25

3. Operators, Eigenfunctions, Eigenvalues

What is an operator?

Define:

Operator  is a rule that transforms a function f(x) of a given function

space into another well-defined function g(x) of the same function space.

Example:

Square integrable functions:

If |f(x)|2dx exists, then

|g(x)|2dx = |Âf(x)|2dx also exists.

Note:

1. An operator always influences functions written to its right hand side.

2. Multiplication of an operator with a constant:

(aÂ)f a(Âf) (a C)

3. Â is called a linear operator if

 (c1f1 + c2f2) = c1(Âf1) + c2(Âf2) (c1, c2 C)

4. Sum of two operators:

(Â + Bˆ) f Âf + B^f

26

5. Product of an operator:

Pf ÂB^ f Â(B^f)

Important: ÂB^ f B^ Âf !!!!

6. The difference

[Â,B^] ÂB^ - B^Â

is called Commutator. If [Â,B^] = 0, we say that  and B^ commute.

Commutator Examples: Let’s define the operators D^=d/dx and â as the

multiplication with a (a C). Then

[D^,â] = d/dx∙a – a∙d/dx = 0

But: [D^,x] = d/dx x – x d/dx = 1 0

D^ = d/dx is a very important operator!

Recall: In order to satisfy the correspondence principle, there must be

formal parallels between quantum theory and classical theory.

27

Example: Remember Schrödinger equation for a free particle:

QM CM

Exmt

i

2

22

2

p2/2m = T = E

Note:

1. Operator of the kinetic energy (for 1-dim. problems):

m

p

xmT

2

ˆ

2

2

2

22

2. Operator for linear momentum: xi

px

in 3D:

ip̂

28

Define:

If  is an operator and  = a (same on both sides!)

we call “eigenfunction” of Â

we call a “eigenvalue” of with respect to Â

Examples:

a. consider a 1-dim. free particle, and assume

/

2

1)( ipxex

Linear momentum:

)(2

1

2

1)(ˆ // xpepe

xixp ipxipx

x

(x) is an eigenfunction of px^ with the eigenvalue p

b. Particle in a potential V(x,t)

Classical physics: kin. E. + pot. E. = Etotal

p2/2m + V = Etotal

Schrödinger equation:

),(),(),(),(2

),(2

22

txEtxtxVtxxm

txt

i

With

29

2

22

2ˆ

xmT

kinetic energy

V^ = V(x,t) potential

Hˆ = T^ + V^

),(),(ˆ),( txEtxHtxt

i

H^ is called Hamilton operator or Hamiltonian, in analogy to the Hamiltonian

function (Sir Rowan Hamilton, 1833) in classical theoretical mechanics.

For a free particle: V(x,t) = const. = V0

0

2//

2

22

22

1),(

2

1

2ˆ V

m

petxVe

xmH ipxipx

is eigenfunction of H^ with eigenvalue (p2/2m + V0)

Note:

1. Every measurable physical property of a system is described by an

operator acting in the state space of that system. Operating on a wave

function is the QM mechanism for measurement:

H^ = (Energy)

2. A physically measurable property in a QM system is called an observable

30

3. If the wave function is an eigenfunction of Â, measurement yields the

eigenvalue A

Are and in the relation  = unique w/ respect to each other?

Each eigenfunction has only one eigenvalue

If there are two eigenfunctions of an operator  that have the same

eigenvalue, the eigenvalue is called degenerate

------------------------ End Lecture 4 (09/03/14) --------------------------

31

Last hour:

Operators are rules transforming a function f(x) of a given function

space into another well-defined function g(x) of the same function space.

