Too Many to Count

Three Notations

The Three Notations of Quantum Mechanics

There are three notations (dialects if you like) commonly used in quantum mechanics

Sometimes they can be used interchangeably and sometimes not Each has a strength and each has a weakness

They are named for the 3 “fathers” of quantum mechanics Schroedinger Heisenberg Dirac

How they compare

Notation Name Type Example Comments

Schroedinger “Wavefunction” Math function

=f(x,y,z,t) Great for integrating, doe

not handle groups

Heisenberg Column matrix

Matrix Handles spin, isospin, and

some flavors, can’t integrate

easily

Dirac “ket” Both matrix and math function

Very ambiguous,

sometimes too ambiguous

0

1

x

label some

a

Postulate 1

Quantum Mechanical States are described by vectors in a linear vector space

Linear vector space means a field of scalars over which the space

From section 4.4 of Liboff’s text

Actually this is nothing new

3

2

1

321

332211

a

a

a

Heisenberg

3a2a1aa Dirac

zayaxaz)y,(x,er Schroeding

3 and ,2 ,1or

1

0

0

and ,

0

1

0

,

0

0

1

or z and ,y ,x

assuch basis a define First,

Postulate 2

A dual space exists with the same dimensionally as the original vector space

AKA “dual continuum” the existence of a dual vector space reflects in an abstract way

the relationship between row vectors (1×n) and column vectors (n×1)

Required to allow the inner product so that vectors can be normalized

Dual Spaces for the notations

To transform a vector from one space to another, a Hermitian conjugation is performed.

ibaiba

aa

baibaaib

aa

aa

** e.g.

Notationer SchroedingIn

Obviously,

real are and if

conjugate seor transpo conjugateHermitian means

bra""a example,For

Postulate 3

An inner product exists.

Back in E&M, we called the inner product: “dot product”

Inner product = dot product = scalar product

In the 3 notations

Schroedinger

Heisenberg

Dirac

dba

2

1*2

*1 b

baa

ba

baba then 0 If

Postulate 4

The dual space is linear and has the following property

21

then

21 If

*2

*1

21

aaa

aaa

Postulate 5

*abba

Postulate 6

vectornull the,0 if 0Only

a

0aa

Postulate 7

Multiplying a ket by a complex number (different from zero) does not change the physical state to which the ket corresponds

Postulate 7 is discussing normalization

state physical same therepresent still and

1

definingby 1such that

ket, normalized a formcan We

Let

2

2

aa'

aa

a'

a'a'

a'

aaa

It is convenient for define an orthonormal basis (and you’ve been doing it all your life!)

mn

mn

δnmzyx

δnm yxxx

z, y, x

where,,

Notation, DiracIn

ˆˆgenerally,or 0ˆˆ1ˆˆ

such that ˆˆˆ usedyou 350, PhysicsIn

Operators

A mathematical operation on a vector which changes that vector into another

This is not mere multiplication (like Postulate 7) but we are actually changing something like its direction or perhaps other quantities.

Example: Let Q be the differential operator with respect to x

mQn

or

mnQ

ˆ

ˆDirection of operation

Direction of operation

Postulate 8

Physical observables (such as position or momentum) are represented by linear Hermitian operators

What does linear mean?

bQaQbaQ

caQcacQ

Q

ˆˆˆ 2)

constant a is whereˆˆ 1)

:conditions twosatisfiesit iflinear is ,ˆ Operator,

What does Hermitian mean?

adjoint" self" called

ˆˆ

ˆˆ

: ifHermitian is ,ˆ Operator,

QQ

bQabQa

Q

A special case for operators

aqQaoraqaQ

baQ

*ˆˆ

following thesee wesometimes,but

ˆ

Mostly

Called“Eigenvector” or“Eigenfunction” or“Eigenket”

Called “eigenvalue”

What does an eigenvalue mean in Schroedinger notation?

function for this eigenvalue theis ,

ˆ

ˆ

ikqik

qik

ikikeex

qQ

eandx

QLet

ikxikx

ikx

What does an eigenvalue mean in Heisenberg notation?

?1

1 ifWhat

function for this eigenvalue theis 1 ,1

1

11

1

01

10

1

1

01

10

ˆ

1

1

01

10ˆ

a

q

aqa

aora

aqaQ

aandQLet

Theorem 1

Eigenvalues of a Hermitian operator are real

i.e.

If Q+=Q then q*=q

Proof of Thm 1

qaQa

aaqaQa

aqaaQa

so

aaandaqaQLet

ˆ

ˆ

ˆ

1ˆ

QED ,

ˆ

ˆˆ

ˆ

ˆ

ˆ

*

*

*

*

*

qq

qaQa

QQbut

aaqaQa

aqaaQa

so

qaQaLet

Theorem 2

Eigenvectors of a Hermitian operator are orthogonal if they belong to different eigenvalues

Proof of Thm 2

QED

babaqq

baqqorbaqbaq

Obviously

baqbQabaqbQa

bqabQabaqbQa

aqQa

aqQa

so

qqqq

bqbQandaqaQLet

0 then 0 Since

0

ˆˆ

ˆˆ

ˆ Hermitian, Since

ˆ

0

ˆˆ

21

2121

21

21

1

*1

2121

21

Note: An operator may have a set of eigenvalues of which 2 or more are equal; this is called degeneracy

Projection operators

Graphically, the inner product represents the project of a onto b or in Dirac notation |a> onto |b>

|a>

|b>

<a|b>

If |a> is considered a unit vector, then the vector which represents projection of |b> onto |a> is written <a|b>|a> or |a><a|b>

Theorem 3

A projection operator is idempotent i.e.

