Baggott J. The Meaning of Quantum Theory: A Guide for Students of Chemistry and Physics
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2
Putting it into practice
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2.1
OPERATORS
IN
QUANTUM
MECHANICS
Now
that
we have satisfied ourselves
of
the
need
for
a
quantum
theory
of
radiation
and
matter,
and
we have seen
the
different
positions taken
up
by
the
major
figures in
the
theory's
early
development,
we
must
interrupt
our
historical
narrative
and
turn
our
attention
to details
of
how
the
theory
works
in practice. Readers
are
reminded
that
this
is
not
a
textbook,
and
so
the
description
we
will give here
is
a highly selective
one.
Firstly, it
is
important
to
see how
the
modern
version
of
the theory
is
constructed
from
a set
of
postulates-
statements
accepted
to
be
true
without
proof
which
are
justified
later
through
agreement with experi-
ment.
We
will
frequently
return
to
the
philosophical
implications
of
such I
an
approach.
Secondly,
in
order
to
appreciate
the
details
of
the
debate
~
between
Bohr
and
Einstein,
and
further
arguments£~ented
by Einstein
and
OIl1n~~lDea.::tQJn~-1:l~:v-aQPJ:ll!TI!P)TTmportant
cxpenmental
test cases,
it
is
necessary
to
know
how
the
!i1eOriTsrouirnely
applied_
.~-
..
---
Finally,
many
of
these crucial
experimental
tests have been
performed
by
exploiting
the
properties
of
pairs
of
photons
with correlated polariza-
tions,
and
so
we have
included
here discussions
of
the Pauli principle
(important
for
any
subsequent
discussion
of
the
behaviour
of
IWO-
particle
quantum
systems),
together
with a section
on
the
polarization
properties
of
photons.
These
properties
can be used 10 illustrate in a
simple
way
the
problems
of
quantum
measurement.
All this material
should
set us
up
nicely
for
the
remainder
of
the
book.
Much
of
the
material
covered
in this
chapter
is tied up with the
mathe-
matical
formalism
of
quantum
theory.
Readers
completely
unfamiliar
with this
formalism
may
therefore
find this
material
somewhat
hard
to
digest.
The
unfamiliar
becomes familiar
with
experience
and
practice,
and
is in
this
case well
worth
the
effort
required.
You will
soon
discover
that
although
the
formalism
of
quantum
theory
presented here may
appear
complicated,
the
complications
often
arise in the language used,
not
in
the
algebra
itself.
Before
we begin,
it
is
important
to
emphasize
that
the
quantum
theory
Operators
in Quantum mechanics
39
described
in this
chapter
is
essentially
that
taught
widely
!O
modern
undergraduate
students
of
chemistry
and
physics.
That
this theory
is
the
best description
of
the
microphysical
world
of
elementary particles,
atoms
and
molecules
is
not
in dispute.
It
is
what
the
theory
means in
terms
of
the
relationship
between its concepts
and
physical reality
that
has
caused
(and
continues
to
cause)
so
much
argument
and debate.
Thus,
in
the
account
which' follows, we
should
note
that
the
concepts
of
the
theory
are
given all
interpretation
that
is not readily accepted by every
scientist.
Mathematical
operators
Despite
their
mathematical
equivalence,
the
quantum
theory
taught
to
modern
undergraduate
students
of
chemistry
and
physics is based mostly
on
Schrodinger's
wave mechanics
rather
than
Heisenberg,
Bom
and
Jordan's
matrix
mechanics.
This
is
because
students
learn
about
the
algebra
of
ordinary
functions
and
their
associated
mathematical
opera-
tors
first;
the
mathematics
of
matrices is
often
regarded
as a topic for
more
advanced
courses.
Consequently,
the
mathematics
of
the
operator
(Schrbdingcr)
form
of
quantum
mechanics
is
more
familiar (although
this
does
notllecessarily
mean
that
it is easy
to
follow). All experienced,
professional
quantum
'mechanic'
will quite
happily
move
between bOlh
descriptions.