(aÂ)f a(Âf) (a C)

Linear operators: Â (c1f1 + c2f2) = c1(Âf1) + c2(Âf2) (c1, c2 C

(Â + Bˆ) f Âf + B^f

ÂB^ f Â(B^f) In general ÂB^ f B^ Âf

Commutator: [Â,B^] ÂB^ - B^Â

Operator of kinetic energy: m

p

xmT

2

ˆ

2

2

2

22

or m

p

mT

2

ˆ

2

22

Operator of linear momentum: xi

px

or

ip̂

If  = a we call “eigenfunction” of Â, “eigenvalue” of with

respect to Â

Example: plane waves are EFs of px^

Hamiltonian: )(2

ˆ2

ˆˆˆˆ2

222

xVxm

Vm

pVTH

Every measurable physical property of a system (=”observable”) is

described by an operator acting in the state space of that system.

32

Operating on a wave function is the QM mechanism for measurement

33

Measurements influence the wave function:

measure first p, then x measure first x, then p:

)(')()()(ˆˆ

)(')()(ˆˆ

xxi

xi

xxxi

xxp

xi

xxxi

xxpx

There is no set of wave functions that can be eigenfunctions of both x^

and p^. This is a consequence of [x^,p^] = iħ 0

Remember HUP: We cannot measure x,p at the same time with infinite

accuracy. In the language of Q.M.: Measurements of observables

corresponding to non-commuting operators interfere with one another

[x^,p^] = iħ 0, but [p^, T^] = 0, [x^,pz] = 0, etc.

If [Â,B^] = 0 and is eigenfunction of Â, B^ is also eigenfunction of Â.

Eigenfunctions of H^:

Stationary States (requires time-independent H^)

Consider time-dependent S.E.

iħ (x,t)/t = H^ (x,t)

Trick: look for special solutions with (x,t) = (x)∙f(t)

34

Separation of variables (x,t) = (x)∙f(t), insert into T.D.S.E.

)()()()()()()(

2)()(

2

22

tfxEtfxxVx

tfx

mtfx

ti

divide by (x,t)

ExVx

x

mtf

tfi )(

)(

)("

2)(

)( 2

E = indep. of x and t = const.

Consequences:

i) From left hand side: f(t) = e-iEt/ħ

|(x,t)|2 = |(x)|2 time independent probabilities

ii) From right hand side: H^(x) = E (x) time independent S.E.

stationary states (x)

Learning goals slide

35

4. The Dirac Delta Function and Fourier Transforms:

The Dirac Delta Function: Slide Dirac

The delta function (r – r0) is a strange but useful function:

In one dimension:

(x-x0) = for x = x0

(x-x0) = 0 for x x0

(x-x0) has the dimension of a length-1.

1)(

)()()(

0

00

dxxx

xfdxxxxf

(x)

xx0

Analogous in 3 dimensions:

(r – r0) = for r = r0

(r – r0) = 0 for r r0

36

(r-r0) has the dimension of a volume-1 !

1)(

)()()(

30

03

0

rdrr

rfrdrrrf

Clicker question: How can we construct a function?

functions come in many shapes and colors:

QMin usedmost 2

1

otherwise 0 a/2,x a/2-for 1

lim

functionr rectangula)/sin(

lim

ondistributi normal a oflimit 1

lim

/)'(

0

0

/

0

22

xxip

a

a

ax

a

edp

a

x

ax

ea

Why is the function useful in QM ?

Point charge q at r0: (r) = q (r – r0)

Localization of a particle using infinitely many plane waves with varying

momentum (i.e. p = )

More uses later

37

Fourier Transforms: Slide Fourier

For a function f(x), we call

ikxexfdxxfkF )(2

1)]([)(

the Fourier transform of f(x).

The inverse procedure (inverse Fourier transform) gives us

ikxekFdkkFxf )(2

1)]([)( 1

(note the + sign in the exp !)

------------------------- End Lecture 5 (09/02/2015) ---------------------

38

Last hour:

Non-commuting operators cannot have the same eigenfunctions

Eigenfunctions of H^ have time-independent probabilities

Solutions of TISE are “stationary states”

The time-dependence of any WF is given by the TDSE

The Dirac Delta Function: In one dimension:

)()()( 00 xfdxxxxf

[(x-x0)] = m-1

Analogous in 3 dimensions:

)()()( 03

0 rfrdrrrf ; (r-r0) has the dimension of a volume-1 !

