Hassani S. Mathematical Physics: A Modern Introduction to Its Foundations
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2.3
CONJUGATION
OF
OPERATORS
61
If
Acommutes with [A, BJ,then the infinite series truncates at the secondterm,
and we have
etABe-
tA
= B +t[A, BJ.
For
instance,
if
A and Bare replacedby D
and
T
of
Example 1.3.4, we get (see
Problem 2.3)
etOTe-
tO
= T +t[D, T] = T +t1.
generator
of
The
RHS
shows that the operator T has
been
translated by an amount t (more
translation
precisely, by t times the unit operator). We therefore call exp(tD) the translation
operator
of
T by t, and we call D the generator
of
translation. With a little mod-
ification T and D become, respectively, the position and
momentum
operators in
momentum
as quantum mechanics. Thus,
generator
of
translation
2.2.6. Box. Momentum is the generator
of
translation in quantum mechan-
ics.
But
more
of
this later!
2.3 Conjugation of Operators
We have discussed the notion
of
the dual
of
a vector in conjunction with inner
products. We now incorporate linear operators into this notion.
Let
Ib), [c) E
'\7
and assume that [c) = T Ib). We know that there are linear functionals in the dual
space
'\7*
that are associated with (Ib))t = (bl
and
(Ic))t
= (c], Is there a linear
operator belonging to
£'('\7*)
that somehow corresponds to T? In otherwords, can
we find a linear operator that relates
(bl
and
(cl
just
as T relates Ib)
and
[c}?
The
answer comes in the following definition.
adjoint
ofan
operator
2.3.1. Definition.
Let
T E
£'('\7)
and
10)
,Ib} E
'\7.
The adjoint,
or
hermitian
conjugate,
ofT
is denoted by
rt
and
defined by
(01
T Ib)* = (blTt
10)
. (2.8)
The
LHS
of
Equation (2.8)
can
be written as
(01
c)*
or
(c] a), in which case
we
can
identify
(2.9)
This equation is sometimes
used
as the definition
of
the hermitian conjugate.
From
Equation(2.8), the reader
may
easilyverifythat 1t = 1. Thus, usingthe unit
operatorfor
T,(2.9) justifies Equation (1.12).
Some
of
the properties
of
conjugation are listed in the following theorem,
whose
proof
is left as an exercise.
62 2.
OPERATOR
ALGEBRA
2.3.2.
Theorem.
Let U,T E .(,(V) and a E C. Then
1. (U +T)t = U
t
+Tt.
3.
(OIT)t
=OI*T
t.
2. (UT)t = TtU
t.
4.
«T)t)t
= T.
The last identity holds for finite-dimensional vector spaces; it does not apply to
infinite-dimensional vector
spaces
in
general.
In previous examples dealing with linear operators T : Ill"
->
Ill", an element
of
Ill" was denotedby a row vector, suchas (x, y) for Ill
2
and (x, y, z) for Ill3.There
wasnoconfusion, because we were
operating
onlyin V.
However,
sinceelements
of
both V and V* are required when discussing T, T*, and
rt,
it is helpful to make
a distinction between them. We therefore resort to the convention introduced in
Example 1.2.3 by which
.
2.3.3.Box. Ketsarerepresentedas
column
vectorsandbrasas rowvectors.
2.3.4.
Example.
Letus findthehermitianconjugateof theoperatorT : C
3
->
C
3
given
by
T
(~~)
=
(at
i::~
~
a
3
) .
{t3
al
- a2 +ia3
Introduce
la}
=
GD
and
Ib) =
(~D
with dualvectors
(al
= (aj a
2
a
3
)
and (bl =
(13j
13
2
13
3
),
respectively.
Weuse
Equation(2.8)tofindTt:
2.4 HERMITIAN
AND
UNITARY
OPERATORS
63
Therefore,
we
obtain
III
hermitian
and
antl-hermltlan
operators
2.4 Hermitianand Unitary Operators
The process of conjugation
of
linear operators looks much like conjugation of
complex numbers. Equation (2.8) alludes to this fact, and Theorem 2.3.2 provides
further evidence.
It
is therefore natural to look for operators that are counter-
parts
of
real numbers. One can define complex conjugation for operators and
thereby construct real operators. However, these real operators will not be inter-
esting
because-as
it turns
out-they
completely ignore the complex character
of the vector space. The following altemative definition makes use
of
hermitian
conjugation, and the result will have much wider application than is allowed by a
mere
complex conjugation.
