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Chapter Eight

The Formal Structure of Quantum Mechanical Function Spaces

Introduction

In this chapter, we introduce the formal structure of quantum theory. This formal

structure is based primarily on the concepts of linear vector spaces. Let's review the

formalism of representing the position of a particle in three-dimensional space using

conventional 3-dimensional vectors. First we establish some arbitrary coordinate system,

which we designate here by the coordinates , , and . We can then write the vector B C D Vt

in terms of vectors (i.e., in terms of predefined vectors which are fixed in directionunit

along the , , and directions and which are of unit length) according to the equationB C D

V œ B 3 � C 4 � D 5t s s s (8.1)

x

y

z

θ

φ

R

These unit vectors , , and are specifically chosen to be perpendicular to each other and3 4 5s s s

of unit length, so that , and , etc. This means that if we take the3 † 3 œ " 3 † 4 œ 3 † 5 œ !s s s s s s

dot product of one of the unit vectors (say ) with the vector we obtain3 Vs t

3 † V œ 3 † 3 B � 3 † 4 C � 3 † 5 D œ Bs s s s s st s (8.2)

Therefore, the of the vector component Vt in a given direction can be written as theprojection of the vector onto the corresponding unit vector.

B œ 3 † Vs t

C œ 4 † Vs t

D œ 5 † Vs t

(8.3)

These unit vectors are said to “span the space” because any vector can be representedVt

by some linear combination of these unit vectors. Although we could span three

Quantum Mechanics (Griffith) 2

dimensional space with three unit vectors which are not co-linear, the choice ofany

orthogonal vectors makes things somewhat simpler.unit

Let's change notation a bit so that we can write things in a form more “normal” to

quantum mechanics. We could write the unit vectors in the form and so thatlBÙß lCÙß lDÙ

the vector is represented by the equationVt

lVÙ œ BlBÙ � ClCÙ � DlDÙ (8.4)

If we wanted to extend our notation to be valid for cases where there are more than three

dimensions, we would need to introduce new coordinates and components in addition to

Bß Cß D and . This is often accomplished by introducing a different notation for our unit

vectors

lBÙ œ l/ Ù

lCÙ œ l/ Ù

lDÙ œ l/ Ù

"

#

$

(8.5)

and for our components

B œ B

C œ B

D œ B

"

#

$

(8.6)

In this notation, we would express the vector with the notationlVÙ

lVÙ œ B l/ Ù � B l/ Ù � B l/ Ù" " # # $ $ (8.7)

where the -component of the vector is given byB"

B œ Ø/ lVÙ" " (8.8)

The notation must be equivalent to a dot product. The unit orthogonal vectors allØ/ lVÙ"

satisfy an equation of the form

Ø/ / Ù œ3 4 34| (8.9)$

For an -dimensional space we simply allow the subscripts to range from 1 to . Thus,R Rthe -dimensional vector can be expressed by the equation8 l Ùα

l Ù œ + l/ Ù � + l/ Ù �â� + l/ Ù œ + l/ Ùα " " # # 8 8 3 3

3œ"

8� (8.10)

where the are simply scalar coefficients (which could be complex), and the 's are+ = l/ Ù3w

3

orthoganal unit vectors. Once a set of unit vectors has been established, the vector can be

represented solely based upon its components, i.e.,

l Ù œ + ß + ßá ß +α � �" # 8 (8.11)

just as you might designate the location of a particle based solely upon its coordinates� �Bß Cß D in a predefined coordinate system.


This looks very similar to the the situation we have when we solve the infinite

square well problem. There are an infinite (though countable) set of eigenfunctions from

which we construct the most general wave function solution. We might write this out

symbolically as

| | (8.12)< <Ù œ + Ù�8

8 8

The individual coefficients can be determined by utilizing the analogy of the dot product

of the unit vector with the vector |<Ù

Ø Ù œ + Ø Ù œ + œ +< < < < $7 8 7 8 8 78 7

8 8

| | (8.13)� �We represented the orthonormality condition for the eigenfunctions of the infinite square

well by the equation

( < < $*7 8 78ÐBÑ ÐBÑ .B œ (8.14)

It should be obvious that the “dot product” notation is similar to the integral

orthonomality conditions, or that

Ø Ù œ ÐBÑ ÐBÑ .B œ< < < < $7 8 8 787| (8.15)( *

This is a valuable “short-hand” way of writing out some of our integral expressions.

However, there is a slight difference between these two notations which we will discuss

later on.

Linear Vector Spaces

Linear vector spaces are defined with regard to the way these vectors can be

manipulated mathematically. We define the vector addition of any two vectors andl Ùαl Ù" by the equations

l Ù � l Ù œ Ùα " #| (8.16)

and

l Ù � l Ù œ + � , ß + � , ßá ß + � ,α " � �" " # # 8 8 (8.17)

This means that the result of adding any two vectors is itself a vector.

