Lecture Notes PH 411/511 ECE 598 A. La Rosa

INTRODUCTION TO QUANTUM MECHANICS ______________________________________________________________________

CHAPTER-10 WAVEFUNCTIONS, OBSERVABLES and OPERATORS 10.5 Properties of Operators

10.5.A Correspondence between bras and kets

Bra Ket

χχ

Ψa

Ψa Ψa

*

**

c ΨΨ c

Ψ cΨ c

Φ

ΦΦ

*b

b b

χ Bm

m

m

m

mmB

m

m

*

mB m

m

m B

10.5.B The action of an operator on “bras” and “kets”

We picture an operator A~

as a mathematical entity that when acting on an arbitrary

wavefunction the result is another wavefunction. For example,

A~

= 3

A~

= dx

d (68)

A~

= I

For a given A~

, we will define its correspondent adjoint operator A

~. Through the definition

we will find out what does A

~do on a wavefunction

A~

= ?

In addition, we would like to know also how to express the action of operators onto wavefunctions in the language of bras and kets. Until now we have defined the action of a linear operator on kets . In this section we will define the action of an operator on bras.

For an operator A~

acting on a ket χ , what does χA~

mean ?

For an operator c acting on a bra , what does A~

mean ?

First, for a given A~

and let’s assume that we know A~

(see (68) for example).

Let’s define now the action of A~

on a ket:

A A ~

~

That is, we define

A~ = A

~ (69)

Action of operator A~

on a ket

The Adjoint operator

Let A~

be an operator. A

~is the adjoint operator of A

~ if it fulfills the following inner product for any arbitrary

wavefunctions and ,

A~

≡ A~ Definition of A

~ (70)

Taking the complex conjugate

A~

A~ (70b)

Exercise: Show that ~A = A

~

A~

≡ A~

Applying the definition to A~

= B~

)~

(B = B~

Using (70b)

= B~

which implies )~

(B = B~

Expression (70b) A~

A~ (where we have interchanged with ) suggests

the following interpretation of an operator acting on a “bra”

A~

A~ (71a)

A A ~

~

A A ~

~

That is, we identify the action of A~

on the bras as follows,

A~

≡ A~ (71b)

Also, since )~

(A = A~

,

A~

≡ A~

Action of operator A~

on a “bra” (72) Notice,

A~ ≡ A

~ = A

~ = A

~ = A

~

Similarly,

A~ = A

~

The two results above indicate there is not ambiguity in the expression,

A~ (73)

on whether the operator is acting on the “bra” or on the “ket”. The results is the same. Summary

A~ = A

~

A~

= A~

(74)

A~

= A~

A~ = A

~ = A

~

Exercise: Show that the adjoint operator A~

is linear.

Exercise. Given the operator dx

dD ~

, what is the operator D~

?

This implies D~

= - D~

Matrix Representation of the adjoint operator

n~m =

~ +nm

= m~ +n

nm ~ = *

~

nm

(75)

Exercise. Show that

)

~

~( BA = B

~ A~

(76)

Example: What is the adjoint operator of the position operator X~

?

( X~

, ) ≡ ( , X~

) =

* x) [X~ ]x d x

=

* x) x x d x =

x * x) x d x

=

[x x) ] * x dx =

[ X~ x) ] * x dx

( X~

, ) = ( X~ )

Since this expression is valid for any arbitrary states and , then

X~

= X~

(77)

That is, the adjoint of the position operator is itself.

10.5.C Hermitian or self-adjoint operators

Many important operators of quantum mechanics have the special property that when you

calculate the Hermitian adjoint you obtain the same operator.

~

= ~

. (78)

Such operators are called the “self-adjoint” or “Hermitian” operators.

Example: The position operator X~

is a self adjoint operator because X~

= X~

, as shown in the

example in the previous section (see expression 77).

Example: Let’s see if the linear momentum operator P~

is self adjoint

( P~

, ) ≡ ( ,P~ ) =

*x) [ i

xd

d

]x dx

= i

* x)xd

ψd

x dx

Integrating by parts

= -i

xd

φd

x) x dx

=

[i

xd

φd

x) ]* x dx

= ( P~ , )

That is,

P

~ = P

~ (79)

Operators associated to mean values are Hermitian (or self-adjoint)

In section 10.4 above we defined quantum mechanics operators associated to a classical

observable quantity. The definition involved the calculation of mean values of observables.