Fourier Transforms:

F.T

ikxexfdxxfkF )(2

1)]([)(

Inverse F.T.

ikxekFdkkFxf )(2

1)]([)( 1

(x)

xx0

39

Properties of Fourier transforms:

Linearity: [f(x)+g(x)] = [f(x)] + [g(x)] = F(k) + G(k)

Complex Conjugation: [f*(x)] = F*(-k)

Integration : f1*(x)f2(x) dx = F1*(k)F2(k) dk

|f(x)|2 dx = |F(k)|2 dk

Most of you know FT’s as a way to extract the frequency spectrum from a

complicated time trace (see applet).

Why is this useful in QM ?

One example (there are more!):

We have seen that “localization” of a free particle involves adding up

many plane waves with many different momenta:

n

xipn

necx

/

2

1)(

But p is continuously variable!

exchange discrete for , get a “wave packet”:

/)(2

1)( ipxepFdpx

draw Gaussian-like curve to visualize!

40

F(p) = p-dependent “weighting factor” analogous to cn, i.e., this is the

momentum distribution of the wave packet.

F(p) is the Fourier transform of (x). If we (somehow) know how (x)

looks, we can get to F(p) that way.

/)(2

1)( ipxexdxpF

Note “-“ sign of exponent

From knowing the momentum distribution F(p), we use the inverse Fourier

transform (like above)

Note:

1. We get the “QM” form of the Fourier transform from the general

form, if we set p=ħk and )(1

)(1

)(

pFkFpF

2. (x) and F(p) contain the same information

3. (x) is the wave function in position space. |(x)|2 = P(x)

draw little diagram

4. F(p) is the wave function in momentum space. |F(p)|2 = P(p)

draw little diagram

41

Example: free particle with momentum of exactly p0 (=plane wave)

)()(2

1)(

2

1)(

2

1)(

0/)(/

/

0

0

ppexdxexdxpF

ex

xppiipx

xip

P(p) = |F(p)|2=0 unless p=p0

Learning goals slide

42

5. Wave function space, Dirac notation, Hermitian operators

We have met operators acting on wave functions, let’s now look at the wave

functions themselves:

One-particle wave function space

(r,t)*(r,t) = probability of finding particle with wavefunction at point

r in space at time t.

Better: (r,t)*(r,t) d3r = probability of finding particle in the volume

element d3r = dx dy dz around r.

Physically meaningful one-particle wave functions must satisfy

normalization condition

1),(23 trrd

, when integrated over all space

square-integrable, or quadratically integrable

space of square-integrable functions is called L2.

Usually, WF’s must also be everywhere defined, continuous, infinitely

differentiable), let’s call them “well behaved”.

They form a subspace of L2, let’s call it F.

43

Note:

(a) F is a linear vector space: if F, then is also F.

(b) A scalar product exists on F. (Analogy in R3: baba),( )

Definition: We associate a complex number with

)()(*, 3 rrrd (always converges if ,F)

Properties:

o (,) = (,)*

o (, 11 + 22) = 1(,1) + 2(,2)

o (11 + 22,) = 1*(1,) + 2*(2,)

(linear w. r. to 2nd function, antilinear w. r. to 1st)

If (,) = 0, we say that and are orthogonal

(analogy to vectors in R3).

(c) The scalar product of a function with itself

233 )()()(*, rrdrrrd

is always real and positive. It is only zero, if (r)=0 for all r.

, is called the norm of .

44

Dirac notation (bracket notation):

In three dimensional space: r=(x, y, z)

Change the axes different coordinates (but the same point!):

r=(r1, r2, r3)

Geometrical vector concept and vector calculation

avoid referring to a specific system of axes

makes life simpler.

Similar for description of QM system:

Quantum state characterized by state vector in state space .

Any element of is called a ket vector or a ket (symbol: |>, e.g., |>).

(r) F |> r.

Scalar Product:

There is a dual space * belonging to , on which we define another set of

vectors, called bra vectors <|.

