stanford qm

8/2/2019 Stanford QM

1/42

Hermitian linear operator Real dynamical variable

represents observable quantity

Eigenvector A state associated with an

observable

Eigenvalue Value of observable associated

with a particular linear operator

and eigenvector

Copyright Michael D. Fayer, 2007


2/42

Momentum of a Free Particle

Know momentum, p, and position, x, cancan predict exact location at any subsequent time.

Solve Newton's equations of motion.

p = mV

What is the quantum mechanical description?

Should be able to describe photons, electrons, and rocks.

Free particleparticle with no forces acting on it.

The simplest particle in classical and quantum mechanics.

classical

rock in interstellar space



3/42

Momentum eigenvalue problem for Free Particle

P P P

momentum momentum momentum

operator eigenvalues eigenkets

Know operator want: eigenvectorseigenvalues

Schrdinger Representationmomentum operator

P ix

LaterDiracs Quantum Condition shows how to select operators

Different sets of operators form different representations of Q. M.



4/42

Try solution

/i xP ce eigenvalue Eigenvalueobservable.

Proved that must be real number.

Also: If not real, function would blow up.

Function represents probability (amplitude)

of finding particle in a region of space.Probability cant be infinite anywhere. (More later.)

Function representing state of system in a particularrepresentation and coordinate system called wave function.



5/42

Schrdinger Representation

momentum operator

P ix

/i xP ce Proposed solution

Check

/ /

/

/i x i x

i x

i ce i i cex

ce

Operating on function gives

identical function back

times a constant.

P P P

continuous range of eigenvalues



6/42

/ipx ikx

pP ce ce momentum

eigenkets

momentum

wave functions

Have called eigenvalues =pp k k is the wave vector.

c is the normalization constant.

c doesnt influence eigenvalues

direction not length of vector matters.

Normalization

a b scalar product

a a scalar product of a vector with itself

1/ 2 1a a normalized Copyright Michael D. Fayer, 2007


7/42

Normalization of the Momentum Eigenfunctionsthe Dirac Delta Function

b ab a d

scalar product of vector functions (linear algebra)

*

a ad .a a scalar product of vector function with itself

*1

0P P

if P PP P dx

if P P

normalization '

orthogonality '

P P

P P

2 ' 2 ( ')ik x ikx i k k xc e e dx c e dx

P kUsing

( ') cos( ') sin( ') .i k k xe dx k k x i k k x dx

Cant do this integral with normal methodsoscillates.



8/42

Dirac d function

( ) 1x dx

Definition

( ) 0 if 0x x

For functionf(x) continuous atx = 0,

( ) ( ) (0)f x x dx f

or

( ) 0

( ) ( ) ( )

x a if x a

f x x a dx f a

d

d



9/42

Physical illustration

x x=0

unit area

double height

half width

still unit area

Limit width 0, height area remains 1



10/42

-40 -20 20 40

-0.2

0.2

0.4

0.6

0.8

1

Mathematical representation ofd function

sin( )g x

x

zeroth order spherical

Bessel function

g any real numberg =

Has valueg/

atx = 0.Decreasing oscillations for increasing |x|.

Has unit area independent of the choice ofg.

As g

dsin

( ) limg

gxx

xd

1. Has unit integral2. Only non-zero at pointx = 0

3. Oscillations infinitely fast, over

finite interval yield zero area.



11/42

Use d(x) in normalization and orthogonality of momentum eigenfunctions.Want p p 1

p p' 0

2 '1 if '

0 if '

ik x ikxk k

c e e dxk k

Adjustc to make equal to 1.

( ') cos( ') sin( ')i k k xe dx k k x i k k x dx

Evaluate

g cos( ') sin( ')limg

g

k k x i k k x dx

Rewrite

g

sin( ') cos( ')lim

( ') '

g

g

k k x i k k x

k k k k



12/42

g

sin( ') cos( ')lim

( ') '

g

g

k k x i k k x

k k k k

2 sin ( ')lim

( ')g

g k k

k k

2 ( ')k kd d function multiplied by 2.

