Representations in Hilbert Space
•
A representation is a choice of orthonormal
basis in Hilbert space
–
analogous to choosing a coordinate system in
3D space
–
A basis is a complete set of unit vectors that
spans the state space
•
Basis sets come in two flavors: ‘discrete’ and
‘continuous’
–
A discrete basis is what we have been
considering so far. The unit vectors can be
labeled by integers, e.g.
{1
!
, 2
!
,…, 
M
!
}
, where
M
can be either finite or infinite
•
The number of basis vectors is either finite or
‘countable infinity’.
–
A continuous basis is a generalization
whereby the unit
vectors are labeled by real
numbers, e.g.
{
x
!
};
x
min
<
x
<
x
max
, where the
upper and lower bounds can be either finite
or infinite
•
The number of basis vectors is `uncountable
infinity’.
Properties of basis vectors
Matrix
element
operator
expansion
projector
component/
wavefunction
state
expansion
normalization
orthogonality
continuous
discrete
property
jk
k
j
!
=
)
(
x
x
x
x
!
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 Fall '08
 MichaelMoore
 Linear Algebra, mechanics, basis, Hilbert space, continuous basis

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