From the fact that mean values are real, we can draw some conclusions concerning the

properties of those operators.

av

f = F~

Since this quantity is real, it will be equal t its complex conjugate

= F~ *

= F~

That is,

F~ = F~

Using definition of adjoint operator

F~ = F~

which implies,

F~ = F~ Operators corresponding to observables (80)

( i.e. operators obtained through the requirement av

f = F~ )

must be hermitians (self-adjoint).

The eigenvalues of a Hermitian (self-adjoint) operator are real.

Let j be and eigenvalue of F~ and j the corresponding eigenvector

F~ j = jj (81)

Since F~ is Hermitian (self-adjoint), F~ = F~ , for any state we will have,

F~ ) = F~ ,

In particular, for =j,

F~ j j = jF~j

Using (81)

jj j = j jj

jj j = j j j

which implies

*

j = j ( for eigenvalues of Hermitian operators) (82)

Eigenfunctions of a Hermitian operator corresponding to different eigenvalues are orthogonal Let,

F~ j = j j and F~ k = k k

where j ≠ k

k F~ j = k j j and F~ k j = k k j

Since F~ =F~ , and j as well as k are real, we obtain,

F~ k j = j k j and F~ k j = k k j

This implies,

j k j = k k , j

( j - k ) k j = 0

Thus,

j ≠ k implies k j = 0 ( for Hermitian operators) (83)

10.5.D Observable Operators

When working in a space of finite dimension, it can be demonstrated that it is always

possible to form a basis with the eigen-vectors of a Hermitian operator. But, when the space is

infinite dimensional, this is not necessarily the case. That is the reason why it is useful to

introduce the concept of an observable operator.

An Hermitian (or self-adjoint) operator ̂ is an observable (84)

operator if its orthonormal eigen-states form a basis. . .

10.5.E Operators that are not associated to mean values

In the previous section we addressed operators associated to i) classical physical quantities

f(x) that depends on the position x, and ii) physical quantities that depend on the momentum p

turn out to be hermitian (self-adjoint) operators.

Here we explore building operators associated to classical physical quantities like xp.

Let’s calculate

*( x) [ PX~

~

] (x ) dx

Naively, let’s call that quantity av

xp

i. e. one would expect that the calculation of the integral will give a positive

number and thus reflect a quantity associated to a classical measurement of xp.

We will see below that this assumption is incorrect. As a matter of fact we will

realize that PX~

~

is not Hermitian.

av

xp ≡

*(x) [ P~

~X ] (x) dx

≡

*(x) [ x )(xdx

d

i

] dx

Rearranging the order of the terms,

≡

x *(x) [ )(xdx

d

i

]dx

dv

Integrating by parts

≡ -

dx

d[ x *(x) ] } [ )(x

i

]dx

≡ -

[ *(x) ] } [ )(xi

]dx

-

[ x dx

d *( x) ] } [ )(x

i

]dx

≡ i -

[x

dx

d *(x) ] [ )(x

i

]dx

≡ i +

[x

*

i

dx

d*(x) ] )(x dx

≡ i +

[ x

i

dx

d (x)]* )(x dx

≡ i +

[ P~

~X ]* (x) )(x dx

≡ i + [

)(* x [ P~

~X ] (x) dx ]*

av

xp ≡ i + *

av xp (85)

This result indicates that av

xp is not a real quantity.

Let’s put the result (85) more explicitly in terms of operators.

Expression (85) can be written more explicitly,

*(x) [ P~

~X ] ( x )dx = i + [

)(* x [ P~

~X ] (x) d x ]*

P~

~X = i + P

~

~X ]*

P~

~X = i + P

~

~X

P~

~X = i +

)~

~

( PX

Although this has been demonstrated for identical bra and ket, the procedure with different

bra and ket would give a similar result. That is,

PX~

~

= i + )

~

~( PX (86)

which shows more explicitly that the operator PX~

~

is not hermitian

Also, since )~

~

( BA = B~ A

~

PX~

~

= i +

~~XP

PX~

~

= i + XP~~

(87)

More properly, this result should be expressed as,

PX~

~

= i I~

+ XP~~

where I~

is the identity operator; ~I

What to do then if a quantity like xp is present in the classical Hamiltonian? How to build the corresponding quantum Operator?

The result in (87), that PX~

~

XP ~~

, indicates that

different quantum mechanics operators

may correspond

to equivalent classical quantities

For example x p, p x, (1/2)( x p + p x ) are equivalent classically.