The
most
common
mathematical
language
of
quantum
theory
is
therefore
that
of
operator
algebra,
and
the
postulates
of
the
theory
are
most
often
presented
to
undergraduates
in
this language.
Operators
are
very
familiar
to
us;
they
arc
simply
instructions
to
do
something
to
a
function
- multiply it,
differentiate
it etc. However, it takes a lillIe while
to
get
used
to
the
idea
of
handling
the
operators
by
themselves, i.e.
without
having
the
functions present in
an
equation.
In fact, in
an
equa-
tion
containing
only
mathematical
operators,
the
existence
of
some func-
tion
in
the
appropriate
coordinates
on
which the operatorS
are
supposed
to
operate
is
implicitly
assumed.
We will discover
later
in this
chapter
that
we
can
go
quite
a long way
in
our
analysis
of
quantum
systems
by
considering
the
properties
of
operators
and
the
assumed
properties
of
the
wavefunctiOIlS
without
actually
having
to
solve
the
appropriate
SchrOdinger
equation.
The
position-momentum
commutation
reilltion
Suppose
I
throw
'an
apple
into
the
air
and
you
photograph
it
as
it falls
to
the
ground
using a
camera
fitted with a rapid
autowind
facility. You
take
a
sequence
of
photographs
at fixed intervals in timc
as
the apple
40
Putting.
It
into
practice
falls
to
the
ground.
Each
photograph
is
a
record
of
the pos,tlOn
of
the
apple
(its height x
above
the
ground)
at
a
particular
time in its
motion.
We
could
analyse
the
sequence
of
photographs
to
find
the
values
of
the
apple's
position
and
velocity
(and
hence
its
momentum
in
the
x
coordinate,
P,)
as a
function
of
time.
Further
suppose
that,
for
some
obscure
reason,
we
need to calculate
the
product
of
the
apple's
position
and
momentum.
I
choose
to
deter-
mine
the
product
by
multiplying
position
by
momentum.
You choose
10
multiply
momentum
by
position.
No
matter,
since we know
that
for
ordinary
numbers
and
their associated units,
xp,
=
p,x,
or
(2.1 )
and
so
we expect
to
get
the
same
answers.
Equation
(2,1)
is
generally
called
a
commutation
relation,
sometimes
abbreviated
using
the
notation
[x,pJ,
This
might
seem trivial,
but
the
results
of
experiments lead
us
to
conclude
that
eqn
(2.1)
does
nol
hold
for
quantum
particles like
eJec-
trons
or
photons.
The
corresponding
quantum
mechanical version
of
cqn
(2.1)
is
(Z.2)
where i = J -
I.
This
is
known
as the
quantum
mechanical posl!!On-
momentum
commutation
relation. As we
have
said
before,
Planck's
constant
h is a very small
quantity
and
so
measurements
made
on
macro-
scopic objects like
apples
will lIever reveal
behaviour
other
thall that
described by
eqo
(2,1).
However,
for
microscopic objects like electrons,
the
magnitude
of
Planck's
constanl
becomes
extremely
important.
lncidentally,
eqn
(2.2)
demonstrates
once
again
that
classical mechanics
can-:be
.J:!£.',ivere§.:,rro!p_'lill.r u
rri
II.'
ecll-"..iircs--
b y
approxima
tlnnTo
zero,
-~--~--~--~-
-
--
-'
--,--
"·-Wehave
alleas!
two
ways
of
proceeding
from
here.
We
cannot
explain
eqn
(2.2)
if
we
treat
x
and
p,
as
ordinary
quantities.
Either we treat
them
as
non-commuting
matrices (ef. Section
1.4)
or
as
non-commuting
mathematical
operators.
I f we
choose
to
use
matrices,
we
are
led to
matrix
mechanics.
