量子力學(xué)學(xué)習(xí)課件第三章英文版

FORMALISMChapter

33.1

Hilbert

Space93Observables

96Eigenfunction

of

a

Hermitian

Operator

100Generalized

Statistical

Interpretation

106The

Uncertainty

Principle

110Dirac

Notation

1183.1

Hilbert

SpaceQuantum

theory

is

based

on

two

constructs:

wave

functions

andoperators.More

general

theoretical

theory

of

quantum

mechanicsWave

functions:Operators:the

state

of

a

systemobservablesPhysicalMathematicalVectorsLinear

transformationsQuantum

mechanics

is

linear

algebra.1.

Vector:In

an

N-dimensional

space,

the

vector

is

represented

by

a

N-numberof

its

components,

with

respect

to

a

specified

orthonormal

basis:the

addition

andWe

can

define

operations

on

vectors:the

inner

productThe

inner

product

of

two

vectors,

which

generalizes

the

dot

productin

three

dimensions,

is

defined

by2.

Linear

transformationsLinear

transformations,

T,

are

presented

by

matrices,

whichcan

act

on

vectors

asfunctionsvectorsBut

the

wave

function

we

encounter

in

QM

are

functions

not

vectors?The

collection

of

all

functions

of

x

constitutes

a

vectors

space.

Butto

present

a

possible

physical

state,

the

wave

functions

must

benormalized:3.

Hilbert

spaceTherefore,

the

set

of

all

square-integrable

functions,

on

a

specifiedinterval,constitutes

a

(much

smaller)

vector

space.Mathematicians

call

it

L2(a,b),

while

physicists

call

it

Hilbert

space.The

inner

product

of

two

functions

is

defined

as

follows:In

particularSchwarz

inequality:Finally,

a

set

of

functions

is

complete

if

any

other

function

can

beexpressed

as

a

linear

combination

of

them:If

a

set

of

functions,

{fn},

is

orthonormal

if

they

are

normalized

andmutually

orthogonal:4.

Normalization,

orthogonal

and

complete3.2.1

Hermitian

Operators(1)

The

expectation

value

of

an

observable

can

be

expressed3.2

ObservablesBut

the

complex

conjugate

of

an

inner

product

reverse

the

order,

soThe

outcome

of

measurement

has

got

to

be

real:Thus

operators

representing

observables

have

the

very

specialproperty

thatWe

call

such

operators

hermitian.orFor

example,

the

momentum

operator3.2.2

Determinate

StatesSome

important

conceptsOn

state

we

measure

an

observable

Q.We

can

not

get

the

same

result

in

each

measurementThis

is

the

indeterminacy

of

QM.Question:

Would

it

be

possible

to

prepare

a

state

such

that

everymeasurement

of

Q

is

certain

to

return

the

same

value

q?That

is,

in

such

determinate

state,

the

standard

deviation

of

Q,would

be

zero,

which

is

to

say,ThereforeThis

the

eigenvalue

equation

for

the

operatoris

an

eigenfunction

of

the

operatorq

is

the

corresponding

eigenvalue,

which

is

a

real

number.Determinate

states

are

eigenfunctions

of

Q.Operator

Q

acting

on

different

eigenfunctions

gives

different

eigenvalThe

collection

of

all

the

eigenvalues

of

an

Operator

Q

is

called

its

specSome

times

two

or

more

linearly

independent

eigenfunctions

share

thesame

eigenvalue:In

this

case

the

spectrum

is

said

to

bedegenerate.Example

3.1:Consider

the

operatorwhere

is

the

usual

polar

coordinate

in

two

dimensions.Question:

Is

the

operator

hermitian?