Still, three different operators PX~

~

, XP~~

, (1/2)( PX~

~

+ XP~~

) can be associated to the same

classical quantity.

The guidance to select the proper operator is:

to manipulate the classical quantity such that the corresponding quantum

mechanics operator result in a Hermitian operator.

In the particular case of building a QM operator associated to the classical quantity xp, it turns

out that, the following selection works well.

~

(1/2) ( PX~

~

+ XP~~

) (88)

Notice this operator fulfills ~

~.

10.6 The commutator

In general, two operator do not commute; that is, BA~~

is different than. The commutator between two operators is defined as,

, A BBA BA~~

~~

]~~

[ (89)

According to (87), the operators X~

and P~

fulfill,

i ]~

, ~

[ PX (90)

Calculation of standard deviations

In this section we will be using terms like 1~

~

22 )( A AA .

What does this term mean? To get familiar first with that terminology, let’s work it out

explicitly for the case in which A~

id the linear momentum operator P~

.

Calculation of ( P~

- < p > )

First, let’s expand in the based formed by the eigenstates of P~

,

=

p p dp (i)

(Note: To indicate the momentum state psometimes we use the notation p = p . )

P ~

=

( P ~

p ) p dp

P ~

=

( p p ) p dp (ii)

Since < p > is a constant, then

< p > =

< p > p p dp (iii)

(ii) – (iii)

P ~

- < p > =

( p - < p > ) p p dp

[ P~

- < p > ] =

( p - < p > ) p p dp (91)

Calculation of ( P~

- < p > )2

In (91), applying again the operator [ P~

- < p > ], one obtains

[ P~

- < p > ]2 =

( p - < p > ) 2 p p dp (92)

This expression indicates that the operator [ P~

- < p > ]2 allows calculating standard deviations.

In effect, from (92) one obtains,

[ P~

- < p > I~

]2 =

( p - < p > )2 p p dp

=

( p - < p > )2 p p dp

[ P~

- < p > I~

]2 =

( p - < p > )2 p 2 dp (93)

Probability with which the term

( p - < p > )2 occurs when taking

measurements in the ensemble

described by

10.6.A The Heisenberg uncertainty relation

For two given observable operators A~

and B~

, let’s define the corresponding standard

deviation (the statistics is taken from an ensemble characterized by the wavefunction ,

1~

~

22 )( A AA (94)

1~

~

2 2)( B BB

To simplify the notation, let’s work with the Hermitian operators a~ and b~

defined as,

1~~

~ A Aa and 1~~

~

B Bb (95)

Notice,

] ~~

[ ]~~[ B ,Ab ,a (96)

2

A and 2

B can then be expressed as,

ψa ψψa ψa 22

Aσ~~~ (97)

ψb ψψb ψb 22

Bσ~~~

Consider the not Hermitian operator C~

,

bλia C~

~~ where is a real constant (98)

Notice: bλia C~

~~ , and

0~~~~

CCC C (99)

0~~

~~ ) () ( ψ bλi abλiaψ

0 ) ]~~ [

~~ ( 222 b,ai b a

Using (97),

0 ~~

222

][ B ,Ai BσσA (100)

Notice that the term ]~~

[ B,A must be a

purely imaginary number.

The function f = f () defined as,

)(f ~~

222

][ B ,Ai BσσA (101)

satisfies, according to (100),

0 )( f

In addition 0)(2 B" f . Therefore the value of at which 0)( 'f is a minimum; such

a value is,

ψ B,A ψ2

i2

Bσ]

~~[min

The value of f at min is ,

2

2

2

2

2

min ]~~

[ 2

1 ]

~

~ [

4

1 )( B,AB,Af

BσσBA

2

2

2 ]

~

~ [

4

1

B,A

BA

According to (100) and (101) this value must be greater or equal to zero.

0 ]~~

[ 4

1 2

2

2

B ,A

BA

222

]~

,~

[ 4

1 BABA Generalized uncertainty principle (102)

where ]~

,~

[ BA , according to (100), is a purely imaginary number.

10.6.B Conjugate observables Standard deviation of two conjugate observables

Two observable operators A~

and B~

are called conjugate observables if their commutator is

equal to i .

iBA ]~

, ~

[ definition of conjugate observables (103)

The result (114) then gives for this type of pair operators the following requirement,

2

BA uncertainty principle for conjugate observables (104)

In particular, the result (90) indicates that X~

and P~

constitutes a pair of conjugate

observables. Hence,

2

px (105)

10.6.C Building a basis out of eigenfunctions common to two observables.