If
we
choose
operators,
we have
to
assume
that
there
mllst exist
some
funclion
which
depends
on
x (let us call it a wavefullc-
tion
.J;)
on
which
the
operators
are
supposed
to
operale,
Putting
this
wavefunclion
into
eqn
(2.2) gives
(2.3)
where we have used
carets
C), to remind
us
that x and
p,
should now be
regarded
as
mathematical
operators.
I
Operators
in
Quantum mechanics
41
Now
we
have
to
decide
what
form x
and
Px
should
take
to
satisfy
eqo (2.3).
We
ca!l
start
by
making
the
operator
x equivale.n!tQ'mul1i:,
plication
by
the value-or
x~just
as
mcIiiSSlcannecIianics
and
setting
px'proISortlonartO;f;'iix'(forureaSOiiSthat
;rrrbecomc'
imme'dialely'
apparent).
We'use
a parti"al
differential
operator
because we need
to
assume
that
the values
of
any
other
coordinates
on
which
if;
might
depend
are
kept
constant
for
the
purposes
of
differentiation.
Let us in fact
assume
that
fix
=
ad/ax,
and
use the
commutation
relation
to
find out
what
the
constant
a
must
be.
From
eqn
(2.3) we
have
xa.!!...
if; -
a!!.
(J0/!)
= iliif;.
ax
iJx
(2.4)
No!e
that
we have
dropped
the use
of
the
caret
on x because it is now
defined
as multiplication
by
a
quantity
rather
than
a
more
elaborate
operator.
This
is a
convention
which we will follow
throughout
this
bOOK.
The
second
term
on
the
left,hand
side
of
eqn
(2.4)
is
lhe
differential
of
the
product
of
two
functions
which
depend
on
x
and
which
we can
expand
using the
product
rule,
iJ(uv)liJx
'"
vo"lax
+
"au/ax:
iJ
raJ
xa ax
if;
- a
L'"
+ x
ax
if;
=
ihif;.
(2.5)
When
lhe
bracket
on
the
left·hand
side
of
eqn
(2.5)
is
expanded,
the
first
and
third
terms
cancel,
a =
-in
and
so
j\
=
-illa/ax.
We can
similiarly
define
py
= -
inil/
ily
and
p,
= - illiJI
oz.
The
square
of
p,
is
readily
deduced
from
p;
=
p,.Px
=
-h'fj'/i)x'
(remember
i'
=
-1).
How
10
~derive'
the
Schrodinger
equation
(again)
According
to
non-relativistic classical mechanics, the [otal energy
of
a
particle is
the
sum
of
its kinetic
and
potential
energies, eqn
(i
.20), which
can
be written
in
terms
of
p
'"
mv
as follows:
p'
2m
+
V",
E.
(2.6)
To
get
the
equivalent
quantum
mechanical
expression. we replace
the
physical Quantity
p'
with its
quantum
mechanical
operator
equiva·
lent
p',
and
introduce
the
wavefunction
f.
We
take
p'
to
be
equal
to
(p! +
iJ;
+ p;)
='
(-/i'lJ';ax'
- /i'a'/ay' - /i'a'/oz')
='
_"'V
'
.
The
result is
(2.7)
42
Putting
it
into practice
which is,
of
course, the non-relativistic three-dimensional Schrodinger
wave
equation.
This exercise
demonstrates
that
in the
equations
of
quan-
tum
mechanics;theva1Uescl-Observablequiintme~osjn()n
aiiO
mi5mentum
are
'i;ep).es~d
~Ylh.~_rr>at.~ematical
ope_ratcir§~l!.Y)£rdihese
vlmren,1leiJThey
operate
Oll
th,,-.~.a-;;erunCt!On.·
.
__
...
-
-.-,-.~~~
...
~.
,,'
...
~---,
.,~
..
--.--~,~
Operators
and
the
uncertainty principle
In 1929, the physicist
Howard
Robertson showed that
the
product
of
the
A ,
standard
deviations
of
two
non-commuting
operators
A and B
is
given
by
(2.8)
A A " A
where C is related
10
the
commutator
[A,
BJ, by
[A,
BJ = iC,
and
I <
C)
I
represents the modulus
of
the average value
of
C.