2.4.1. Definition.
A linear operatorH E
L(V)
is calledhermitian,or self-adjoint,
ifHt
= H.Similarly, A E
L(V)
is calledanti-hermitian if
At
=
-A
Charles Hermite (1822-1901), one of the most eminent
French
mathematicians
of the
nineteenth
century,
was
par-
ticularly distinguishedforthecleanelegance andhigh artis-
tic quality of his work. As a student, he courted disaster
byneglecting his
routine
assigned worktostudytheclassic
masters
of
mathematics;
and
though
he
nearly
failedhisex-
aminations,
he
became
a
first-rate
creative
mathematician
whilestillinhisearly
twenties.
In 1870hewas
appointed
to
a
professorship
atthe
Sorbonne,
where
he
trained
a whole
generation
of well-known
French
mathematicians,
includ-
ing
Picard,
Borel,
and
Poincare.
The
character
of
his mindis suggestedbya
remark
of
Poincare:
"Talk
withM.
Hermite.
Heneverevokesa
concrete
image,yetyousoonperceivethatthemost
abstract
entities areto
him likeliving
creatures."
Hedisliked
geometry,
butwas
strongly
attracted
to
number
theory
andanalysis,andhis
favorite
subject
was ellipticfunctions,
where
thesetwo fieldstouch
in manyremarkable
ways.
Earlier in the centurythe Norwegian genius
Abel
had proved
that
the
general
equation
of the
fifth
degreecannotbe solvedby
functions
involvingonly
rational
operations
androot
extractions.
Oneof
Hermite's
most
surprising
achievements
(in
1858)wasto show
that
this
equation
canbesolvedby elliptic
functions.
His 1873proofof the
transcendence
of e was
another
highpointof his career," If he
hadbeenwillingtodigeven
deeper
intothisvein,he could
probably
havedisposedof
tt
as
4Transcendental
numbers
arethose
that
arenot
roots
of
polynomials
with
integer
coefficients.
64 2.
OPERATOR
ALGEBRA
well,butapparently hehadhadenoughofa goodthing.Ashewrotetoa
friend,
"Isballrisk
nothing
onan
attempt
toprovethe
transcendence
of the
number
n,
If
others
undertake
this
enterprise,
no onewill be
happier
than
I at
their
success,butbelieveme,my
dear
friend,
it
willnotfail tocostthemsome
efforts."
Asitturned out,
Lindemann's
proofnine yearslater
restedon extendingHermite's method.
Several
of
his
purely
mathematical
discoveries hadunexpected
applications
many
years
later
to
mathematical
physics.For
example,
theHermitian
forms
and
matrices
that
he in-
ventedin connectionwith
certain
problems
of
number
theory
turned
out to be
crucial
for
Heisenberg's
1925
formulation
of
quantum
mechanics, and
Hermite
polynomials (see
Chap-
ter7) areusefulin solving
Schrodinger's
waveequation.
The following observations strengthen the above conjecture that conjugation
of complex numbers and hermitianconjugation of operators are somehow related.
expectation
value
2.4.2.Definition. The expectation value (T)a
of
an operator T in the "state" la)
is a comp/ex numberdefined by (T)a
=
(aITla)
..
The complex conjugate
of
the expectation value is
5
(T)* = (al T la)* = (al
rt
la).
In words,
r',
the hermitian conjugate
of
T, has an expectation value that is the
complex conjugate ofthe latter'sexpectationvalue.
In
particular,
ifT
is
hermitian-
is equal to its hermitian
conjugate-its
expectation value will be real.
What
is the analogue
of
the known fact that a complex number is the sum of a
real number and a pure imaginary one? The decomposition
shows
that
any operator can be written as a sum of a hermitian operator H =
~
(T +
Tt)
and an anti-hermitian operator A =
~
(T -
Tt).
We can go even further, because any anti-hermitian operator Acan be written
as
A =
i(-iA)
in which
-iA
is hermitian:
(-iA)t
=
(-i)*At
=
i(-A)
=
-iA.
Denoting
-iA
by H', we write T = H+iH
',
where both H and H' are hermitian.
This is the analogue of the decomposition z = x +iYin which both x and y are
real.
Clearly, we should expect some departures from a perfect correspondence.
This is due to a lack
of
conunutativityamong operators.
For
instance, althoughthe
product of two real numbers is real, the product
of
two hermitian operators is not,
in general, hermitian:
5Whennoriskofconfusionexists,it is
common
to
drop
the
subscript
"a"
andwrite
(T)
forthe
expectation
valueofT.