Properties of observable operators that do commute

Some of the theorems listed below are valid even if the operators are not observables. But

we will assume the latter, since the main objective of this section is to show that it is possible to

build a basis out of eigenfunctions common to two observables.

Let’s start with the assumption that the eigenvalues { a1, a2, ,… } and the eigenfunctions

{ 1a , 2a , … } of the operator A~

are known.

Theorem-1:

Let A~

and B~

two observable operator such that 0]~

, ~

[ BA

If a is an eigenfunction of A~

then aB ~

is also an eigenfunction of A~

, with the same eigenvalue.

Proof:

A~ a = a

a

B~

A~ a = a B

~ a

Since A~

and B~

commute

A~

( B~ a ) = a ( B

~ a ) (106)

Thus, the theorem is proven.

This result is illustrated graphically,

1

~a ,A

2a

~

1a

~

2a is the space

generated by the eigenfunctions of

A~

that have

eigenvalue2a

2

~a ,A

1~ a

B

2a

2~ a

B

1a is the space

generated by the eigenfunctions of

A~

that have

eigenvalue1a

2a

1a

2a

The operator B~

acting on an eigenfunction in the space a of A~

gives a state that remains in the space a .

Two cases arise:

i) The eigenvalue a is non-degenerate.

i.e. the eigenvalue-space a associated to a is one dimensional.

B~ a is therefore proportional to a ; that is,

B~ a = b a (107)

In this case a is also an eigenfunction of B~

.

ii) The eigenvalue a is ag -fold degenerate.

That is, there exists ag independent eigenvectors associated to the same eigenvalue a.

A~

a

j

= a a

j

for j = 1, 2, … , ag

The result (106) indicates that the following wavefunctions

B~ a

j

for j = 1, 2, … , ag

are also eigenfunctions of A~

and with the same eigenvalue a.

But, in general, we cannot ensure that the states B~ a

j

are also eigenfunctions of B~

.

1a

2a is the space generated

by the eigenfunctions of

A~

that have eigenvalue2a

1~ a

B

2a

~

2a B

1a is the space generated

by the eigenfunctions of

A~

that have eigenvalue1a

1a

2a

We cannot assert that this is eigen-function of

the operator B~

1~ a

B remains in the space 1

a .

All we can say at the moment is that B~ a

j

belong to the space a ,

For a

j

a we also have B~

a

j

a , for j = 1, 2, … , ag (108)

Or more general,

For any a we also have B~ a .

and the space a is said to be globally invariant under the action of B~

.

Hence, theorem-1 can also be stated as,

If two operators A~

and B~

commute, every subspace of A~

(109)

is globally invariant under the action of B~

.

Theorem-2:

If two observables A~

and B~

commute, and if 1a and 2a are two eigenvectors with

different eigenvalues a1 and a2 respectively, then the matrix element 21~ aa B = 0.

Proof: Since 2 ~ aB

2a and 1

a 1

a then they are orthogonal.

(Eigenvector of hermitian operators that have different eigenvalue are orthogonal; see

expression (83) above).

Theorem-3:

If two observables A~

and B~

commute, one can construct an orthonormal basis with

eigenvectors common to A~

and B~

.

Proof:

Let A~ i

n

= an in

; n = 1, 2, 3, …

i = 1, 2, … , gn

Here gn is the degeneracy of the eigenvalue an.

That is, gn is the dimension of the spacen

a .

Also, n'ni'ii

ni'

n'

How does the matrix representing B~

in the basis { i

n

} look like?

Let’s arrange the basis { i

n

} in the following order

1

1 , 2

1 , … , 11

g ; 1

2 , 2

2 , … , 22

g ; … ; …

We know: For n'n we know that 0~ ii'

nn' B (see theorem 2 above).

For n'n we can say nothing a priori about in

i'n' B ~

(see below).

The matrix representing B~

is then a “block-diagonal” matrix

1

000

000

000

000

0000

0000

0000

0000

0000

0000

0000

0000

000

0000

0000

000

1 2 3 4 5

1

2

3

4

5

Two cases arise:

i) The eigenvalue an is non-degenerate.

Then the block n is a 11 matrix

ii) The eigenvalue an is ng -fold degenerate.

Then B~

i

n

n , but i

n

is not necessarily eigenstate of B~

The block n is a nn gg matrix

Notice, however, the matrix representation of A~

in the sub-base { i

n

; I = 1, 2, …, ng } is

diagonal. Also, it will remain diagonal if the elements of the sub-base are replaced by any

linear combination of its elements.