Find

the

eigenfunctions

andeigenvalues.Solution:(1)

hermitian?In

this

case:As

is

the

usual

polar

coordinate:On

the

interval(2)

The

eigenvalue

equation,The

general

solution

isBy

using

periodic

boundary

conditionwe

haveTwo

categories:3.3

Eigenfunctions

of

a

Hermitian

OperatorContinuous

spectrumDiscrete

spectrum:The

eigenfunctions

are

normalizable.The

eigenfunctions

are

physically

realizable

statesContinuous

spectrumThe

eigenfunctions

are

not

normalizable.The

eigenfunctions

do

not

represent

possible

wave

states,

butthe

linear

combination

of

them

may

be

normalizable.The

eigenfunctions

of

a

Hermitian

Operator

fall

into

two

categories.Discrete

spectrumof

the

operatorMathematically,

the

normalizable

eigenfunctions

of

a

hermitianoperator

have

two

important

properties:Theorem

1:

Their

eigenvalues

are

real.Proof:

Supposeand3.3.1

Discrete

SpectraPhysical

sense:

If

you

measure

an

observable

on

a

particle

in

adeterminate

state,

you

will

at

least

get

a

real

number.Theorem

2:

Eigenfunctions

belonging

to

distinct

eigenvalues

areorthogonal.Proof:

Supposedistinct

eigenvalues:Axiom:The

eigenfunctions

of

an

observable

operator

are

complete.Any

function

(in

Hilbert

space)

can

be

expressed

as

a

linearcombination

of

them.If

the

spectrum

of

a

Hermitian

operator

is

continuous,

the

eigenfunctionsare

not

normalizable,

and

the

proofs

of

Theorems

1

and

2

fail,

because

the

inner

products

may

not

exist.

Nevertheless,

there

is

a

sense

in

whichthe

three

essential

properties

(reality,

orthogonality,

and

completeness)still

hold.Specific

examples:Example

3.2:

Find

the

eigenfunctions

and

eigenvalues

of

themomentum

operator.3.3.2

Continuous

SpectraLet

fp(x)

is

the

eigenfunction

of

momentum

operator

and

theeigenvalue

is

p.Thenthe

general

solution

isAbove

function

is

not

square-integrable,

for

any

value

of

p,

that

isthe

momentum

operator

has

no

eigenfunctions

in

Hilbert

space.However

if

we

restrict

p

to

be

real

eigenvalues,

we

can

get

thefollowing

orthonormality.If

we

set,

so

thatandAt

last

we

get

a

Delta

orthonormality

equation,

which

can

be

calledDirac

orthonormality.Further

important,

the

eigenfunctions

of

momentum

operator

arecomplete,

in

the

sense

that

any

function

f(x)

can

be

expressed

byThat

isWhich

is

nothing

but

a

Fourier

transformation,

and

cp

is

determined

byUse

Dirac

orthonormality:Discussion

:The

wave

length

isDe

Broglie

formulaExample

3.3:

Find

the

eigenfunctions

and

eigenvalues

of

theposition

operator.Let

gy(x)

is

the

eigenfunction

of

position

operator

and

the

eigenvalue

is

y.That

clearly

isAbove

eigenfunction

is

not

square-integrable,

but

again

they

admitfollowing

Dirac

orthonormality,If

we

pick

A=1,

soThenThese

eigenfunctions

are

also

complete:Conclusion:If

the

spectrum

of

a

hermitian

operator

is

continuous

(so

theeigenvalues

are

continuous

and

labeled

by

a

continuous

variable),the

eigenfunctions

are

not

normalizable,

they

are

not

in

theHilbert

space

and

they

do

not

represent

possible

physical

states;nevertheless,

the

eigenfunctions

with

real

eigenvalues

are

Diracorthonormalizable

and

complete.3.4

Generalized

Statistical

InterpretationIf

you

measure

an

observable

Q(x,p)

on

a

particle

in

the

stateyou

are

certain

to

get

one

of

the

eigenvalues

of

the

hermitian

operatorIf

the

spectrum

of

Q

is

discrete,

the

probability

of

getting

the

particulareigenvalue

qn

associated

with

the

orthonormalized

eigenfunction

fn(x)

isIf

the

spectrum

is

continuous,

with

realeigenvalues

q(z)

and

associatedDirac

orthonormalized

eigenfunctions

fz(x),

the

probability

of

getting

aresult

in

the

range

dz

isUpon

measurement,

the

wave

function

“collapses”

to

thecorresponding

eigen-state!Here

is

the

probability

that

a

measurement

of

Q

wouldyield

the

value

qn

.◆Discrete

spectrum1.

