This preview shows pages 1–2. Sign up to view the full content.
Physics 580
Handout 7
14 September 2010
Quantum Mechanics I
webusers.physics.illinois.edu/
∼
goldbart/
Homework 3
Prof. P. M. Goldbart
University of Illinois
1) Schwarz inequality; triangle inequality
: Let

S
⟩
and

T
⟩
be two vectors.
a) By considering the positivity of the norm of the vector

S
⟩ −
⟨
T

S
⟩
⟨
T

T
⟩

T
⟩
establish the validity of the Schwarz inequality, i.e.,
⟨
S

S
⟩⟨
T

T
⟩ ≥ ⟨
S

T
⟩
2
.
b) Give, and describe the origin of, the condition under which the inequality becomes an
equality.
c) By considering the norm of the vector

S
⟩
+

T
⟩
, and then using the Schwarz inequality,
prove the triangle inequality
√
{⟨
S

+
⟨
T
}{
S
⟩
+

T
⟩} ≡ 
S
+
T
 ≤ 
S

+

T
 ≡
√
⟨
S

S
⟩
+
√
⟨
T

T
⟩
.
d) Give, and describe the origin of, the condition under which the inequality becomes an
equality.
2) Bras as linear functionals
: The twodimensional linear vector space over the real
numbers
V
(2)
(
R
) is spanned by the two orthonormal basis vectors

1
⟩
and

2
⟩
.
a) What are the values of the elements of the matrix of inner products
(
⟨
1

1
⟩ ⟨
1

2
⟩
⟨
2

1
⟩ ⟨
2

2
⟩
)
?
An arbitrary ket

ψ
⟩
is a linear combination of these basis vectors:

ψ
⟩
=

1
⟩
a
1
+

2
⟩
a
2
.
b) Show that the appropriate expansion coeﬃcients,
a
1
and
a
2
, are given by
a
1
=
⟨
1

ψ
⟩
and
a
2
=
⟨
2

ψ
⟩
.
Recall that a linear functional is an object
⟨
ϕ

, called a bra, that acts on a ket

θ
⟩
and returns
the scalar
⟨
ϕ

θ
⟩
. It is natural to choose as a basis for the set of linear functionals on
V
(2)
(
R
)
the bras
⟨
1

and
⟨
2

. For example, when we act with
⟨
1

on

θ
⟩
the result is
⟨
1

θ
⟩
. A general
bra may be expanded in terms of the basis as
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
 Fall '10
 Staff
 mechanics, Work

Click to edit the document details