1
PHYS 3143 QUANTUM MECHANICS I: REMARKS ON THE VECTOR SPACE FORMALISM
Brian Kennedy, School of Physics, Georgia Tech
The notes below are not intended to be complete, but I hope you will find them useful.
They are supposed to
supplement the class notes and other reading.
A.
Euclidean and abstract vector spaces.
We are used to dealing with vectors in one, two or three dimensional Euclidean space. Consider then a set of abstract
vectors
S
=
{
vectoru,vectorv,vectorw,...
}
. The algebra of vectors (due to W.R. Hamilton) involves the operation of vector addition
vectoru
+
vectorv
 by the parallelogram rule in the case of Euclidean vectors  giving another vector in the same space (this is
called closure under addition). The operation of addition is commutative and associative (
vectoru
+
vectorv
) +
vectorw
=
vectoru
+ (
vectorv
+
vectorw
)
.
There exists a zero vector
vector
0 and for every vector
vectoru
there is an inverse
vector
−
u
also in the set:
vectoru
+
vector
−
u
=
vector
0. The set of vectors
thus forms what is called an Abelian group under vector addition (Abelian means that addition is commutative). We
get a vector space when we give the set of vectors an additional structure, namely multiplication by (the field of)
real numbers,
λ
∈
R
:
λvectoru
is in the set of vectors.
In quantum theory we consider vector spaces over the field of
complex numbers rather than the reals. The scalar multiplication properties include
λ
(
vectoru
+
vectorv
) =
λvectoru
+
λvectorv
, 0
vectoru
=
vector
0 and
(
−
1)
vectoru
=
vector
−
u
.
In summary, for a vector space think of a set of geometrical objects, the vectors, and operations of vector addition
and multiplication by scalars. We will henceforth omit the arrow symbol from the abstract vectors, but retain it when
we refer to Euclidean vectors, in order to simplify the notation.
B.
Inner product of vectors: the inner product space
To introduce an idea of length, we need a norm for the vectors. This in turn follows from the inner (scalar) product
of two vectors. Abstractly an inner product is an operation which maps two vectors to a complex scalar: (
u,v
)
∈
C
,
satisfying (
i
)
(
u,v
+
w
) = (
u,v
) + (
u,w
)
,
(
ii
)
(
u,λv
) =
λ
(
u,v
)
,
(
iii
)
(
u,v
) = (
v,u
)
∗
and
(
iv
)
(
u,u
)
≥
0
.
The
equality in (iv) is true if and only if
u
= 0. The norm of a vector
bardbl
u
bardbl
=
radicalbig
(
u,u
). You can readily verify that the
familiar real scalar product of Euclidean vector algebra
vectoru
·
vectorv
∈
R
satisfies all these properties. We derived in class two
important inequalities that follow from (i)(iv), the CauchySchwarz or Schwarz inequality

(
u,v
)
 ≤ bardbl
u
bardblbardbl
v
bardbl
,
and the triangle inequality
bardbl
u
+
v
bardbl ≤ bardbl
u
bardbl
+
bardbl
v
bardbl
.
The latter was derived with the help of the former. A complete vector space with an inner product is called a Hilbert
space, and it may have finite or infinite dimension. The Schwarz inequality is used in the derivation of the famous
Heisenberg Uncertainty principle. The triangle inequality provides a test that a sequence of vectors (called a Cauchy
sequence) converges to a limit without having to know what the limit vector is !