On the other hand, particular linear combination can be work out in n to obtain

eigenvectors of B~

; that is, we can always diagonalize the block-matrix

Let’s work out explicitly the simple case of ng =2.

a

1 and a

2 are known.

B~ a

1 and B~ a

2 are known.

Let’s express B~ a

1 and B~ a

2 as linear combinations of a

1 and a

2

B~ a

1 = 11 a

1 + 12 a

2 (110)

B~ a

2 = 21 a

1 + 22 a

2

The coefficients jk are known.

We have to find p, q and such that,

b = p a

1 + q a

2 ( p and q are unknown) (111)

is an eigenstate of the operator B~

B~

b = b ( is unknown) (112)

Using (111) in (112),

B~

( p a

1 + q a

2 ) = ( p a

1 + q a

2 ) (113)

Since B~

is a linear operator,

p B~

a

1 + q B~

a

2 = p a

1 + q a

2

Using (110)

p(11 a

1 + 12 a

2 ) +

+ q ( 21 a

1 + 22 a

2 ) = p a

1 + q a

2

( p11 + q 21 ) a

1 +

+ ( p12 + q 22 ) a

2 = p a

1 + q a

2

This implies,

11 p + 21 q = p (114) 12 p + 22 q = q

which can be rewritten as,

q

p

q

p

2212

2111

(115)

The coefficients jk are known

This is an eigenvalue problem. We expect then to obtain a couple of independent solutions,

(p1 q1) with eigenvalue and (116) (p2 q2) with eigenvalue

Using these solutions in expressions (111) and (112) above, gives a couple of eigensates for

the operator B~

.

1

b = p1 a

1 + q1 a

2

Eigenstates of B~

(117)

2

b = p2 a

1 + q2 a

2

It is straightforward to verify that these two states are also eigenstates of A~

.

We have thus shown a procedure to obtain eigenstates common to A~

and B~

. That is,

there exist in n eigenvectors of B~

.

Therefore, for the degenerate case, it is possible to find (118)

in n common eigenvectors to A~

and B~

.

Notice,

The eigenvalues ,gn, may end up all being different.

In that case the n-th block representing B~

will be a diagonal matrix, and thus a unique

basis of eigenfunctions common to A~

and B~

for that block would have been

constructed.

But it may occur that some (or all) values in the set ,gn are equal; that is, the

degeneracy would still persist. The matrix representation for B~

would then still have blocks of dimension greater than 1.

In either case, let’s denote by i n,s

the eigenvectors common to A~

and B~

.

A~ i

n,s = an i

n,s (119)

B~ i

n,s = bs

i n,s

n,s specify the eigenvalues an and bs of A~

and B~

.

i = 1,2, …, nsg allows to distinguish the different basis eigenvectors corresponding to the

same eigenvalues an and bs of A~

and B~

.

The reciprocal of theorem-3 is also valid.

If there is a base composed of eigenvectors common to A~

and ~B , (120)

then 0]~

, ~

[ BA

Proof:

Let { un

} the common base of eigenvectors;

and

~

nnn ua uA ,

~nnn ub uB .

For any state =

nn

n

u c

A~ =

~

nn

n

uA c =

n nn

n

ca u ; ~B =

~

nn

n

uB c =

nnn

n

b u c

AB~~ =

~n nn

n

a uB c =

n nnn

n

ab u c ; ~~BA =

~

nnn

n

b uA c =

nnnn

n

ba u c

That is,

AB~

~

= ~~BA , for any state

Hence, 0]~

, ~

[ BA

Let’s put theorem-3 and its reciprocal in just one statement;

Theorem-4:

Two observables A~

and B~

satisfy 0]~

, ~

[ BA .

If and only if there exists a basis composed of states (121)

that are mutual eigenfunctions of A~

and ~B . .

10.6.D How to uniquely identify a basis of eigenfunctions? Complete set of commuting observables

Let A~

be an observable. Its eigenfunctions { in , i = 1, 2, … , ng / for n=1, 2, 3, … … }

constitute a basis in space . Here ng specifies the degeneracy of the eigenvalue an.

If all ng =1: Then specifying an determines in a unique

way the corresponding eigenfunction n .

That is, there exists only one basis formed by the eigenfunctions of A~

.