As

the

eigenfunctions

fn

of

an

observable

operator

Q

arecomplete,

the

wave

function

can

be

written

as

a

linearcombination

of

them:2.

The

total

probability

has

got

to

be

one:This

comes

from

the

normalization

of3.

The

expectation

value

of

Q

should

be

the

sum

over

all

possibl

outcomes

of

the

eigenvalue

times

the

probability

of

getting

th

eigenvalue:Proof:◆Continuous

spectrum1.

Position

measurementA

measurement

of

position

x

on

a

particle

in

state

Ψ

must

return

one

of

the

eigenvalues

of

the

position

operator

.

In

Example

3we

know

that

every

real

number

y

is

an

eigenvalue

of

,

and

its

corresponding

eigenfunction

isEvidently,

the

measurement

probability

of

getting

y

is

relateThen,

the

probability

of

getting

a

result

around

y

in

the

rangedy

is2.

Momentum

measurementThe

eigenfunction

of

the

momentum

operatorp

iswith

eigenvaluesoThen,

the

probability

of

getting

a

result

around

p

in

the

ranof

dp

isBecause

c(p)

is

such

an

important

quantity

that

we

give

it

aspecial

name

and

symbol:the

momentum

space

wave

function,Actually,

the

momentum

space

wave

function

isthe

Fourier

transform

of

the

position

space

wave

functionAccording

to

the

generalized

statistical

interpretation,

tprobability

that

a

measurement

of

momentum

would

yield

aresult

in

the

range

dp

isExample

3.4:

A

particle

of

mass

m

is

bound

in

the

delta

functiowell

V(x)=-αδ(x).

What

is

the

probability

that

a

measurement

oits

momentum

would

yield

a

value

greater

than

p0=mα/h

?Solution:

The

(position

space)

wave

function

is

thereforeThen

the

momentum

space

wave

function

is

thereforeThe

probability

greater

than

p0

is3.5

The

Uncertainty

Principle3.5.1

lead-in

of

uncertainty

principleWerner

Karl

Heisenberg

1927Earl

Kennard

1927Because

of

measurement?Or

intrinsic

property?????Makinga

particle

out

of

waves440

Hz

+439

Hz440

Hz

+439

Hz

+438

Hz440

Hz

+439

Hz

+438

Hz

+437

Hz

+436

HzSpatial

extent

of

localized

wavex

=

spatial

spread

of

‘wave

packet’

Spatial

extent

decreases

as

the

spread

inincluded

wavelengths

increases.3.5.2

Proof

of

the

general

uncertainty

principleFor

any

two

observables

F

and

GDefine:An

integral

with

a

parameter

ξ:Conclusion:The

standard

deviation

of

any

observable

A

is:This

is

the

generalized

uncertainty

principle.As

an

example,There

is,

in

fact,

an

“uncertainty

principle”

for

every

pobservables

whose

operators

do

not

commute.We

call

them

incompatible

observablesIncompatible

observablesDo

not

have

shared

eigenfunctions.

At

least

they

can

not

ha

complete

set

of

common

eigenfunctions.Compatible

observablesHave

shared

eigenfunctions.