2.4
HERMITIAN
AND
UNITARY
OPERATORS
65
We have seen the relation between expectation values and conjugatiou properties
of operators. The following theorem completely characterizes hermitian operators
in terms
of
their expectation values:
2.4.3.
Theorem.
A linear transformation H on a complex inner product space is
hermitian
ifand only if(al H la) is real
for
allla).
Proof
We have already pointed out that a hermitian operator has real expectation
values. Conversely, assume that
(al H la) is real for all ]e). Then
(al H la) =(al Hla)* =(al Ht la) {} (al H - H
t
la) =0
By Theorem 2.1.4 we must have H - Ht =
o.
Via)
.
D
2.4.4. Example. In this
example,
we
illustrate
the
result
ofthe
above
theorem
with2 x 2
matrices.
The
matrix
H=
C?
c/)
is hermitian'' andactson
«:2.
Letus
take
an
arbitrary
vector la) =
(~P
and evaluate (al H la). We have
Therefore,
(al H la) =(ai a
z
)
(-.ia
2)
=
-iaia2
+
iazat
,a!
=
iaial
+ (ia20:1)* = 2Re(io:iat),
and (al Hla) isreal.
Forthemost
general
2 x 2 hermitian
matrix
H=
(;*
~),
where
a
and
y arereal, we
have
Hla)
=
(;*
and
(al H la) = (Oli
az)
(pOl*OI!
++P(
2)
=
ai(aa!
+
P0l2)
+
OIZ(P*OI!
+
y(2)
at
y
a
2
=
ala!1
2
+
ai
P0I2
+aZp*a! +Y
la21
2
=
0I1a!1
2
+
YI0I212
+2
Re(aip0I2).
Again (al Hla)is real. III
2.4.5. Definition. An operator A on an inner product space is called positive
positive
operators
(written A:::: 0) ifA is hermitian
and
(al
Ala)
::::
Ofor
allla).
6We
assume
thatthe
reader
hasacasual
familiarity
withhermitianmatrices. Thinkof ann x n
matrix
asalinear
operator
that
actsoncolumnvectorswhoseelementsare
components
of vectorsdefinedinthe
standard
basisof
en
or
R".
Ahermitian
matrix
thenbecomesa hermitian
operator.
66 2.
OPERATOR
ALGEBRA
positive
definite
operators
2.4.6.
Example.
Ali example of a positive
operator
is the squareof a hermitian opera-
tor.? Wenote
that
forany
hermitian
operator
Handanyvector[c), we have (c] H
2
1a} =
(al HtH la) = (Hal Ha)
~
obecauseofthepositivedefiniteness oftheinnerproduct. III
An
operatorT satisfying the extracondition that (aIT la) = 0 implies la) = 0
is called positive definite. Fromthe discussion
of
the example above, we conclude
that the square
of
an invertible hermitian operator is positive definite.
The
reader
may
be familiarwith two- and three-dimensionalrigid rotations and
the fact thatthey preserve distances and the scalarproduct. Can this be generalized
to complex inner product spaces? Let jz) ,
Ib) E V, and let Ube an operator on V
that preserves the scalar product; that is, given
Ib') = U Ib) and la') = Ula), then
(a'i b') = (al b). This yields
(a'l b') =
«al
Ut)(U
Ib»
= (al UtU Ib) = (al b) =
(aI1Ib).
Sincethis is
truefor
arbitrary la) and Ib), we obtain u'u= 1.
In
the nextchapter,
when we introduce the concept
of
the determinant
of
operators, we shall see that
this relation implies that U and
ut are both Inverrible," with each
one
being the
inverse
of
the other.
2.4.7. Definition.
Let
V be afinite-dimensional innerproductspace.
An
operator
unitary
operators
U is called a unitary operator
ifU
t
=
U-
I.
Unitary operatorspreserve the inner
product
o/V.
2.4.8.
Example.
Thelineartransformation T : C
3
-->C
3
givenby
T
I:~)
= (
(al
~li~2i'::~~:0./6
)
\;;3
{al
- a2 +a3 +
i(al
+cz+
(3)}/./6
is
unitary.