Vector addition is both (i.e., you can reverse the order of additioncommutative

without changing the result)

l Ù � l Ù œ l Ù � l Ùα " " α (8.18)

and associative

l Ù � l Ù � l Ù œ l Ù � l Ù � l Ùα " # α " #� � � � (8.19)


We define the as that vector satisfying the equationnull vector l!Ù

l Ù � l!Ù œ l Ùα α (8.20)

so that

l!Ù œ !ß !ßá ß !� � (8.21)

and the or or a vector by the vector which satisfies theinverse negative l Ù l" Ùα αequation

l Ù � l" Ù œ l!Ùα α (8.22)

so that

l" Ù œ "+ ß"+ ßá ß"+α � �" # 8 (8.23)

is defined as the multiplication of a scalar times a vector.Scalar multiplications

This is just another vector defined by the equationproduct

-l Ù œ l Ùα # (8.24)

or

-l Ù œ -+ ß -+ ßá ß -+α � �" # 8 (8.25)

Obviously the multiplication of the scalar times a vector gives the inverse of that""vector. Scalar multiplication is with respect to vector additiondistributive

+ l Ù � l Ù œ +l Ù � +l Ù� �α " α " (8.26)

and with respect to scalar addition

� �+ � , l Ù œ +l Ù � ,l Ùα α α (8.27)

It is also associative with respect to multiplication of scalars

+ ,l Ù œ +, l Ù� � � �α α (8.28)

In three dimensions we define two types of vector multiplication - the dot product

and the cross product. Only the dot product (called the inner product), however, has a

natural extension into multi-dimensions. In three dimensions, the inner product of two

vectors

E † F œ lEllFlt tt t cos (8.29))

is a scaler which can be thought of geometrically as the length of one vector multiplied by

the projection of the other vector onto the first, as seen in the diagram below. If the two

vectors and are to each other, then , and if they are E F E † F œ !t tt tperpendicular parallel

to each other, and of unit length, then . This means of course that theE † F œ lEllFlt tt t


absolute magnitude of a vector can be expressed as

lEl œ E † Et t tÈ (8.30)

θ

A

B From this geometrical definition, we can see that

E † Ft t

lEllFlt tœ Ÿ "cos (8.31))

or, in our new notation,

Ø+ ,Ù

Ø+ +Ù Ø, ,ÙŸ "

|

| |(8.32)È

Although the inner product | may be complex in general, the absolute value of thisØ+ ,Ùlast equation gives

lØ+ ,Ùl Ÿ Ø+ +Ù Ø, ,Ù| | | (8.33)#

which is known as the Schwartz inequality.

The Formal Structure Of Quantum Mechanics In previous chapters, we have looked at the solutions to several one-dimensional

problems. In each case, we started with the time-dependent Schrodinger equation

" ÐBß >Ñ � Z ÐBÑ ÐBß >Ñ œ 3h ÐBß >Ñh ` `

#7 `B `>

# #

#< < < � �8.34

which can be written in operator form as

L Bß >Ñ œ I ÐBß >Ñs s< <( 8.35� �where the Hamiltonian operator is given byLs

L œ � Z ÐBÑs :s

#7s

# � �8.36

and the total energy operator is given by I I œ 3h `Î`>Þs s


If the potential energy is time-independent, we showed that the solution to the

time-dependent Schrodinger equation is given by

< <ÐBß >Ñ œ ÐBÑ/"3I>Îh 8.37� �where is the solution to the time-independent Schrodinger equation<ÐBÑ

" ÐBÑ � Z ÐBÑ ÐBÑ œ I ÐBÑh `

#7 `B

# #

#< < < � �8.38

This last equation can also be expressed in operator form as

L ÐBÑ œ I ÐBÑs < <E E � �8.39

which is the so-called energy equation.eigenvalue

The subscript on the eigenfunction is there to remind us that the eigenfunctions are

functions of the energy of the system, and that for each different value of the energy, we

have a different eigenfunction. In some cases the energy spectrum is discrete (such as the

infinite square well and the simple harmonic oscillator), but in others the spectrum is

continuous (such as the free particle). The most general solution to the Schrodinger

equation (since it is a linear equation) is the sum (or integral) over all possible states, or

G < <ÐBß >Ñ œ E ÐBß >Ñ œ E ÐBÑ /� � � �8 8

8 8 8 8"3 >=8 8.40

where we have explicitly written the expansion as if it were of a discrete set of functions

with . In the case where the energies form a continuum, we write=8 8œ I Îh

G <ÐBß >Ñ œ 1ÐIÑ ÐBÑ / .I( � �-

+E /

∞

∞

I"3 > h 8.41

In both of these expansions, it is assumed that the eigenfunctions (both discrete and

continuous) form a of solutions so that any “vector” in the vectorcomplete set orthogonal

space can be represented by a linear combination of these “basis vectors”. For the case

where the energies are discrete, the orthonormality condition is expressed by the equation