45

Scalar product of two ket vectors

(|>, |>) = <|> = d *

(bra and ket come from “bracket”)

Rules for the scalar product in bracket notation:

<|> = <|>*

<|11 + 22> = 1 <|1> + 2 <|2>

<11 + 22|> = 1* <1|> + 2* <2|>

<|> real, positive; zero only if |> = 0

A linear operator A^ acting on a ket |> produces another ket:

|> = A^|>

----------------------- End Lecture 6 (09/04/2015) -----------------------

46

Last hour:

We get the “QM” form of the Fourier transform from the general form,

if we set p=ħk and )(

1)(

1)(

pFkFpF

(x) and F(p) contain the same information

The WF space F is a linear vector space.

A scalar product exists on F, with )()(*, 3 rrrd

If (,) = 0, we say that and are orthogonal

Dirac notation (bracket notation): Quantum state characterized by state

vector (“ket”) in state space . Every function F is mapped onto a

ket |> . There is a dual space * belonging to , on which we define

another set of vectors, called bra vectors <|. Scalar product of two ket

vectors (|>, |>) = <|> = d *

A linear operator A^ acting on a ket |> produces another ket: |> =A^|>

47

Consider a countable set of function in F, labeled by an index j

(j = 1,2,3,..). If

<uj|uk> = 1 for j=k, 0 for jk

then we call {uj} orthonormal.

This condition can also be written as

<uj|uk> = jk (jk is called the Kronecker delta)

If every function in F can be expanded in one (and only one) way in terms of

the ui:

)()( rucrj

jj

we call {uj} a basis.

Analogy from R3: (1,0,0), (0,1,0), and (0,0,1) constitute a basis in R3.

How do we get the components ci of a wave function?

)()(,

),(,,

*3 rrurduc

ccuucucuu

uc

jjj

kj

kjjj

jkjj

jjkk

jjj

If , are wave functions, we can write:

48

jjj

kjjkkj

kjkjkj

kkk

jjj

kkk

jjj

cb

cbuucbucub

uc

ub

*

,

*

,

*

,

),(,,

What about plane waves?

/

2

1)( ipx

p exv

= plane wave w/ momentum p F because L2

Remember “wave packet”:

/)(2

1)( ipxepfdpx

Let’s write this a bit different:

)()(),()(

)()()(

* xxvdxvpF

xvpFdpx

pp

p

Note:

1. vp(x) is a plane wave, i.e., not square integrable!

2. Compare

49

)()()( xvpFdpx p )()( rucrj

jj

)()(),()( * xxvdxvpF pp

)()(, *3 rrurduc jjj

continuous basis discrete basis

We see that f(p) is the analog of cj and that vp is the

analog of uj !

3. It looks as if plane waves form a basis! However, vp cannot represent a

physical state! They are just mathematical constructs to describe a

wave function.

------------------------ End Lecture 7 (09/10/14) -------------------------

50

Adjoint operators:

Definition:

)(*)](ˆ[)(ˆ)(* xgxfAdxgAxfd

or

<f|Âg> = <Â+f|g>

Note that  acts on g, Â+ acts on f. Â+ is called adjoint operator of Â.

Hermitian operators:

Definition:

An operator  is called Hermitian (or symmetric) if

<f|Âg> = <Âf|g>

or d means generalized “integration over all spatial coordinates”

)(*)](ˆ[)(ˆ)(* xgxfAdxgAxfd

Why is that important? All operators representing observables in QM are

Hermitian!

51

Implications of an operator being Hermitian:

1. Â+ = Â (self-adjoint) Proof in Levine’s book

2. If Ân = nn (n is eigenvalue of n w.r.t Â)

<n|Ân> = <n|nn> = n <n|n>

also = <Ân|n> = <nn| n> = n* <n|n>

n = n* n is real number

3. All eigenfunctions of a Hermitian operator with different eigenvalues are

orthogonal (see homework).

4. The eigenfunctions of a Hermitian operator form a basis. Special

example: If we can solve the TISE, the solutions (EF’s of H^) form a

basis!