This result
is
quite
general,
independent
of
the significance
of
A
and
Ii or their interpreta-
tion
as
operators
for physical quantities.
Some
physicists have argued
that
eqn
(2.8)
justifies the view that the
Heisenberg
uncertainty
relation should be regarded as a fundamental
law
of
nature.
Since D.xAp.;;, h141f, for x
and
P.
the
term equivalent to
I
(C>
I in eqn
(2.8)
has the value fl. Hence
Ix,
fix
J =
in.
We used this I
position-momentum
commutation
relation
to
deduce
the
form
of
f;
needed for the
quantum
mechanical version
of
the
equation
for the
total'
energy
of
a particle,
and
arrived at the Schrodingcr equation.
Thus,
in
principle, all
of
quantum
mechanics
can,
via
eqn
(2.8), be deduced from
the uncertainty relation.
2.2
THE
POSTULATES
OF
QUANTUM
MECHANICS
Contrary
to
popular
belief, science
is
a very
untidy
discipline. This
is
seen
to
be
true nowhere
more
than in the historical development
of
quantum
mechanics.
The
revolutionary new theory that was to replace the classical
mechanics
of
Galileo
and
Newton in the microphysical world was
born
amid
confusion
and
desperation and grew
up
amid confusion
and
argu-
ment.
By
the end
of
the 1920, most physicists accepted
that
quantum
theory
had
quite
a lot going for it,
but
were unsure
about
how it should
be interpreted.
The
theory
had
tremendous predictive power, and it
scored
success
after
success when
put
to
the test
in
the laboratory. Where
it failed the test, subtle but mathematically logical modifications
10
the
theory
were
introduced
which
made
it even more powerfuL
It
became
clear
that,
although
the
theory was difficult to understand, it was cnor-
The
postulates
of
Quantum mechanics
43
mously useful
and
the
gains from using it
more
than
outweighed the
doubts
about
what it
meant.
Although
the
arguments
about
the interpretation
of
quantum
theory
were
far
from resolved,
by
the end
of
the 19205 there were few problems
with its mathematical
structure.
Wave mechanics
and
matrix
mechanics
had been shown
to
be equivalent,
and
there could be
no
doubt
that
physicists knew
how
the
theory
should
be
applied.
For
some
problems,
wave mechanics
offered
the most accessible
route
to
the solutions; for
others matrix mechanics was the preferred
choice,
We saw in Section 1.3 how Einstein, faced with the
problem
of
explaining a fixed speed
of
light, raised that fact
to
the
status
of
a
postu·
lat~
and
used it
to
deduce the special theory
of
relativity. Schrodinger's
version
of
quantum
mechanics makes use
of
the
mathematical
properties
of
the wavefunctions
and
their relationships
to
observable
quantities
such
as
posilion,
momentum,
and
energy
~Although
phlSiicis!s'l'i!lhl
aE~uea~!.!!t':.
~njgg:2Lili~avefu!!£!io.ns,
there
w~o
dispu.l~
a_t:.0ut.how the
.",a.v.!'fun<:!lgn3uEo'~L'Lb'£"!Tl'!!l!f'ulated
liLa.brain
result~.
th<l~cp_'!IQ..Q~
~o!Tl.p<lr.eg_\V}t.l1.<:.xperil:n_en
I.a1
!Tl.$.as.urern..!n
t~:
.
.c:~q
uelltl~
,
!he""~,,"istence"
of
the
wavefuncti()!l~_aE9.!)1",
.s~Cj.ll..e.Il£~.()L
op~_a!lons
that
has
to
be
done
on the"nilo
obtain
the results were elevated to'lh-e status"
-"-_._,---,,.
-"
.~
. -
,-"'
---"-.~
,---
-
.-~
",-
of
postulates. This was
done
principally by the
mathematician
John
von
"Neumann,and
the details were described in his
book
Mathematic(11
found(1tions
of
quantum
mechanics, first published in Berlin in 1932.