( � �"∞

�∞

7 8 7ß8< < $* ÐBÑ B .B œ( ) 8.42

and for the case of continuous energy levels, by

( � �"∞

�∞

I Iw< < $*

wÐBÑ B .B œ ÐI " I Ñ( ) 8.43

A short-hand notation (called the “bracket”, or “bra” “ket” notation) was introduced"by P. A. M. Dirac to help simplify these complex expressions. (However, there is the

possibility of sometimes getting lost in the simplification!) To understand this notation,


we begin with an arbitrary eigenvalue equation which may be written as

U? ÐBÑ œ ; ? ÐBÑs8 8 8 � �8.44

where is an eigenfunction of the operator with corresponding eigenvalue .? ÐBÑ U ;s8 8

Dirac proposed that we write an eigenvector in terms of the eigenvalues associated with

that eigenvector, i.e.,

U l; Ù œ ; ; Ùs8 8 8| 8.45� �

Therefore, we can write the eigenvalue equation of the Hamiltonian

L lI Ù œ I lI Ùs8 8 8 � �8.46

where is an eigenvector of enegy. (The implication of the subscript is that thislI Ù8eigenvalue equation forms a discrete set of eigenvectors, and this will be our assumption

throughout our development of the Dirac notation.) We write the expression for

representing the orthonormalization condition by

ØI lI Ù œ7 8 78$ � �8.47

where | is to be the complex conjugate of | , and where theØI I Ù8 8understood

combination of the two is similar to our normalization integral. The complex conjugate

vector | is called the “bra”, while the eigenvector | is called the “ket”.ØI I Ù8 8

The most general solution to the time-dependent Schrodinger equation, in terms of

the eigenfunctions of position, is given by

G <ÐBß >Ñ œ E ÐBÑ /� � �8

8 8"3 >=8 8.48

If we associate the time-dependence with the expansion constant, as follows

E Ð>Ñ œ E /8 8"3 >=8 � �8.49

we can write our expansion of the eigenfunctions in the form

G <ÐBß >Ñ œ E Ð>Ñ ÐBÑ� � �8

8 8 8.50

The Dirac notation for eigenvectors is similar in form to this equation, and is given by

l Ð>ÑÙ œ E > lI ÙG � � � � �8

8 8 8.51

where the time-dependent notation in the “ket” tips us off that the expansion constants

contains the time factor!

As mentioned earlier, Dirac's notation for the combination of eigenvectors is not

quite equivalent to the position-time equations which we have been using. You may

remember that we stated in an earlier chapter that the state of the system could be

represented either in position (configuration) representation or in momentum


representation. This is specifically denoted in the Dirac notation in the following way.

The state of the system expressed in configuration representation and in momentum

representation are denoted, repectively, by

G <ÐBß >Ñ ´ ØB l Ð>ÑÙ � �8.52

and

9 <Ð:ß >Ñ ´ Ø: l Ð>ÑÙ � �8.53

in Dirac notation.

The “ket” | (t) is not to ( ), but is even more fundamental. This< GÙ Bß >equivalent

“ket” is called the state of the system. This state may be ontovector vector projected

position space by taking a of | with | , an eigenvector of the positiondot product <� �> Ù BÙoperator ! It may likewise be projected onto momentum space by taking a Bs dot product

of with , an eigenvector of the momentum operator (The state vector may bel Ð>ÑÙ l : Ù :Þs<projected onto any other basis set as well.) In a similar manner we will take the position

representation of the eigenfunctions of the Hamiltonian to be

<8 8ÐBÑ ´ ØB l I Ù � �8.54

and we will represent the complex conjugate of this eigenfunction with the notation

<8 8*ÐBÑ ´ ØI l BÙ 8.55� �

We have previously expressed the normalization condition for the energy eigenstates in

position representation by the equation

( � �"∞

�∞

7 8 78< < $* ÐBÑ ÐBÑ .B œ 8.56

In Dirac notation, this equation becomes

( � �"∞

�∞

7 8 78ØI l BÙØB l I Ù .B œ 8.57$

But we have already stated that the orthonormality condition for the eigenvectors satisfies

the equation

ØI lI Ù œ7 8 78$

This means that the integral over given byB

” œ l BÙØB l .B œ "( � �"∞

�∞

8.58

must somehow act like a unity operator.


Examples of Dirac Notation

To gain some familiarity with the use of the Dirac notation, let's examine the

expectation value of the energy, . The eigenvalue equation for the Hamiltonian isØLÙs

given by

L lI Ù œ I lI Ùs8 8 8 � �8.59

where the energy eigenvectors representation-dependent. Now, the expectationare not

value of the Hamiltonian for a system in a given eigenstate of that system is given by

ØLÙ œ ØI lL lI Ù œ I ØI lI Ù œ Is s7 8 8 7 8 8 78$ � �8.60

regardless of the specific representation which is used! Since the energy eigenvectors

derived from the time-independent Schrodinger equation form a complete set of vectors,

we can express a general state of the system in terms of these eigenvectors according to

the equation

l Ð>ÑÙ œ E Ð>Ñ l I Ù< � � �8

8 8 8.61

The expectation value for the energy in this case is given by

ØLÙ œ Ø lL l Ù E Ð>Ñ ØI L E Ð>Ñ l I Ùs s s< <(t) (t) = | 8.62 � � � �7 8

7 7 8 8*

Now, since the coefficients are just constants times the time-factor, and since is notLs

explicitly time-dependent, the Hamiltonian operator simply acts on the eigenvector ,l I Ù8giving

ØLÙ œ E Ð>ÑE Ð>ÑØI lL lI Ùs s

œ E Ð>ÑE Ð>Ñ I ØI lI Ù

œ E Ð>ÑE Ð>Ñ I

œ lE Ð>Ñl I

��

� �7ß8

7 8 7 8

7ß87 8 8 7 8

7ß87 8 8 78

77

#7

*

*

* $

8.63

So we see that the expectation value of the Hamiltonian is just the weighted average of

the energies of the different states of the system under study.