Q2 =Q

QEDˆˆ

ˆˆ

ˆ

ˆ

ˆ and 1Let

:Proof

2

2

2

2

QQ

QaaQ

aaaaQ

aaaaQ

aaQaa

Theorem 4

matrixidentity the,ˆ

thenbasis, orthogonalan form ,3,2,1 If

Ikk

k

k

Proof of Thm 4

1

so j ofdimension same hask But

on operate , operator, projection a Have

: vectorsbasis of

set a of in terms written becan andtor chosen vecy arbitrarilan be Let

,

k

k

kk

kk

k

kj

jkjj

j

jj

kk

aakk

akc

kcakk

So

kckcjkkcakk

akk

jca

a

Creating a set of orthogonal vectors from a set of normalized linear independent kets

Let |a>, |b>, and |c> be a set of normalized linear independent kets

We are going to create a new set of kets (|1>, |2>, |3>) from these which will be orthogonal to one another i.e. <1|2>=0, <1|3>=0 and <2|3>=0

First, pick one of the original set and build the rest of the set around it

|1>=|a>

Constructing |2>

Geometrically

|b>

|1>|1><1|b>-|1><1|b>

|b>-|1><1|b>

|2>=|b>-|1><1|b>

Test that |2> is orthogonal to |1>

QEDbb

bb

bb

bb

1121

111121

11121

112

Normalizing |2>

2

2

11

112 is 2 Normalized

1122

111111122

1111111122

11111122

111122

112 and 112

b

bb

b

bbbbbb

bbbbbbbb

bbbbbb

bbbb

bbbb

|3>

1

1

2

1

1

22

22

1

211

22113

211ofconstant ion normalizat awith

ed)(unormaliz 22113

3 of structure theguessalmost may You

k

L

k

L

jL

jLLjkngGeneralizi

cc

ccc

cc

ccc

Postulate 9

Eigenvalues are the only possible outcome of physical measurements

If physical observables are represented by Hermitian operators and these have real eigenvalues, it is reasonable to assume that there is a connection between their eigenvalues and the results of experiments.

Theorem 5

Operators representing simultaneously observable quantities commute

Proof of Thm 5

commute they therefore0ˆ,ˆ

0ˆˆˆˆ

0

ˆˆˆˆˆˆˆˆ

ˆˆˆˆˆˆ

ˆˆˆˆˆˆ

ˆˆ

QR

aRQQR

arqqrarqaqr

aRQQRaRQaQR

arqaQraRQaqraRqaQR

arQaRQaqRaQR

araRaqaQ

Let

Commutator Brackets[a,b]=(ab-ba)If [a,b]=0 then a and b commuteQM analog of Poisson brackets

An Example of non-commuting operators

1,

,

,

,

evaluate to, function,dummy a Use

,Consider

xx

xx

xx

xxx

x

xxx

xx

x

xx

Postulate 10

The average value in the state |a> of an observable represented by an operator Q, is

aa

aQaQ

ˆˆ

Called an “expectation value” or called the “mean”

In Schroedinger Notation

dxxx

*

position of valueAverage

In Heisenberg Notation

11

111

2

1

1

1

2

1

01

1011

2

1

112

1

statein of n valueexpectatio theFind

1

1

2

1 and

01

10Let

x

x

x

x

aa

aa

a

a

a

Defining Standard Deviation

Let Q= operatorQ= standard deviation of measurement of

Q(Q)2= variance of that measurement

Sometimes called mean square deviation from the mean

(Q)2 =<(Q-<Q>)2>Or, more compactly (Q)2 =<Q2>-<Q>2

The Uncertainty Principle

If two observables are represented by commuting operators then you can measure the physical observables simultaneously

If the operators DO NOT COMMUTE then a SIMULTANEOUS measurement will NOT BE EXACTLY REPEATABLE

There will be a spread in the measurement such that the product of the standard deviations will exceed a minimum value; the size of the minimum depends on the observable

To calculate this, we first have to build some mathematical machinery.

Theorem 6

Schwartz’s Inequality

2kjkkjj

Proof of Thm 6

QED

00 and 0

ket aConstruct

2

2

2

22

2222

kjkkjj

kjkkjjjjff

kjkkjjjjff

kkjjjjkjff

kkjjjjkjjjkjjjkjff

kjjjjkjkjjjk

kjjjkjjkjjkjff

jjkjkjf

kjjjkjf

Theorem 7

Let = A-<A> and

=B -<B> then

[] =[A,B]

QED

Proof

A,BBAABα,β

BAABBABA

BABAABABα,β

AABBBBAAα,β

Derivation of the Uncertainty Principle for any Operator

aaaa

BB

AA

2222

2222

22

222

222

BA

BA

BA calculate sLet'

B

A

7, Thmin sdefinition theUsing

Derivation of the Uncertainty Principle … page 2

222

22

2222

BA

BA

aa

Let

BA

kjkkjj

aaaa

so

kandj

aaaa

Need more power!

Now the absolute square of any complex number, z, can be written as |z|2 = (Re(z))2 +(Im(z))2

Of course, |z|2 (Im(z))2

22222

222

ImReBA

BA

kjkj

kj

An Aside

b

i

ib

i

ibaiba

i

zz(z)

then

ibaz

2

2

22Im

Let

*

So we can now start having fun…

2

22

222

2222

222

2BA

ImBA

parts,

itsan greater thImaginary and Real of sum Since

ImReBA

BA

i

jkkj

kj

kjkj

kj

The final slide

2

22

2

22

2

22

2

22

2

,BA

2

,BA

2BA

2BA

i

BA

i

i

aaaa

i

jkkj

Does it work?

2x

squares gEliminatin4

x

2ix

i,,

i1, Recall

2

,x

iALet

2BA

222

2

22

2

22

2

22

x

x

x

x

xx

x

p

p

ip

xpxx

xx

i

xpp

xB

px

i

jkkj