Learning goals slide

52

Last hour:

If every element of state space can be expanded in one (and only one)

way in terms of a set of countable, orthonormal functions ui:

)()( rucrj

jj

we call {uj} a discrete basis.

If every element of state space can be written in terms of functions with

a continuously varying parameter p as

)()()( xvpFdpx p , we call the

p a continuous basis.

Adjoint operator Â+: <f|Âg> = <Â+f|g>

Hermitian operator: <f|Âg> = <Âf|g>

All operators representing observables in QM are Hermitian!

Hermitian Operators are self-adjoint: Â+ = Â

All eigenvalues of Hermitian operators are real

All eigenfunctions of a Hermitian operator with different eigenvalues are

orthogonal

The eigenfunctions of a Hermitian operator form a basis

53

6. Average values:

We know how to predict probabilities for

1. position |(x)|2

2. momentum |f(p)|2

We can predict average values (expectation values) for measurements which

depend on x or p:

Position:

)()()( * xxxdxxxPdxx

Kinetic energy:

)(2

ˆ)(

2)(

2

2*

22

pfm

ppfdp

m

ppPdp

m

pE

In general (Ô is Hermitian!):

)(ˆ)(ˆ * xOxdO

In Dirac notation, we write this as

<Ô> = <|Ô|>

Why?

If is eigenfunction of Ô, the measurement yields the corresponding

eigenvalue.

54

If is not EF of Ô, we can expand in terms of the eigenfunctions:

n nnn nn cc ***;

n nn

n nnn

mnn m mmn

m mmmn nn

m mmn nn

cO

cc

dcc

ccd

cOcdOdO

2

*

**

**

***

ˆ

ˆˆˆ

In other words, the probability of measuring a particular eigenvalue in a set

of many identical systems is given by |cn|2. The average value is the weighted

average of the EV’s (weighted by their probability!)

Note:

A common definition for uncertainties is

2222 ; pppxxx

From this definition, one can obtain the rigorous formulation of Heisenberg’s

uncertainty principle xp ≥ ½ ħ

Clicker Question “Ehrenfest’s theorem”

Time evolution of average values:

55

t

AdAdAdAd

dt

d

dt

Ad ˆˆˆˆ

ˆ****

From the T.D.S.E: ** ˆ1

:ˆ1 H

iH

i

t

AHAd

iAHd

idt

Ad ˆˆˆ1ˆ)ˆ(

1ˆ

**

H^ is Hermitian, therefore

t

AAH

i

t

AHAd

iAHd

idt

Ad

ˆ

]ˆ,ˆ[

ˆˆˆ1ˆˆ1

ˆ**

If [H^, Â] = 0 and  f(t), then <Â> is a constant of motion, independent of

time.

Ehrenfest’s Theorem: “QM averages obey classical mechanics“

ppxxppm

i

m

ppxx

m

ppi

xVxTi

t

xxH

i

dt

xd

ˆˆˆˆˆˆ22

ˆˆˆˆ

2

ˆˆ

]ˆ,ˆ[]ˆ,ˆ[ˆ

]ˆ,ˆ[

Use ixppxpx ˆˆˆˆ]ˆ,ˆ[

ixppx ˆˆˆˆ and ipxxp ˆˆˆˆ

56

Bring x^ between the p^’s:

m

ppixpipxp

m

i

dt

xd ˆˆ)ˆˆ()ˆˆ(ˆ

2

----------------- Skipped ------------------

2

2

*

*

*

)(

)()(

)(

)()()(

)()(

]ˆ,ˆ[]ˆ,ˆ[ˆ

]ˆ,ˆ[

dt

xdmxF

xFx

xV

dxx

xV

i

i

dxxVxixi

xVxi

xVi

dxxVxixi

xVi

pVpTi

t

ppH

i

dt

pd

Note: Careful! <F(x)> F(<x>)

except for F=0

or F=const.

or F=kx

--------------------------- End Skipped ----------------------------

57

---------------------- END Lecture 08 ---- 09/11/15 ------------------