In the same way, we can show that the coefficients are probability amplitudes for

finding the system in a particular state of the system. We can see this by looking at the


normalization condition as it is applied to a general state of the system. We have

Ø l Ù œ E Ð>Ñ Ø I E Ð>Ñ l I Ù

œ E Ð>ÑE Ð>ÑØI lI Ù

œ E Ð>ÑE Ð>Ñ

œ lE Ð>Ñl œ "

< <

$

(t) (t) | 8.64 � ��

� �7 8

7 7 8 8

7ß87 8 7 8

7ß87 8 78

77

#

*

*

*

This shows that the sum of the probability of finding the system in each particular

eigenstate of the system is always equal to unity. This will be true even if the system is in

a single eigenstate, say , in which case the coefficient would be unity.l I Ù E; ;

We next consider how we are to find the individual coefficients for the expansion

of a general wavefunction. The general expansion is given, as we stated above, by the

equation

l Ð>ÑÙ œ E Ð>Ñ l I Ù< � � �8

8 8 8.65

If we now take the dot-product of our general vector onto a particular energy eigenvector

(which represents one of the basis vectors which span the solution space of our particular

problem) we obtain

ØI l Ð>ÑÙ œ E Ð>Ñ ØI lI Ù

œ E Ð>Ñ

œ E Ð>Ñ

7 8 7 8

8

8

8 78

7

<

$

��

� �8.66

This means we can express the general wave equation as

l Ð>ÑÙ œ E Ð>Ñ l I Ù

œ ØI l Ð>ÑÙl I Ù

<

<

��

� �8

8 8

8

8 8

8.67

This last equation can be written in a slightly different fashion as:

l Ð>ÑÙ œ lI ÙØI l l Ð>ÑÙ< < � � �8

8 8 8.68


Written in this way, we see that the expression in parentheses is a special kind of unity

operator

” œ lI ÙØI l œ "� � �8

8 8 8.69

much like the unity integral we introduced above. This last equation is called the closure

relationship. It implies that the entire solution space is spanned by the set of basis vectors

l I Ù8 .

Next, we examine the Dirac notation for the free-particle wave function in terms

of the various energy (or momentum) components. We have shown earlier that the most

general form for the free-particle wavefunction must be given by

< 91

ÐBß >Ñ œ Ð:ß >Ñ .:/

# h( È � �"∞

�∞ 3:BÎh

8.70

so that the wavefunction is normalizable. We can express the position dependent and the

momentum dependent wavefunctions, respectively, in Dirac notation using the equations

< <

9 <

ÐBß >Ñ œ ØB l Ð>ÑÙ

Ð:ß >Ñ œ Ø: l Ð>ÑÙ

� �8.71

Now, if we begin with a simple statement of equality of the wavevector with itself, we

have

l Ð>ÑÙ œ l Ð>ÑÙ

ØB l Ð>ÑÙ œ ØB l Ð>ÑÙ

ØB l Ð>ÑÙ œ ØB l l Ð>ÑÙ

ØB l Ð>ÑÙ œ ØBl :ÙØ: l Ð>ÑÙ.:

< <

< <

< ” <

< <(

� �

"∞

�∞

8.72

where we have used the in momentum spaceunity operator

” ´ l :ÙØ: l .:( � �"∞

�∞

8.73

This definition is analogous to the definition we stated earlier for position space and for

the energy eigenvectors,

” ´ lI ÙØI l� � �8

8 8 8.74

You will notice that for eigenvectors which take on discrete values we utilize a

summation, while for eigenvectors which take on a continuous range of values we utilize


an integral. Thus, comparing

ØB l Ð>ÑÙ œ ØBl :ÙØ: l Ð>ÑÙ.:< <( � �"∞

�∞

8.75

with our previous expression for the free particle wave function

< 91

ÐBß >Ñ œ Ð:ß >Ñ .:/

# h( È � �"∞

�∞ 3:BÎh

8.76

which we now write as

ØB l Ð>ÑÙ œ Ø: l Ð>ÑÙ .:/

# h< <

1( È � �"∞

�∞ 3:BÎh

8.77

we see that we must identify the expression , which is the projection of theØBl :Ùmomentum eigenvector onto the position eigenvector, with the equation

ØBl :Ù œ/

# h

3:BÎh

È � �1

8.78

This also implies that

ØBl :Ù œ Ø: l BÙ œ/

# h

*"3:BÎh

È � �1

8.79

In working with Dirac notation, the expressions | and | will often occur.ØB B Ù Ø: : Ùw w