The
postulates
promote
the
view
that,
rather
than
worry
overmuch
about
where the wavefunctions come
from,
we might
as
well accept thaI they
exist and use them to tell us interesting things
about
the microscopic
world
of
molecules,
atoms,
electrons,
and
photons.
The
postulates
of
quantum
mechanics
are
the
foundation
stones
upon
which the most widely accepted version
of
the
theory
is built.
From
now
on,
we
will
refer
to
this
most
widely accepted version as the
'orthodox'
interpretation
of
quantum
theory.
For
the present, we will accept the
postulates without question -
to
a certain extent, their justification
comes from the fact that
the theory which flows from
them
is
arguably
the most highly successful
theory
of
physics ever devised, We will reserve
our
discussion
of
their validity until later
chapters.
The
wavefunclion
Postulate
1,
The
stale
of
a
quantum
mechanical system
is
completely
described
by
the wave function "'".
Here
the subscript n serves
as
a
shorthand
for the set
of
one
or
more
quantum
numbers
on
which the wavefunction will depend,
For
example,
the wavefunctioll
of
an
electron in a hydrogen atOm
is
completely
44
Puttrng it into practIce
specified by
the
set
of
quantum
numbers
n,
I,
m,
and
m,.
The
values
n = 2, I = I,
m,
= 0
and
m, = + y
refer
to
one,
and
only one, state
of
the
electron (recognizable, by
convention,
as
a 2p, electron with spin
up). We will use,p"
(or
,pm'
,p"
etc)
to
denote
wavefunctions that refer
to specific
states.
of
a
quantum
system
and
will
continue
to
use'"
to
denote
a general
wavefunction.
The
wavefunction will be a
mathematical
function
in whatever coordi-
nate
space
is
appropriate
for
the
system
under
study,
and
there will
be
as
many
quantum
numbers as there are
coordinates.
For
example,
electron wavefunctions for
the
hydrogen
atom
can be represented in
three
spherical
polar
ceordinates
and
an
'internal'
coerdinate
associated
with the spin
ef
the electron. For thcj2J:esent·,.-wewi!Ueave
open
the exact
-.~-
----.-~-
-
-_.-
~lLa!ule
of
the wavefunction,
Dill
will accept
that
it somehow
'conlains'
all
~t~~egabQ1!L!l1e
gUf!!lwms:Ystemit
d~cnf:;es::--
\ According
to.
Bern's
interpretation,
the
w3vefunctions represent
prob-
\ ability 'amplitudes',
and
the
square
of
the wavefunction gives the
prob~
ability density
of
the
quantum
particle in a
particular
region
Of
space.
Because the wave functions
may
be complex,
it
is necessary
to.
calculate
f this probability density from
the
product
I'"
I' = ",'';'
if
the result is
geing
to
be a real
quantity
("'.
is
the
cemplex
conjugate
of
f).
If
we wish
to know the probability
of
finding the particle
in
a specific region
of
space, it is necessary to multiply the probability density by the volume I
element
to
which we refer
(much
as
We
would determine the mass
of
some
material by multiplying its mass
densi1Y
by
its
volume). Because
the
'space'
in which the wavefunction is represented
may
be
a complicated,
many-dimensional
configuratien
space, we replace the volume element
by
a generalized spatial element
dr.
Thus,
the probability
of
finding
the
. particle in the region
d,
is given by ,p·';'dr. Because there
is
only
one
~
particle
and
it must be
found
somewhere,
the
integral over all spatial
elements
dr
must
be
unity, i.e.;
(2.9)
This
is
known as the
condition
of
normalization.
Operators
for
observables
I Postulate 2. Observable quantities
are
represented by
mathematical
I operators. These
operators
are
chesen
to
be consistent with the
position-
t momenWm
commutation
relation .