They

do

admit

complete

setssimultaneous

eigenfunctions.Another

proof,

for

any

observable

A,

we

haveHermitianDefineHermitianDefineLikewise,

for

any

observable,

B,Therefore,

by

using

the

Schwarz

inequality,For

any

complex

number

z,LettingAnd

we

haveSimilarlyWe

also

haveWhereis

the

commutator

of

the

two

operators.Conclusion:This

is

the

generalized

uncertainty

principle.As

an

example,3.5.2

The

Minimum-Uncertainty

Wave

PacketFor

harmonic

oscillator,

we

have:Problem:

What

is

the

most

general

minimum-uncertainty

wave

packet?Therefore,c

must

be

purely

imaginary.Then

the

necessary

and

sufficient

condition

for

minimum

uncertainty

isRecallIf

we

set

and

then,

for

the

position-momentum

uncertainty

principle

this

criterion

becomes:Evidently

the

minimum-uncertainty

wave

packet

is

gaussian.The

position-momentum

uncertainty

principle

is

oftenwritten

in

the

form：Similarly,

the

energy-time

uncertainty

principle

can

bewritten

byIn

non-relativistic

QM,we

must

derive

the

energy-timeuncertainty

principle。3.5.3

The

Energy-Time

Uncertainty

PrincipleRight

or

wrong?quid

est

ergo

tempus?si

nemo

ex

me

quaerat,

scio;si

quaerenti

explicare

velim,

nescio------The

Confessions

of

Saint

AugustineWhat

is

time?Time

is

not

an

observable,

but

standard

of

change

of

observablDerivation

of

the

energy-time

uncertainty

relation:let

us

compute

the

time

derivative

of

the

expectationvalue

of

some

observable,

Q(x,p)

:assume

that

Q

does

not

depend

explicitly

on

t,and

then:Time

is

standard

of

change

of

observable,then:Example:Example

3.5:

In

the

extreme

case

of

a

stationary

state,

for

which

theenergy

is

uniquely

determined,

all

expectation

values

are

constant

intime

(

)——as

in

fact

we

noticed

some

time

ago.To

make

something

happen

you

must

take

a

linear

combination

of

atleast

two

stationary

states——say:In

this

state,

the

period

of

oscillation

is3.6

Dirac

Notation1.

Expansion

of

vector

in

Hilbert

space

by

different

basis(1)

Imagine

an

ordinary

vector

A

in

two

real

dimensions.In

the

system

S

:

coefficientsTwo

independent

bases,

two

components.In

the

basis

S’

:

coefficients(2)

Imagine

an

state

vector

in

Hilbert

space.

We

can

also

express

itwith

respect

to

any

number

of

different

bases.

The

dimension

of

the

Hilbertspace

is

determined

by

the

number

of

the

independent

bases.In

the

basis

of

position

eigenfunctions:

coefficientswhere

is

the

eigenfunction

of

the

with

eigenvalue

x.In

the

basis

of

momentum

eigenfunctions:

coefficientswhere

is

the

eigenfunction

of

the

with

eigenvalue

p.In

the

basis

of

energy

eigenfunctions:

coefficientswhereis

the

eigenfunction

of

thewith

eigenvalue

En.(3)

Equivalence

of

them(4)

TransformationIn

Hilbert

space,

operator

are

linear

transformation

acting

on

a

vector

toget

another

vector,

that

is,With

respect

to

a

particular

basis,

they

can

be

expanded

byTaking

the

inner

product

withHere

the

operators

are

represented

(with

respect

to

a

particular

basis)

bytheir

matrix

elements

Qmn.Then

we

haveThat

means

the

matrix

elements

tell

you

how

the

components

transform.In

Hilbert

space,

if

we

choose

a

certain

basis

which

admits

only

a

finitenumber

N

of

linearly

independent

bases,

then

any

vector

will

live

inan

N-dimensional

vector

space,

which

can

be

represented

as

a

column

of

Ncomponents

with

respect

to

this

basis:Vector

(state/wave

function)OperatorsThe

operators

will

take

the

form

of

ordinary

matrices

with

N-dimension.But

for

many

systems,

there

are

often

exist

infinite

dimensional

vector

spacExample

3.5:

Imagine

a

system

in

which

there

are

two

linearly

independentstates:The

most

general

state

in

this

space

is

a

normalized

linear

combination

ofthem:The

Hamiltonian

can

be

expressed

as

a

Hermitian

matrix;

suppose

it

has

thespeci