We now want to determine just what these expressions are equivalent to. We begin with

the expression | and insert the momentum unity operator, obtainingØB B Ùw

ØB B Ù ØB :ÙØ: B Ù .:

h h.:

h.:

B " B

| = | | (8.80)

= e e

2 2

= e

2

= ( )

w w

∞

∞

∞

∞:B h :B h

∞

∞: B B h

w

(

( È È(

-

-

i / -i /

-

i ( - )/

1 1

1

$

w

w

Thus,

ØB B Ù ´ B " B| ( ) (8.81)w w$


By a similar process we can show that Likewise, we could show that

Ø: : Ù ´ : " :| ( ) (8.82)w w$

Now, let's examine the expectation value for the momentum using Dirac notation.

We begin with the expectation value defined by the equation

Ø:Ù Ø > : > Ùs = ( )| | ( ) (8.83)< <

Utilizing two distinct unity relationships for the momentum, the expectation value can be

expressed by

Ø:Ù Ø > : > Ù Ø > :ÙØ: : : ÙØ: > Ù .: .:s s = ( )| | ( ) = ( )| | | | ( ) (8.84)< < < <(-∞

∞

w w w

Now we know the result of operating on the eigenvector | : it's just the value of the: :Ùsmomentum, , or:

: : Ù : : Ùs | = | (8.85)w w w

Thus, the expectation value becomes

Ø:Ù Ø > : > Ù Ø > :Ù : Ø: : ÙØ: > Ù .: .:s = ( )| | ( ) = ( )| | | ( ) (8.86)< < < <(-∞

∞

w w w w

Using the identity | = ( ), this becomesØ: : Ù : " :w w$

Ø:Ù Ø > : > Ù Ø > :Ù : : " : Ø: > Ù .: .:s = ( )| | ( ) = ( )| ( ) | ( ) (8.87)< < < $ <(-∞

∞

w w w w

or

Ø:Ù Ø > : > Ù Ø > :Ù : Ø: > Ù .:s = ( )| | ( ) = ( )| | ( ) (8.88)< < < <(-∞

∞

which is equivalent to

Ø:Ù : > : : > .: = ( , ) ( , ) (8.89)(-

*

∞

∞

9 9

This is the expectation value of the momentum as represented in momentum

representation (or momentum space).


To obtain the corresponding expression in configuration space, we make use of two

distinct unity operators in position space

Ø:Ù Ø > : > Ù Ø > BÙØB : B ÙØB > Ù .B .Bs s = ( )| | ( ) = ( )| | | | ( ) (8.90)< < < <( (- -∞ ∞

∞ ∞

w w w

In this equation, we have the quantity | | . But we don't know the result of ØB : B Ù :s sw

operating on the eigenvector . However, we can make use of the unity operator inlBÙmomentum space and write

ØB : B Ù œ ØB : : ÙØ: B Ù .:s s

œ : ØB : ÙØ: B Ù .:

| | | | | (8.91)

| |

w w w w w

∞

∞

∞

∞

w w w w w

(

(-

-

Now, the projection of the momentum vector onto position is given by

ØBl :Ù œ/

# h

3:BÎh

È 1(8.92)

and the derivative of this function with respect to position gives

` 3: 3:

`B h hh hœ œ ØBl:Ù

e e

2 2(8.93) È È

3:B h 3:B h/ /

1 1

Similarly,

` 3: 3:

`B h hh hœ " œ " Ø: BÙ |

e e

2 2 È È

"3:B h "3:B h/ /

1 1

so we can write

: ØBl: Ù œ " h ØBl: Ù`

`B

: Ø: lB Ù œ h Ø: lB Ù`

`B

w w w

w w w w w

i (8.94)

i w

(which means than the number is real!).:

ØB : B Ù ØB :Ù h Ø: B Ù .:s`

`B| | = | i | (8.95)w w

∞

∞

w( Œ -


Substituting this back into the equation for the expectation value of and separating out:the dependence on we haveBw

Ø:Ù Ø > : > Ù Ø > BÙØB :Ù h Ø: B Ù ØB > Ù.B .: .Bs`

`B = ( )| | ( ) = ( )| | i | | ( ) < < < <( ( (œ Œ

- -∞ ∞

∞ ∞

"∞

�∞

ww w w

We must integrate the term in braces by parts with respect to Bw

( Œ Ÿº ( Œ

"∞

�∞

ww w w

w w w w w

"∞

�∞

"∞

�∞

w

i | | ( )

| | ( ) | | ( )

(8.96)h Ø: B Ù ØB > Ù.B`

`B

œ 3h Ø: B ÙØB > Ù " Ø: B Ù ØB > Ù .B`

`B

<

< <

The wavefunction | ( ) goes to zero at infinity, so that this equation becomesØB > Ùw <