• I
The
term observable
quantity
(or
just
'observable') is used
to
signify
all
the
quantities
that
we
could,
in prinCiple, measure, such
as
position,
The
postulates
of
quantum
mechanics
45
momentum,
energy etc.
By
choosing
operators
that
are
consistent with
[x,p,]
= iii, we
are
laking the
commutation
relation (or, as
we
have
seen, the
uncertainty
principle depending
on
your
point
of
view) to be a
fundamental
'truth'
which
is
deemed
not
to
be in need.of
proof.
This
is
similar
to
Einstein's elevation
of
the speed
of
light to a
fundamental
cons-
.
tant
whose
in variance
is
accepted
but
cannot
be proved.
According
to
this postulate, every observable must have a correspond-
ing
operator
in the theory. We have seen in the previous section that the
position
operator
can
he chosen to be simply 'multiplication by
x',
as
.in classical mechanics.
This
choice for x forces the linear
momentum
operator
fix
to
take
the
form
-iha/ax
in
order
to
satisfy
the
commuta-
lion relation.
The
operator
for the total energy
is
called
the
Hamiltonian
operator
if.
It
consists
of
kinetic
and
potential energy
operators,
usually
A A
given the symbols
Tand
V respectively. T is usually a differential opera-
tor
(it
is
obviously closely related to
fi')
and
V
is
usually
just
'multi-
plication
by V'.
Thus,
the Schrodinger
equation,
eqn
(2.7),
can
be
written succinctly
as
follows:
where
A A
H=T+
A
iN
=
E>/I
h'
V=
- - 17' +
V.
2m
(2.1O)
(2.11)
Equation
(2.10)
has a
form
characteristic
of
an
eigenvalue
equation:
a
mathematical
operator
(in
this
case
if)
operates
on
some
function
to
give
the
same
function multiplied by
the
value
of
some
quantity
(in this
case
E).
Functions
that
satisfy such
an
equation
are
said
to
be eigenfunc-
tions
of
the
operator
(if)
and
the corresponding values
of
the quantities
(the energies)
are
called eigenvalues. We
can
say that
if
the specific
A A
wave function
>/In
is
an
eigenfunction
of
H,
then
H"'.
=
E."'.
and its
eigenvalue is
E,.
Different specific eigenfunctions
-.;"
"'m
-
mayor
may
not
have
different
energy
eigenvalues-E"
Em.
For
example,
the
energies
of
the electron wavefunctions (or orbitals)
of
the
hydrogen
atom
depend
only
on
the
principal
quantum
number
n. In the absence
of
a
magnetic
field, states with the
same
value
of
n
but
different values
of
I,
m,
or
m, will therefore have the
same
energy.
The
values
of
observables
Postulate 3
The
mean value
of
an
observable is equal
to
the expectation
value
of
its corresponding
operator.
46
Putting
it
into practice
For
a specific
wavefunction
,p"
the
expectation
value
of
some
opera-
tor
A is
defined
by
the
expression:
,
(A
) _
f""'~A""'!,~:
n - lv':I/;,dT
(2.12)
This
expression
appears
to
have
arrived
out
of
the
blue, but it is, in
fact, related
to
an
equivalent
expression
from
probability
calculus.
Lei us
examine
how
the
expectation
value is related
to
the
value
of
an
observable
for
the
case when
v',
is
an
eigenfunction
of
A.
Denoting
the
corresponding
eigenvalue
as
0"
we
have
Av',
= 0,1/;,.
If
we
multiply
both
sides
of
this
equation
from
the left
by
1/;::
and
integrate
over all
spatial
elements,
we
obtain:
(2.13)
Note
that
1/;:
on
the
left-hand side
of
eqn
(2.13) multiplies
the
result
of
A "
the
operation
of
A
on
>/;,
and
I/;!A"', * AI/;!I/;,.
There
is
no
such
rest ric-
lion on
the
right-hand
side
of
eqn (2.13): a, is
just
the value
of
some
quantIty which
is
independent
of
the
coordinates
and
which can tllcre,
fore be taken
out
of
the
integral.