In fact,let
and
withdual vectors
(al =
(at
ai
a3') and (hi = (Pi
Pi
P3'),respectively. Weuse
Equation(2.8)and
the procedureof Example2.3.4to find
rt,
Theresultis
al
a2 a3(1 -
i)
v1+./6+
./6
ial
ia2
a3(l
+i)
v1-./6-
./6
~This
is
further
evidencethathermitian
operators
are
analogues
ofreal
numbers:
The
square
of anyreal
number
ispositive.
Thisimplication holdsonlyfor
finite-dimensional
vectorspaces.
2.5
PROJECTION
OPERATORS
67
and we can verify that
Thus
TIt
= 1. Similarly, we can show that
TtT
= 1 and therefore that Tis unitary. II
2.5 Projection Operators
We have already considered subspaces briefly. The significance
of
subspaces is
that physics frequently takes place not inside the whole vector space, but in one
of its subspaces. For instance, although motion generally takes place in a three-
dimensional space, it may restrict itselfto a plane either because
of
constraints or
due to the nature of the force responsible for the motion. An example is planetary
motion, which is confined to a plane because the force of gravity is central. Fur-
thermore, the example of projectile motion teaches us that it is very convenient
to "project" the motion onto the horizontal and vertical axes and to study these
projections separately.
It
is, therefore, appropriate to ask how we can go from a
full space to one of its subspaces in the context
of
linear operators. Let us first
consider a simple example. A point in the plane is designated by the coordinates
(x,
y).
A subspace of the plane is the x-axis. Is there a linear operator.? say P
x
,
that acts on such a pointand somehow sends it intothat subspace? Of course, there
are many operators from
]Rz
to R However, we are looking for a specific one.
We want P
x
to projectthe point onto the x-axis. Such an operator has to act on
(x, y) and produce (x, 0): Px(x, y) = (x, 0). Note that
if
the pointalready lies on
the
x-axis, P
x
does not change it.
In
particular,
if
we apply P
x
twice, we get the
same result as
if
we apply it ouly once. And this is true for any point in the plane.
Therefore, our operator must have the property
P~
= P
x.
We can generaIize the
above discussion in the following deflnition.I''
projection
operators
2.5.1. Definition. A
hermitian
operator
PEl:.,
(V) iscalledaprojectionoperator
ifP
Z
= P.
From this definition it immediately follows that the only projection operator
with an inverse is the identity operator. (Show this!)
Considertwo projectionoperators
PI and Pz. Wewantto investigate conditions
under which
PI +Pz becomes a projection operator. By definition, PI +Pz =
(PI +Pz)z =
pi
+PI Pz +PZPI +
P~.
So PI +Pz is a projection operator
if
and
ouly
if
(2.10)
9We want this operator to preserve the vector-space structure
of
the plane and the axis.
l°lt
is sometimes useful to relaxthe condition ofherrniticity. However, in this part
of
the book, we demand that P be hermitian.
68 2.
OPERATOR
ALGEBRA
MUltiply this ou the left by PI to
get
PfP2 +PIP2PI =0
=}
PIP2 +PIP2PI =O.
Now multiply the same equatiou ou the
right
by PI to get
PIP2PI +P2Pf =0
=}
PIP2PI +P2PI =
O.
These last two equatious yield
(2.11)
orthogonal
projection
operators
compieteness
relation
The
solutiou to Equatious (2.10)
aud
(2.11) is PIP2 = P2PI = O.We therefore
have the following result.
2,5.2.
Proposition.
Let PI, P2 E
.G(V)
be projection operators. Then PI +P2
is a projection operator ifand only if PIP2 = P2PI =
O.
Projection operators
satisfying this condition are called orthogonalprojection operators.
More geuerally, if there is a set
{P;}i~1
ofprojectiou
operators satisfyiug
{
Pi
ifi
=
j,
p.p.
-
, J - 0
ifi
'"
j,
theu P = I:7::1 Pi is also a projectiouoperator. Giveua uormalvector Ie),oue cau
show easily that P = [e) (e]is a projectiou operator:
•
Pishermitiau:
pt
= (Ie) (el)t =
«el)t(le»t
= Ie) (e].
• P equals its square: p
2
= (le) (el)(le)
(el) = Ie) (el e) (e] = lei (e],
---
=1
ill
fact, we cantake au orthouormal basis B = {lei)
}~I
andcoustruct a set
of
projectiouoperators {Pi = lei)
(eil}~I'
The
operators Pi are mutually orthogoual.
Thus, their sum
I:~I
Pi is also a projectiou operator.
2.5.3.
Propositiou.