If a particular ng is greater than 1:

Specifying an is not sufficient to characterize a basis eigenfunction. Several set

of ng vectors in the sub-set n can be chosen as a base.

That is, the base of eigenfunctions of A~

is not unique. (Therefore we choose an

additional observable and attempt to obtain a unique basis.)

Let’s choose then another observable B~

that commutes with A~

.

Let’s construct a basis comprising eigenfunctions common to A~

and ~B ,

{ i n,s

, i = 1, 2, … , nsg / for n=1, 2, 3, … }.

If all nsg =1: Then specifying an and bs determines in a unique

way the corresponding eigenfunction i n,s

.

That is, there exists only only one basis formed by the eigenfunctions of A~

and ~B .

If a particular nsg >1:

Specifying an and bs do not characterize a basis eigenfunction uniquely. The

diagonalization process of B~

in the subspace n may have rendered an

eigenvalue with multiplicity (degeneracy) nsg greater than 1. Hence several set

of nsg vectors in the sub-set ns can be chosen as a base.

That is, the base of eigenfunctions of A~

and B~

is not unique.

Let’s add then another observable C~

that commutes with A~

and B~

. Then, we repeat the

procedure of diagonalization of C~

in the sub-space ns , … , . And so on.

We repeat adding more operators that commute with the previous ones, until we obtain all

subspaces with degeneracy equal to 1.

Thus, we will have a set of commuting operators { A~

, B~

, … ,Q~

} such that to any set of

eigenvalues { na , sb , … , zq } there exists exactly one eigenfunction. We call such an

observation maximal or complete, and the corresponding set of observables a complete set of

commuting observables.

A set of observables { A~

, B~

, C~

, … } is a complete set of commuting

Observables (CSCO) if there exists a unique orthonormal basis of (122)

common eigenfunctions.

Example. For the case of a particle in a central field, H~

, 2L~

and zL~

constitute a complete set of

commuting observables.

Notice, after an observation with a complete set of commuting observables { A~

, B~

, … ,Q~

} we

do not know the values of all possible observables.

- We cannot ascribe a definite value to an observable W~

that does not commute with all

the observables in the set { A~

, B~

, … ,Q~

}

- Only if W~

commutes with { A~

, B~

, … ,Q~

} can we ascribe a definite value to it.

However, after an observation with a complete set of commuting observables

{ A~

, B~

, … , Q~

} we do of course know the exact wavefunction of the system (the latter is the

common eigenstate). Hence,

- We can calculate the statistics (mean values, probability distributions, etc.) of any

observable W~

.

It should be clear that there is not one unique complete set of commuting observables.

- If we had taken an observable W~

for the first measurement, and W~

does not commute

with every operator of the set { A~

, B~

, … ,Q~

}, we could still obtain a different complete

set of commuting observables.

Example. For the case of a particle in a central field, { H~

, 2L~

, zL~

} form a complete set of

commuting observables.

Since xL~

nor yL~

commute with zL~

, we cannot determine their values simultaneously

with zL~

.

But { H~

, 2L~

, xL~

} or { H~

, 2L~

, yL~

} would be equally acceptable complete set of commuting

observables.

10.7 Simultaneous measurement of observables1

According to classical mechanics:

The state of a system with N degrees of freedom can be determined, at

the time t, by measuring the 2N position and momentum coordinates

)(tqi , )(tpi ; i= 1, 2, …, N .

We can think of these measurements as performed simultaneously, or as carried out in very rapid succession so that the coordinates have not varied appreciable (on accounts of their development according to the equations of motion) during the time required for all the measurement.

But it is inherent in the concepts of classical physics that, the measurements do not affect the values of the variables describing the

system (i.e. the measurements do not affect the state of the system); that

the order of measuring the 2N coordinates is immaterial.

Once the 2N coordinates are known, all conceivable observables can be calculated.

In quantum mechanics the situation is entirely different. It is no longer possible to specify the simultaneous values of all observables of a system. In general, observables are mutually incompatible; i.e. measuring one of them completely destroys any knowledge about the others.

For if two observables A~

and B~

possess a mutual eigenfunction b a, , i.e. b a,b a, a A

~ and b a,b a, b B

~, then in the state b a, ,

A~

has the value a and B~

the value b .

It follows that successive measurements (performed sufficiently quickly or the

time variation of the system to be unimportant) would give these results.

However,

if A~

and B~

have no common eigenfunctions, successive alternate

measurements of A~

and B~

, which necessarily leave the system in eigenstates

if A~

and B~

respectively, destroy the information obtained earlier.