2 ( ') 0 if 'k k k kd The momentum eigenfunctions areorthogonal. We knew this already.Proved eigenkets belonging to differenteigenvalues are orthogonal.function

To find out what happens fork = k' must evaluate .2 ( ')k kd But, whenever you have a continuous range in the variable of a vector function

(Hilbert Space), can't define function at a point. Must do integral about point.'

'

( ') ' 1 if

k k

k k

k k dk k k

d



13/42

if

if2

2 '1'

0 '

P PP P

c P P

Therefore

P are orthogonal and the normalization constant is

2 1

21

2

c

c

The momentum eigenfunction are

1( )

2

ikx

p x e k p momentum eigenvalue



14/42

Wave Packets

P P p P

momentum momentum momentum

operator eigenvalue eigenket


Free particle wavefunction in the Schrdinger Representation

1 1( ) cos sin2 2

ikx

pP x e kx i kx k p

State of definite momentum

2k

wavelength


15/42

1 1( ) cos sin2 2

ikx

pP x e kx i kx k p

What is the particle's location in space?

x

real imaginary

Not localized Spread out over all space.

Equal probability of finding particle from .

Know momentum exactly No knowledge of position.

to



16/42

Not Like Classical Particle!

Plane wave spread out over all space.

Perfectly defined momentum

no knowledge of position.

What does a superposition of states of definite momentum look like?

( ) ( ) ( )p p

p

x c p x dp indicates that wavefunction is composed

of a superposition of momentum eigenkets

coefficienthow much of each

eigenket is in the superposition

must integrate because continuous

range of momentum eigenkets



17/42

0

0

k k

ikx

k

k k

e dk

Then Superposition ofmomentum eigenfunctions.

Same as integral for normalization but

integrate overk aboutk0

instead of overx aboutx = 0.

02sin( )

ik x

k

k xx

xe

Rapid oscillations in envelope ofslow, decaying oscillations.

Work with wave vectors - /k p

k0k k0 k0 + kk

Wave vectors in the range

k0

k to k0 +

k.In this range, equal amplitude of each state.

Out side this range, amplitude = 0.

0k k

2sin( )At 0 have lim 0 2

k xx x k

x

First Example



18/42

0 0 02sin( ) cos sink kx k x i k x k kx

Writing out the 0

ik xe

X

k

This is the envelope. It is filled in

by the high frequency real and imaginary

oscillations

Wave Packet

The wave packet is now localized in space.

As | | large, small.kx

Greater k more localized.Large uncertainty inp small uncertainty inx.



19/42

Born Interpretation of Wavefunction

Schrdinger Concept of Wavefunction

Solution to Schrdinger Eq.

Eigenvalue problem (see later).

Thought represented

Matter Waves

real entities.

Born put forward

Like E field of light amplitude of classical E & M wavefunction

proportional toE field, .

Absolute value square ofE field Intensity.

E

2 *| |E E E IBorn Interpretationsquare of Q. M. wavefunction proportional to probability.

Probability of finding particle in the regionx + x given by

*Prob( , ) ( ) ( )

x x

x

x x x x x dx

Q. M. Wavefunction Probability Amplitude.

Wavefunctions complex.

Probabilities always real.



20/42

Localization caused by regions of constructive and destructive interference.

5 waves Cos( ) 1.2, 1.1, 1.0, 0.9, 0.8ax a

-30 -20 -10 10 20 30

-4

-2

2

4

Sum of the 5 waves



21/42

Wave Packetregion of constructive interference

Combining different ,

free particle momentum eigenstates

wave packet.

P

Concentrateprobabilityof finding particle in some region of space.

Get wave packet from

0

0

( )( )k

ik x

f k k

x

x e dk

Wave packet centered around

x = x0,

made up ofk-states centered aroundk = k0.

0 weighting fu c n) o( n tif k k Tells how much of eachk-state is in superposition.

Nicely behaved dies out rapidly away fromk0

.

k

f(k-k0)

k0 Copyright Michael D. Fayer, 2007


22/42

0 weighting fu c n) o( n tif k k Only havek-states neark = k0. 0 0

1ik x x

e e At x = x0

Allk-states in superposition in phase.

