PHY4604
R. D. Field
Department of Physics
Chapter3_2.doc
University of Florida
Obsevables and Vector Spaces
Eigenvalue Equation:
Consider the case of an hermitian operator
A
op
acting on a set of wave functions,

ψ
n
>,
such that
A
op

ψ
n
>
= a
n

ψ
n
>
where
a
n
is a constant. Note
a
n
is real and
<
ψ
i

ψ
j
>
=
δ
ij
(
i.e.
orthonormal
set).
Completeness:
If the ensemble of eigenkets

ψ
n
>
spans the entire space (
i.e.
any “ket” of finite norm can be expanded in a series of these “kets”), then
they are said to form a
complete set
and the hermitian operator
A
op
is called
an
observable
.
Vector Space (ordinary vectors):
In threedimensional space any vector
can be written as a superposition of the three unit vectors,
z
a
y
a
x
a
a
ˆ
ˆ
ˆ
3
2
1
+
+
=
r
,
where a
1
, a
2
, a
3
are real numbers.
The three unit vectors form an
orthonormal bases. The simplest way to represent a vector is to specify the
three constants a
i
with respect to the basis,
=
3
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This note was uploaded on 05/29/2011 for the course PHY 4064 taught by Professor Fields during the Spring '07 term at University of Florida.
 Spring '07
 FIELDS
 mechanics

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