Such an incompatibility of operators is related to the uncertainty principle

(expression (102) above) 2

22 ]

~,

~ [

4

1 BABA .

The concepts of complete set of commuting observables (CSCO) addresses the conditions to

ascribe simultaneous values respectively to several observables.

10.7.A Definition of compatible (or simultaneously measurable) operators

If A~

and B~

are measured on a system (in quick succession) with the

results a and b, and an immediate re-measurement of A~

or B~

(123) necessarily reproduces these results, whatever the initial state of the

system, then we say that A~

and B~

are compatible or simultaneously measurable.

10.7.B Condition for observables A~

and B~

to be compatible

It is rare for two arbitrary observables to be compatible. It is interesting to establish the

condition under which two observables are compatible. One clue comes from the fact that if

two observables A~

and B~

possess a mutual eigenfunction b a, (i.e b a,b a, a A ~

and b a,b a, b B ~

) then, successive measurements with A~

and B~

would give the results

a and b and the system will remain in the same state i n,s

. It is straightforward to infer then

that if A~

and B~

were to have a common basis of mutual eigenfunctions, then they would

display their compatibility when measuring any state (not only the eigenfunctions). Indeed, we

have a theorem of fundamental importance:

A necessary and sufficient condition for A~

and B~

to be compatible

is for them to possess a complete orthonormal set of simultaneous (124)

eigenfunctions.

Proof:

i) Necessary condition

If A~

and B~

are compatible then B~

A~ = A

~B~ ; that is ][

~

~B ,A = 0. From theorem-4

(expression (121) above) a basis of common eigenfunctions exists.

ii) Sufficient condition

Let { i n,s

, i = 1, 2, … , nsg / for n, s = 1, 2, 3, … } be a complete set of simultaneous

eigenfunctions (with an and bs specifying the corresponding eigenvalues) of A~

and B~

.

Let’s assume the system is initially in a state .

= is,n,

ss Ψi n,

i n,

A~

= is,n,

ss ΨA i n,

i n,

~

= is,n,

ss Ψa i n,

i n, n

B~

A~

= is,n,

ss ΨaB i n,

i n, n

~

= is,n,

ss Ψab i n,

i n, ns

(125)

Similarly,

A~

B~

= is,n,

ss Ψba i n,

i n, sn

(126)

From (125) and (126).

][~

~

B ,A = 0.

Therefore B~

( A~ )= A

~( B

~ ), which implies that A

~and B

~are compatible.

Putting together the results (121) and (124):

Theorem: Any one of these conditions implies the other one;

(1) A~

and B~

are compatible

(2) A~

and B~

possess a basis composed of simultaneous eigenvalues. (127)

(3) A~

and B~

commute

Summary

When two observables A~

and B~

are compatible, the measurement of B~

does not cause

any loss of information previously obtained from a measurement of A~

(and vice versa) but,

on the contrary, adds to it. Moreover, the order of measuring the two observables is of no

importance.

On the other hand, if A~

and B~

do not commute, the preceding arguments are no longer

valid.

10.8 How to prepare the initial quantum states2

In the previous sections we have introduced i) wave functions as mathematical entities that

represent states, and ii) operators as representing the physical quantities of systems. We have

described how to predict the probable outcomes of a measurement once a wave function is

known. We have also seen the association between operators and the average values of the

physical quantities.

What has not been addressed yet is how to prepare quantum state ensembles upon which

measurement can be made. We raised this question in in Section 6.4 of Lecture 6, but we

postponed its discussion until we were armed with better mathematical tools and expanded

concepts (like QM operators, observables, and, in particular, commutation properties). It is

time then to retake and explain those questions.

Let’s start mentioning,

A system, classically described as one of n degrees of freedom, is

completely specified quantum mechanically by a normalized wave

function ( q1, q2, … , qn ) which contains an arbitrary factor of

modulus 1. All possible information about the system can be

derived from this wave function.

(128)

How to build a QM wave function?

How to use the classical measurements of the physical properties of a

system to build a QM wave function?

It turns out,

“the state of a system, specified by a wave function in the Hilbert space,

represents a theoretical abstraction. One cannot measure the state directly

in any way. What one does do is to measure certain physical quantities such

as energies, momenta, etc., which Dirac referred to as observables. From

these observations one then has to infer the state of the system.”

We have introduced wavefunctions as representing states and operators as representing the

physical observables of the system. We must now see how these quantities are related to

actual measurements.

