
matrix representing a Hermitian operator are real.

- The set {
**i**,**j**} forms a basis for a two-dimensional vector space. Let**i**= |x_{1}>,**j**= |x_{2}>.

The vector**A**may be written as**A**= A_{x}**i**+ A_{y}**j**= A_{x}|x_{1}> + A_{y}|x_{2}>.

with A_{x}= A_{1}= |A|cosφ and A_{y}= A_{2}= |A|sinφ.

In the {|x_{i}>} basis |A> is represented by the matrix

Let Ω be the projector onto the x axis. We know the action of Ω on the basis vectors.

Ω|x_{1}> = |x_{1}>, Ω|x_{2}> = 0. We therefore know what Ω does to |A>.

Ω|A> = |B>. Ω|A> = A_{1}Ω|x_{1}> + A_{2}Ω|x_{2}> = A_{1}|x_{1}> = B_{1}|x_{1}> + B_{2}|x_{2}>. B_{1}= A_{1}, B_{2}= 0.

.

is the matrix of Ω in the {|x_{i}>} basis.

Ω_{11}= <x_{1}|Ω|x_{1}>, Ω_{12}= <x_{1}|Ω|x_{2}>, Ω_{21}= <x_{2}|Ω|x_{1}>, Ω_{22}= <x_{2}|Ω|x_{2}>.

Ω is a Hermitian operator, Ω_{ij}= Ω_{ji}^{*}.