Let B =
{lei}}~1
be an orthonormal basis for VN. Then the
set
{Pi = lei)
(eil}~1
consists
of
mutually orthogonal projection operators, and
I:~I
Pi =
I:~I
lei) (eiI=1. This relation is called the completeness relation.
Proof
The mutual orthogouality
of
the Pi is au immediate cousequeuce
of
the
orthouormality
of
the lei). To show the secoud part, consider au arbitrary vector
la), writteu in terms
of
lei): la) =
I:f=1
aj leji. Apply Pi to both sides to obtain
2.5
PROJECTION
OPERATORS
69
Therefore, we have
11a)
=
tai
lei)
=
tPi
la)
=
(tPi)
la).
Since this holds for an arbitrary
la),
the two operators must be equal. D
If
we choose only the first m < N vectors instead of the entire basis, then
the projection operator
p(m)
'"
I:~=I
lei)
(eil
projects arbitrary vectors into the
subspace spanned by the first
m basis vectors (lei)
li':"I'
In other words, when
p(m)
acts on any vector
la)
E V, the result will be a linearcombination of only the first
m vectors. The simple proof
of
this fact is left as an exercise. These points are
illustrated in the following example.
2.5.4.
Example.
Considerthree orthonormal vectors
(lei)
1;=1 E
lW,3
given by
I (I I
0)
10)=2:
1 1
0.
. 0 0 0
The
projection
operators
associated witheachof thesecanbe
obtained
bynoting
that
(eiI
is a row vector.Therefore,
PI = leI) (eIl =
~
(D
(I
Similarly,
pz=H~I)(l
-I
2)=H~1
~:
~2)
and
(
I
-I
I)=~
-I
I
3
-I
I
-I)
I .
I
Note thatPi
projects
ontotheline along lei}. This canbetestedby lettingPi acton
an
arbitrary
vectorand
showing
that
the
resulting
vectoris
perpendicular
to the
other
two
vectors.
For
example,
let P2actonan
arbitrary
column
vector:
la)
==
Pz
(;)
=
~
(~I
~I
!2)
(;)
=
~
(:x-:/_2~z).
z 2
-2
4 z
2x-2y+4z
We verify
that]e}
is perpendicularto both let)
and
[eg}:
I
(ella)=
vrz(1
I(X-
Y+2Z)
I 0) (;
-x
+Y -
2z
= O.
2x-2y+4z
70 2.
OPERATOR
ALGEBRA
Similarly,
(e31
a) =
O.
So
indeed,
la) is atongle2).
We can find the operator that projects
onto
the plane
fanned
by leI)
and
le2). This is
t
(2
t
PI + P2 = - I 2
3 I
-t
When
this operator acts on an arbitrary column vector, it produces a vector lying in the
planeof
1eJ)
and le2), orperpeodicularto le3):
(
X)
t
(2
It)
(X) t
GX+Y+Z)
Ib)
==
(PI +P2) Y =
3"
I 2
-I
Y =
3"
X +2y - Z •
Z I
-t
2 Z - Y+ 2z
It is
easy
to show that
(e31
b) = O.
The
operators
that
project
onto
the othertwo planes are
obtained
similarly.
Finalty,
weverifyeasilythat
PI + P2 + P
3
=
(~
~
~
=1.
o 0
~)
2.6 Operators in Numerical Analysis
11II
forward,
backward,
and
central
difference
operators
In
numerical calculations, limitiog operations involving infinities and zeros are
replaced with finite values. The most natural setting for the discussion of such
operations is the ideas developed in this chapter.
In
this section, we shall assume
that all operators are invertible, and (rather sloppily) manipulate them with no
mathematicaljustification.
2.6.1 Finite-Difference Operators
In
all numerical manipulations, a function is considered as a table with two
columns. The first column lists the (discrete) values of the independent variable
Xi,
and the second column lists the value of the function I at
Xi.
We often write
Ii for I(xi).
Threeoperators that areinuse in numericalanalysis are the
forward
difference
operator
~,the
backward
difference
operator
V (not to be confused with the
gradient), and the
central
difference
operator
6. These are defined as follows:
~Ii
sa 11+1 - fi,
Vfi es Ii
-fi-I,
(2.12)
The last equation has only theoretical significance, because a half-step is not used
in the tabulation of functions or in computer calculations. Typically, the data are
equally spaced, so
Xi+1
-
Xi
=h is the same for all i, Then
Ii±1
=I(xi ± h),
and we define Ii±lj2sa I(xi ±
hI2).