Given a system in a definite state, described by a given wavefuntion, how can we predict its physical properties, which are to be compared with experiments?

How to predict

energy linear momentum

angular momentum

of the system

?

Given

Conversely, how do we incorporate information about a system, gained as a result of measurement, into the wavefunction? How, for example, do we determine the initial state

0 a system upon which we can predict, using the Schrodinger, the state at a later time t?

Having measured Energies Linear momenta Angular momentum

?

How to infer the state of the system?

10.8.A Knowing , what can we predict about eventual outcomes from the measurement?

Consider F~ to be an observable operator associated to the observable f, which has a

complete orthonormal set of eigenfunctions { 1, 2, … } and corresponding eigenvalues

{ f1, f2, … }

If a system is in an eigenstate n of F~ , then measuring f gives a definite result f n.

In general, a system will be in a state , which is not an eigenstate of F~ . But it can be

expanded in the form = m

m cm . Measuring f on a system in a state no longer

leads to a definite result; rather there exists a probability distribution to find anyone of the values f1, f2, …

P ( f n ) = 2

nc Postulate (129)

The most important implication of this postulate is that, in contrast to classical mechanics, a

system with the maximal specification of its state (namely with a given wavefunction )

still shows dispersion; measuring f does not necessarily leads to one definite result.

This means, if we measured f on a large number of systems (the ensemble characterized by

) then the different possible results f1, f2, … would occur with a frequency given by

(129).

10.8.B After a measurement, what can we say about the state of a system ?

We come now to the converse question of how information obtained about a system

through measurements is incorporated in the wavefunction.

Let’s assume the observable f is measured on a system in the (130)

state , and the value f n (an eigenvalue of F~ ) is observed.

If this statement is to mean anything, it must imply that,

If one repeats the measurement of f sufficiently quickly one (131) necessarily again finds the same value f n.

[In general, this only holds for a sufficiently short interval of time between the two measurements (for if the interval is too long the system will have changed appreciably, according to the Schrodinger equation).]

If the second measurement is to have the definite result f n,

then the wavefunction ' (representing the system immediately after the first

measurement) must be an eigenfunction of F~ belonging to the eigenvalue f n.

Let’s make some predictions about ':

Case: The state before the first measurement is unknown

If f n is non-degenerate, then the state after the measurements ' is n .

If fn is gn-fold degenerate (i.e. r n for r =1,2, … gn have the same eigenvalue

fn), then we only know that ' lies in a gn -dimensional sub-space,

n

nnn

n

g

gr

r r c

c 1

1

1

2 r

r

' ( rnc unknown) (132)

The amplitudes rnc are in general unknown.

Case: The state before the first measurement is known

We calculate

rrnnc ( r

nc known) (133)

and postulate that the wavefunction ' is given by (132) as well. That is, before the measurement, we have

n

nnn

n

k

grr

grnk

k c

c

c1

1

1

2 r

r

(134)

n

nnn

n

grr

grnk

kk

c1

1

1

2 r

r

If after a measurement on that measurement renders the eigenvalue fn, then:

n

nnn

n

grr

gr

c

c1

1

1 '

2 r

r

(135)

n

nnn

n

grr

grc

1

1

1

2 r

r

In the particular case that before the measurement we knew that is an eigenfunction belonging to the eigenvalue f n then, the above simply means that,

' = (136)

That is, the wavefunction is unchanged by the measurement.

We have just described how to incorporate information about a system, obtained as a result of

experiments, into the wavefunction describing that system.

In general, before the measurement, there may be a variety of possible results. But once the

experiments have been carried out and one particular result obtained, we can “discard” most of

the wavefunction (corresponding to all the results which were not observed) and retain only

that part which on immediate measurement would lead to the same result.

We are thus partially able to determine the state of a quantum system. This determination is

complete only if the eigenvalue measured is non-degenerate. But in general it will be

degenerate and further measurements are required to determine the state completely. The

latter is achieved with simultaneous measurements of several observables: With a complete set

of commuting operators.

For the state of the system after a measurement to be completely

defined uniquely by the result obtained, this measurement must (137)

be made on a complete set of commuting observables (CSCO).

Note: The measurement of a CSCO enables us to prepare only states comprising the unique

basis associated with this CSCO. However, changing the set of observables allows us to obtain

other states of the system

1 This section follows closely the book by Mandl, “Quantum mechanics” 2 This section follows closely the book by Mandl, “Quantum mechanics”