( (Œ Œ "∞ "∞

�∞ �∞

w ww w w w w wi | | ( ) | ( ) | (8.97)h Ø: B Ù ØB > Ù.B œ "3h ØB > Ù Ø: B Ù.B

` `

`B `B< <

The expectation value of can now be expressed by the equation:

Ø:Ù Ø > BÙØB :Ù " h ØB > Ù Ø: B Ù .B .: .B`

`B = ( )| | i | ( ) | (8.98)( ( ( Š ‹- - -∞ ∞ ∞

∞ ∞ ∞

ww w w< <

With a slight rearrangement, we write

Ø:Ù Ø > BÙØB :ÙØ: B Ù " h ØB > Ù .B .: .B`

`B = ( )| | | i | ( ) (8.99)( ( ( Š ‹- - -∞ ∞ ∞

∞ ∞ ∞

w w ww

< <

We have eliminated all dependence upon the momemtum, except for what you will

recognize as the unity operator in momentum space. This equation, then reduces to

Ø:Ù Ø > BÙØB B Ù " h ØB > Ù .B .B`

`B = ( )| | i | ( ) (8.100)( ( Š ‹- -∞ ∞

∞ ∞

w w ww

< <

Here we recognize | = ( ), so that integration over finally givesØB B Ù B " B Bw w w$

Ø:Ù B > " h B > .B`

`B = ( , ) i ( , ) (8.101)( Š ‹-

*

∞

∞

< <

our previous result for the expectation value of the momentum in configuration space.

Hermetian Operators in State Vector Notation

The basic interpretation of the wave function as it relates to the probability of

measuring a particle's position, momentum, etc., requires us to normalize the wave


function. This same condition must also apply to the wavevector for a state of the system

giving

Ø > > Ù œ< <( )| ( ) 1 (8.102)

This normalization condition must also be time-independent (provided the particles are

not created or destroyed), so that

`

`>Ø > > Ù ( )| ( ) = 0 (8.103) Ÿ< <

In our new notation this last equation becomes

Œ Œ � � � �` `

`> `>Ø > l > Ù � Ø > l > Ù< < < <( )| ( ) | (8.104)

The Schrödinger equation in our new notation can be expressed as

L > Ù œ I > Ùs s

L > Ù œ 3h > Ù`

`>

ØL > œ "3h Ø >`

`>

| |

| |

| |

(8.105)< <

< <

< <

� � � ��

s

s

where | | Note: When an operator |Š ‹� � � �L > Ù œ ØL > Þ Ùs s s< < <*

b operates on a state vector

it produces a new state vector which is represented by the equation

b bs s| | (8.106)< <Ù œ Ù

This means that we can write

` "

`> 3h> Ù œ L > Ùs

` "

`> 3hØ > œ " ØL >s

| | (8.107)

| |

< <

< <

� � � ��

Our requirement of normalization, therefore, becomes

" ØL > > Ù � Ø > L > Ù œ !" "

3h 3hs s< < < <� � � � � � � �| | (8.108)

or

Ø > L > Ù œ ØL > > Ùs s< < < <� � � � � � � �| | (8.109)


Since the complex conjugate is represented by “reversing the direction of the state

vectors” we can write this last equation as

Ø > L > Ù œ ØL > > Ù œ Ø > L > Ùs s s< < < < < <� � � � � � � � � � � �Š ‹| | | (8.110)*

which is often expressed in the more compact form

ØLÙ œ ØLÙs s * (8.111)

This means that the expectation value of the Hamiltonian is ! Just as we pointed outreal

earlier, for physically measurable quantity the expectation value must be real. Anany

operator which satisfies this condition is called an Hermitian operator. Thus, all

physically measurable quantities must be represented by Hermitian operators whose

expectation values are real useful ! But this does not mean that all quantum mechanical

operators are Hermitian.

Hermitian Operators and the Adjoint

For any operator , we define the of that operator according to theU Us sadjoint †

following equation

(8.112)( )| (t) ( )| (t) (t)| (t) ( )| (t)Ø > Ul Ù œ Ø > U Ù œ ØU Ù œ Ø > U Ùs s s s< < < < < < < <† † *

Let's compare this definition with what we learned about the Hamiltonian operator. First,

using the definition of the adjoint, we obtain

Ø > Ll Ù œ Ø > L Ù ØL Ù œ Ø > Ùs s s s< < < < < < < <( )| (t) ( )| (t) = (t)| (t) ( )|H (t) (8.113)† † *

But from what we learned above,

Ø > L > Ù œ ØL > > Ù œ Ø > L > Ùs s s< < < < < <� � � � � � � � � � � �Š ‹| | | (8.114)*

This obviously means that the adjoint of the Hamiltonian operator must be equal to the

Hamiltonian operator itself. The Hamiltonian is said to be . This must be trueself-adjoint

for Hermitian operators which represent physically measureable quantities. Thisall

means that !all Hermitian operators are self-adjoint

________________________________________________________________________

Problem 8.1

Show that the definition of the adjoint of an operator is given by the equationgeneral Es

Ø lE Ù œ ØE l Ù œ Ø lE Ùs s s9 9 9 9 9 9" # " # # "† † *

Start with the definition of the adjoint operator operating on a vector | composed of two<Ùother vectors such that | | | .< 9 9Ù œ + Ù � + Ù" " # #