From
eqll (2.13)
we
have
(2.14)
Equation
(2.14)
applies
only
if
t,
is an
eigenfunction
of
A.
Although
postulale
3 refers
to
the
'mean
value'
of
an
observable, we can see
from
eqn
(2.14)
that
when
v',
is an
eigenfunction
of
A
the
expecta-
tion value is exactly an' We will see later
that
the
use
of
the
word
'mean'
becomes necessary when we consider functions which
are
not
eigen-
functions.
Obviously,
if
t,
is normalized, J
v'il/;,dT
= I
and
(A,>
=
,
J
",:A""
dT
= a
•.
We have seen
how
the wavefuflctions
may
sometimes be complex
functions.
The
operators
too
can
sometimes be complex (as in
Px
=
-iliil/ilx),
However,
if
postulate
3
is
to
make
any
sense,
the
eigenvalue
of
an
operator
representing
an
observable
must
be a real
quantity
if
we
are
to
interpret it as the value
of
the observable, since this
is
something
that
wecan,
in principle,
measure
in
the
laboratory.
This is an
important
restriction. OJ:!erators
whose
eigenvalues
are
exclusively re'al
are
c;;llea
h--er-m-i"'tianopera!Ors:-Their-intcgrafS-ha"ve-th-e
i:iropertytliat
--
---~;
I
~:~I/;:~~---D;;~~:~:r--
(2.15)
,
where
tm
and
1/;,
are
eigenfunctions
of
A,
The postulates
of
quantum mechanics
47
Furthermore,
as a result
of
this property
of
the
operator
(called
hermiticity) any two eigenfunctions are also orthogonal,
i.e.
(2.16)
If
the eigenfunctions are also separately normalized, so that
I
"':;"'m
dr
=
jlf:""d7
=
J,
then
(2. J
7)
where
oorn
= 0 when n * m and I when n =
m.
Eigenfunctions that lire
both
orthogonal
and
normalized
are
said
to
be
orthonormal.
The
first three postulates capture the essence
of
the operator form
of
quantum
mechanics. There are further postulates,
but
these three are
the most
important.
They
firmly establish the existence
of
the wavefunc-
lion
and
its status as a complete description
of
the state
of
a quantum
particle, the replacement
of
the values
of
observable quantities (classical
mechanics) with their corresponding operators (quantum mechanics)
and
they provide a recipe for using the wavefunctions and operators to
calculate the values
of
the observables. All the rest follows.
Complementary observables
We
saw above
that
if
the
expectation value
of
an operator is calculated
using
one
of
its eigenfunctions, the result
is
equal to the correspond-
,ing eigenvalue.
If
the
quantum
system
of
interest is specified
by
the
. normalized wavefunction
"'.,
then the measurement
of
some properly
•
I
(A,
> (position,
momentum,
energy etc) requires the evaluation
of
the
. integral I .;,:)ilfn
dr
which,
as
we
have seen, is equal to
Q,
if
"'n
is an
eigenfunction
of
)i.
There
is
nothing inherent in this process to limit the
precision with which
(An)
(which is equal to
a.)
can
be
determined, i.e.
there is
in
principle
no
uncertainty associated with
(A.).
Suppose
we
wish to determine a second property, say
(B,),
of
tne
state described by the wavefunction
"'n.
Clearly, from postulate 3,
A A
(B
n
> =
I.;,:
B""
d,
where B
is
the operator corresponding
to
this second
observable (we have
assumed
that
If, is normalized).
If
we
wish
to
determine
(Bn)
simultaneously with the same arbitrarily high preci-
sion
as
we determined
(A,),
then
we
require
(En)
= b, where b,
is
the
corresponding eigenvalue. Hence,
"'n
must be a simultaneous eigenfunc-
. A
lion
of
B.
We
condude
that
in
order
to measure simultaneously two different
observables
of
a
quantum
state to arbitrary precision,
tne
wavefunction