Interfere constructively add up.

All contributions to integral add.

For large (x

x0) 0 0 0cos sinik x xe k x x i k x x Oscillates wildly with changingk.

Contributions from differentk-states destructive interference.

0| | 0kx x Probability of finding particle 02

k wavelength



23/42

Gaussian Wave Packet

Gaussian distribution ofk-states

2

0

22

( )1( ) exp

2( )2

k kf k

kk

Gaussian in k, with

normalizedk

2 2

02

1( ) exp[ ( ) / 2 ]

standard deviat

2

ion

G y y y

Gaussian Function

2o

2o

( )

( )2( )

2

1

( ) 2

k k

ik x xk

k x e e dkk


0

0( )

ik x xef k k

Written in terms ofxx0 Packet centered aroundx0.



24/42


Looks like momentum eigenket atk = k0

times Gaussian in real space.

Width of Gaussian (standard deviation)

1

k

The bigger k, the narrower the wave packet.Whenx = x0, exp = 1.

For large |x - x0|, ifk large, real exp term


25/42

x = x0

1

k

k smallpacket wide.

Gaussian Wave Packets

1

k

x = x0

k largepacket narrow.

To get narrow packet (more well defined position) must superimpose

broad distribution ofk states. Copyright Michael D. Fayer, 2007


26/42

Brief Introduction to Uncertainty Relation

Probability of finding particle

betweenx &x + x*Prob( ) ( ) ( )

x x

x

x x x dx

Written approximately as

( ) ( )x xFor Gaussian Wave Packet

* 2 2

0exp ( ) ( )k k x x k (noteProbabilities are real.)This becomes small when

2 2

0( ) ( ) 1x x k 0Call , then

1x

x x x

k



27/42

1x k

1

pk

x p

x p

Superposition of states leads to uncertainty relationship.

Large uncertainty inx, small uncertainty inp, and vis versa.

(See later that

differences from choice of 1/e point instead of)/2x p

spread or "uncertainty" in momentum

momentumk p

k p


Ob i Ph t W P k t


28/42

-0.1

0.1

0.3

0.5

0.7

0.9

1.1

1850 2175 2500 2825 3150

frequency (cm-1

)

In

tensity(Arb.

Units)

368 cm-1 FWHM

Gaussian Fit

/p h

-0.3

-0.1

0.1

0.3

0.5

0.7

0.9

1.1

-0.2 -0.1 0 0.1 0.2

Delay Time (ps)

Intensity(Arb.

Units)

40 fs FWHMGaussian fit

collinearautocorrelation

Observing Photon Wave Packets

Ultrashort mid-infrared optical pulses

1

29

28

0

/ /

2.4 10 kg-m/s (14%)

1.7 10 kg-m/s

p h p h

p

p

-15

5

40 fs = 40 10 s

1.2 10 m 12 m

t

x c t

x



29/42

CRT - cathode ray tube

-

cathode

filament

controlgrid

screenelectronsboil off

electrons aimed electronshit screen

accelerationgrid

+

e-

Old style TV or computer monitor

Electron beam in CRT.

Electron wave packets like bullets.

Hit specific points on screen to give colors.

Measurement localizes wave packet.


El t diff ti Electron beam impinges


30/42

Peaks line up.

Constructive interference.

in coming

electron

wave"

out goingwavescrystal

surface

Electron diffraction

many grating

different directionsdifferent spacings

Low Energy Electron Diffraction (LEED)

from crystal surface

Electron beam impingeson surface of a crystallow energy electron diffractiondiffracts from surface.Acts as wave.Measurementdetermines

momentum eigenstates.


G V l iti


31/42

Group Velocities

Time independent momentum eigenket

1

2pikxP x e p k

Direction and normalization ket still not completely defined.

Can multiply by phase factor realie

Ket still normalized.

Direction unchanged.


F ti i d d t E


32/42

For time independent Energy

In the Schrdinger Representation (will prove later)

Time dependence of the wavefunction is given by

/i E t i te e E Time dependent phase factor.