________________________________________________________________________

Problem 8.2

An operator can be formed from a combination of Hermitian operators. This operatorSs


may or may not be Hermitian. Determine the necessary conditions for the following

operators to be Hermitian:

a) The operator is a linear combination of Hermitian operators and G E Fs s s

G œ E� Fs s sα "

Determine the necessary conditions on the constants and .α "

b) The operator is a product of two Hermitian operators and H E Fs ss

H œ EFs s s

________________________________________________________________________

Problem 8.3

The so-called step-up and step-down operators

+ œ : � 37 Bs s s"

#7

+ œ : " 37 Bs s s"

#7

�

"

È e fÈ e f

=

=

are utilized in solving the quantum oscillator problem using operator methods.

(a) Determine the adjoint of both of these operators.

(b) Are these two operators Hermitian?

________________________________________________________________________

Commutator Relationships and Measurement Theory

When two operators operate on a wave function, the order of operation may be

important. As we have demonstrated earlier, if we operate on an arbitrary function 0ÐBÑwith and then expressed in position representation, we obtainB :s s

:B0ÐBÑ œ " 3h B0ÐBÑ œ " 3h0ÐBÑ " 3hBss` `0ÐBÑ

`B `B� � (8.115)

whereas if we operate on this same function first with the operator and then the : Bs soperator, we have

B:0ÐBÑ œ B " 3hss`0ÐBÑ

`BŒ (8.116)

Clearly, these two operations are not identical. We define the of twocommutator c dBß :s soperators and by the equationB :s s

c dBß : ´ B: " :Bs s ss ss (8.117)

If this commutator were zero, then the two operators would commute, i.e., it would make

no difference in what order they operate - you would get the same results either way.

Clearly, the operators and .B :s s do not commute

The value of the commutator is determined by allowing the commutator to

operate on an arbitrary function. From our previous results, we can see that


c dŒ Œ

Bß : 0ÐBÑ œ B:0ÐBÑ " :B0ÐBÑs s ss ss

œ B " 3h " " 3h0ÐBÑ " 3hB`0ÐBÑ `0ÐBÑ

`B `B

œ � 3h 0ÐBÑ

(8.118)

so that we have for the commutator of and :B :s s

c dBß : œ 3hs s (8.119)

Notice that the commutator

c d:ß B œ " 3hs s (8.120)

The commutation properties of operators are extremely important. As we mentioned

previously, these commutation relationships are valid no matter what representations you

are using, position or momentum, and therefore the commutation relationships must be

very fundamental in nature. The commutator relationships are valid for state vectors as

well.

________________________________________________________________________

Problem 8.4 Show that the commutator gives the same result when operating onc dBß :s s

the function in position representation, or on the function g in momentum0 B :� � � �representation.

________________________________________________________________________

Notice that the non-commutativity property of the operators B :s s and is of the

order of h, i.e., it is extremely small. This means that the order of operation (ormeasurement) is unimportant when is so small that it is neglegible - i.e., in the classicalhlimit. But for quantum systems the order of operation is important. The fact that B :s s and

do not commute is linked to the uncertainty principle. In general, one can show that

Š ‹ Š ‹ º? ?E F Ø E Ùs ss s# #

ßF"

%’ “ º# (8.121)

where

Š ‹?E Ø E ØEÙ Ù œ ØE Ù " ØEÙs s s s s# #œ "Š ‹#

# (8.122)

is the root-mean-square deviation in the measurable quantity designated by . Thus, ifEtwo operators commute, there is no uncertainty in the respective measurements. For

position and momentum we obtain

� � � � º? ?B : Ø B Ù œ l3hl œ" h

% %s s s s

# # ß :

"

%c d º# #

#

(8.123)

which gives us

? ?B : s s hÎ# (8.124)


As you work with commutators you may find the following relationships useful

c d� �c dc d c d� � � �

0 B ß : œ 3

Bß 1Ð:Ñ œ 3

Bß 0 B œ :ß 1 : œ !

s s ss

s s ss

s s s s

h 0ÐBÑ`

`B

h 1Ð:Ñ`

`:

(8.125)

________________________________________________________________________

Problem 8.5

Prove the commutation relations given above. You may do these in position or

momemtum representation.

________________________________________________________________________

Time-Dependent Expectation Values and the Correspondence Principle

In wave vector notation, the expectation value of an operator Es is given by

Ø > E > Ù< <� � � �| | (8.126)s

If we take the time derivative of this expectation value we obtain

.

.>Ø > E > Ù œ Ø > E > Ù �â

`

`>

� Ø > E > Ù � Ø > E > Ù` `

`> `>

< < < <

< < < <

� � � � � �Œ � � � � � � � �Œ

| | ( )| | (8.127)

| | | |

s s

s s

Just as we did when we examined the normalization condition, we use Schrödinger's

equation to write out the time derivative of the wave vectors

L > Ù œ I > Ùs s

L > Ù œ 3h > Ù`

`>

ØL > œ "3h Ø >`

`>

| | (8.128)

| |

| |

< <

< <

< <

� � � ��

s

s

Thus, the time derivative of the expectation value of Es can be written as

.

.>Ø > E > Ù œ ØL > E > Ù �â

3

h

� Ø > E > Ù � Ø > E L > Ù` "3

`> h

< < < <

< < < <

� � � � � � � �Œ � � � � � � � �Œ

| | | | (8.129)

| | | |

s ss

s s s


Factoring out the constant and utilitzing the fact that the Hamiltonian is self-adjoint, we

can simplify this expression to obtain

.

.>"Ø > E > Ù œ Ø > E > Ù � Ø > LE EL > Ù

` 3

`> h< < < < < <� � � � � � � � � � � �Š ‹| | | | | | (8.130)s s s ss s

which can be written utilizing the commutation relationship

.

.>ßØ > E > Ù œ Ø > E > Ù � Ø > L E > Ù

` 3

`> h< < < < < <� � � � � � � � � � � �’ “| | | | | | (8.131)s s ss

This last equation is an extremely useful relationship. It states that if the operator Es isnot a function of time, and if it commutes with the Hamiltonian operator, thenexplicitly

the expectation value of the operator is a constant - it does not change in time.

Examples

Let's look at the time-derivative of the expectation value of position. Using the equation

above, we obtain

.

.>ßØ > B > Ù œ Ø > B > Ù � Ø > L B > Ù

` 3

`> h< < < < < <� � � � � � � � � � � �’ “| | | | | |s s ss

This is often written without the explicit representation of the wave vector, as

.

.>B ßØ Ù œ Ø BÙ � Ø L B Ù

` 3

`> hs s ss’ “

We pointed out earlier that the commutator

’ “ – — – — L B œ œs s s s s s ss s

ß � Z ÐBÑ ß B ß B � Z ÐBÑß B: :

#7 #7

# # c dThe second term on the right is zero, and the first term can be written as

– — – — : : : :

#7 #7 : #7 7ß B œ " B œ " 3

s s s ss s

s

# # #

ß h œ " 3h`

`

So we have, for the time derivative of the expectation value of the position operator

. : :

.> 7BØ Ù œ ! � Ø " 3h Ù œ

3 Ø Ù

h 7s

s s

________________________________________________________________________

Problem 8.6


Using this same proceedure, show that

. BÑ

.>Ø:Ù œ " Ø Ù

`Z Ð

`Bs

s

s

which is compatible with Newton's second law.

________________________________________________________________________

Problem 8.7

Show that the time derivative of the expectation value of the Hamiltonian is zero, unless

the Hamiltonian is explicitly time-dependent. This is just a statement of the conservation

of energy.

________________________________________________________________________

Measurement and Complete Sets of Basis Vectors

We interpret the equation

E+Ù œ + +Ùs | |

as representing the process of measuring a physical quantity represented by Es (forexample the position, or momentum, or spin, or angular momentum, or color, or

temperature, or whatever). This equation indicates that when we make such a

measurement on an eigenvector of the operator we obtain a certain value corresponding to

that particular eigenstate and no other. That is, when we make a measurement of

position, or momentum, or spin, or angular momentum, or color, or temperature - we

obtain a certain value depending upon the particular state of the system.

Similarly, if we represent the process of measuring a separate physical quantity by

the operator , we writeFs

F ,Ù œ , ,Ùs | |

Now, let's assume that we have a quantum system in some state | which is<Ù

simultaneously an eigenvector of the two operators E Fs s and . We can, therefore, express

this eigenvector more specifically by writing

| |<Ù œ +,Ù

where we know that

E+,Ù œ + +,Ù

F +,Ù œ , +,Ù

s

s

| |

| |

If we now make a measurement of the state corresponding to the operator and then| +,Ù Es

follow that with a measurement corresponding to the operator , we haveFs

F +,Ù œ F +,Ù œ +F +,Ù œ +, +,Ùs s ssE +| | | |


If we make our measurements in the reverse order, we obtain

EF +,Ù œ E +,Ù œ ,E +,Ù œ ,+ +,Ùs s ss | | | |,

Now, since and are just numbers associated with a measurement, and since the+ ,

numbers commute, the order of the operations are unimportant - i.e., the operators andEs

Fs commute, or

’ “E Fs sß œ !

Thus, if two operators commute, we can describe the state of the system in terms

of the two eigenvalues associated with these operators. For example, in the case of the

free particle the Hamiltonian commutes with the momentum operator. This means that a

state vector designated only by its energy eigenvalue | is , i.e., there are<IÙ degenerate

actually two different states which have the same energy; one for eachmomentum

direction of the momentum of the particle. We can remove this degeneracy by

designating the state vector | . Thus, to completely specify the state of a system one<IßT�Ù

must find all operators which commute. A degeneracy in any observable quantity implies

that there is some other measurement which can be made which will unambiguoously

define the quantum state of the system.

chapter eight the formal structure of quantum mechanical ... structure... · chapter eight the...

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