1,i t i t

e e Still normalized

The time dependent eigenfunctions of the momentum operator are

0( ) ( )

,k

i k x x k tx t e



33/42

Time dependent Wave Packet

0, ( )exp ( )k x t f k i k x x k t dk

time dependent phase factor

Photon Wave Packet in a vacuum

Superposition of photon plane waves.

Need c c 2k

c

2


U i k


34/42

Using ck 0( , ) ( )exp ( )k x t f k ik x x ct dk

Have factored outk.Att = 0,

Packet centered atx = x0.

Argument of exp. = 0. Max constructive interference.

At later times,

Peak atx = x0 + ct.

Point where argument of exp = 0 Max constructive interference.

Packet moves with speed of light.

Each plane wave (k-state) moves at same rate,c.

Packet maintains shape.

Motion due to changing regions of constructive and destructive

interference of delocalized planes waves

Localization and motion due to superposition

of momentum eigenstates.Copyright Michael D. Fayer, 2007

Photons in a Dispersive Medium


35/42

Photons in a Dispersive Medium

Photon enters glass, liquid, crystal, etc.

Velocity of light depends on wavelength.

n() Index of refraction

wavelength dependent.

n()

bluered

glassmiddle of visible

n ~ 1.5

2 c / n( ) 2 / c

n( ) 2

V k

V

Dispersion, no longer linear ink.( )k


W P k t


36/42

Wave Packet

, ( )exp ( ) ( ) )k ox t f k i k x x k t dk

Still looks like wave packet.

0At 0,t x xMoves, but not with velocity of light.

Different states move with different velocities

blue waves move slower than red waves.

Velocity of a single state in superposition

Phase Velocity.


V l it l it f t f k t


37/42

Velocityvelocity of center of packet.

00,t x x max constructive interferenceArgument of exp. zero for allk-states at 0 , 0.x x t The argument of the exp. is

0( ) ( )k x x k t (k) does not change linearly withk.

Argument will never again be zero

for allk in packet simultaneously.

Most constructive interference when argument changes withk

as slowly as possible.

This produces slow oscillations.

k-states add constructively, but not perfectly.


No longer perfect constructive interference at peak


38/42

-3 -2 -1 1 2 3

-1

-0.5

0.5

1

Red waves get ahead.

Blue waves fall behind.

No longer perfect constructive interference at peak.

The argument of the exp. is 0( ) ( )k x x k t When changes slowly as a function ofk within range ofk determined byf(k)

Constructive interference.

Find point of max constructive interference. changes as slowly as possible withk. 0

00

k

x x tk k

0

0

k

x x tk

max of packetat time, t. Copyright Michael D. Fayer, 2007

Distance Packet has moved at time t


39/42

Distance Packet has moved at time t

0

0( )

k

x x tk

d Vt

Point of maximum constructive interference (packet peak) moves at

0

Group VelocitygVk

k

Vg speed of photon in dispersive medium.

Vp phase velocity = /konly same when (k) = ck vacuum (approx. true in

low pressure gasses)


Electron Wave Packet


40/42

Electron Wave Packet

de Broglie wavelength

/p h k (1922 Ph.D. thesis)A wavelength associated with a particle

unifies theories of light and matter.

2 2

( ) /2 2

p kk E

m m E

Non-relativistic free particleenergy divided by .

p kUsed


V Free non zero rest mass particle


41/42

VgFree non-zero rest mass particle

2( )

/2

g

k k kV p m

k m k m

/ /p m mV m VGroup velocity of wave packet same as Classical Velocity.

However, description very different. Localization and motion due tochanging regions of constructive and destructive interference.

Can explain electron motion and electron diffraction.

Correspondence PrincipleQ. M. gives same results as classical mechanics

when in classical regime.



42/42

Q. M. Particle Superposition of Momentum Eigenstates

Partially localized Wave Packet /2x p

PhotonElectron

Photon wave packet description of light same as

wave packet description of electron.

Therefore, Electron and Photon can act like wavesdiffract

or act like particleshit target.

WaveParticle duality of both light and matter.

stanford qm

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