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Physics 2D Lecture SlidesLecture 17: Feb 8th 2005

Vivek SharmaUCSD Physics

A PhD Thesis Fit For a Prince

• Matter Wave !– “Pilot wave” of λ = h/p = h / (γmv)– frequency f = E/h

• Consequence:– If matter has wave like properties then there would be

interference (destructive & constructive)• Use analogy of standing waves on a plucked

string to explain the quantization condition ofBohr orbits

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De Broglie’s Explanation of Bohr’s Quantization

Standing waves in H atom:

s

Constructive interference when

n = 2 r

Angular momentum

Quantization condit

ince h

=p

......

io !

( )

2

n

h

mNR

nhr

m

n mvr

v

v

! "

!

"#

#

=

=

=

!

n = 3

This is too intense ! Must verify such “loony tunes” with experiment

Just What is Waving in Matter Waves ?

For waves in an ocean, it’s the water that “waves”For sound waves, it’s the molecules in medium For light it’s the E & B vectors that oscillate

• What’s “waving” for matter waves ?– It’s the PROBABLILITY OF FINDING THE PARTICLE

that waves !– Particle can be represented by a wave packet

• At a certain location (x)• At a certain time (t)• Made by superposition of many sinusoidal

waves of different amplitudes, wavelengths λand frequency f

• It’s a “pulse” of probability in spacetime

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What Wave Does Not Describe a Particle

• What wave form can be associated with particle’s pilot wave?• A traveling sinusoidal wave?• Since de Broglie “pilot wave” represents particle, it must travel with same speed

as particle ……(like me and my shadow)

cos ( )y A kx t!= " +#

cos ( )y A kx t!= " +#

x,t

y2, 2k w f

!!

"= =

p

2 2

p

2

p

In Matter:

h( ) =

Phase velocity of sinusoid

E(b) f =

a l wave: (v )

v

h

!

v

E mc cf c

p

ha

p mv

v

m

m

h

f

v

c

!"

"

"

!

!"

= = = =

=

=

>

=

#

Conflicts withRelativity Unphysical

Single sinusoidal wave of infinite extent does not represent particle localized in space

Need “wave packets” localizedSpatially (x) and Temporally (t)

Wave Group or Wave Pulse• Wave Group/packet:

– Superposition of manysinusoidal waves with differentwavelengths and frequencies

– Localized in space, time– Size designated by

• Δx or Δt– Wave groups travel with the

speed vg = v0 of particle• Constructing Wave Packets

– Add waves of diff λ,– For each wave, pick

• Amplitude• Phase

– Constructive interference overthe space-time of particle

– Destructive interferenceelsewhere !

Wave packet represents particle prob

localized

Imagine Wave pulse moving along a string: its localized in time and space (unlike a pure harmonic wave)

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How To Make Wave Packets : Just Beat it !•Superposition of two sound waves of slightly different frequencies f1 and f2 , f1 ≅ f2

•Pattern of beats is a series of wave packets

•Beat frequency fbeat = f2 – f1 = Δf

• Δf = range of frequencies that are superimposed to form the wave packet

[ ]

2 1

1 2

1 1 2 2

2 1 2 1 2 1

22 1

Resulting wave's "displacement " y = y :

cos( ) cos( )

A+B A-BTrignometry : cosA+cos B =2cos( )cos( )

2 2

2 cos( )2 2

since , k

cos( )2 2

ave

k

y

y A k

k w wx

k k w

x w t k x w t

k k w wy A x t t

+ +! "#$

+

= # + #

% # # &! "' = #$ (() *+ ,-

+, .-

/-

'

1

y = A cos( )

' 2 cos( ) = modulated amplit

cos( ) A' oscillates in x,t

ud

2 cos( ) ,

e2 2

,

2

,

2

ave

kx wtk w

y A x kx w

w w k k w

t

A A x

w

k wt

t ##

0 0! "= #

% 0 0 &! "

$ (+ ,

' = # 1$ () *+ ,/ .

- 0 0! !

g

Phase Vel V

Group Vel V

: Vel of envelope=

avep

ave

g

w

k

w

k

dwV

dk

=

!=!

Wave Group Or packet

Addition of 2 Waves with slightlydifferent wavelengths and slightly different frequencies

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Non-repeating wave packet can be created thru superposition Of many waves of similar (but different) frequencies and wavelengths

Wave Packet : Localization•Finite # of diff. Monochromatic waves always produce INFINTE sequence of repeating wave groups can’t describe (localized) particle•To make localized wave packet, add “ infinite” # of waves with Well chosen Ampl A, Wave# k, ang. Freq. w

localized

vgt

x

0

( )

( ) Amplitude Fn

diff waves of diff k

have different amplitudes A(k)

w = w(k), depends on type of wave, media

( , )

Group Velocity

( ) i k

g

x

k

wt

k

e dk

A

x t

dwV

k

dk

k

A!"

#

#"

=

=

=

=

$

%

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Group, Velocity, Phase Velocity and Dispersion

0

0

0

p

In a Wave Packet: ( )

Group Velocity

Since V ( )

g

k k

p

p

g p k kk k

w w k

dwV

dk

wk def w k

dVdwV V k

dk dk

V

=

=

=

=

=

= ! =

= = +"

p

p

p

Material in which V varies with are said to be Dispersive

Individual harmonic waves making a wave pulse travel at

different V thus changing shape of pulse an

usu

d b

ally V (

ecome spread out

)pV k or!

!

=

g

g

In non-dispersive media, V

In dispersive media V ,depends on

p

p

p

V

dVV

dk

=

"

1ns laser pulse disperseBy x30 after travelling1km in optical fiber

Group Velocity of Wave Packets: Vg

2

g

2Energy E = hf = mc

Consider An Electron:

mass = m velocity = v, momentum = p

; 2

= 2 mch

2 2k

hWavelength = ; =

Group Velocity /

/

:

p

2

Vdw dw dv

dk dk dv

dw d

dv

f

k mvh

dv

!" ! #

! !#

$$

#

%

=

=

=

=

=

2 1/

g

2

2 1/ 2 2 3/ 2 2 3/ 22

/V

mc2 mv 2 mh &

v v v[1-

Group velocity of electron Wave packet "pilot wave"

( ) ] h

2

v[1-( ) ][1-( ) ] h[1-( )

]

/

c cc c

dk d

dv dv

dw dw dvv

dk dk

m

h

dv

v

!! !!

& '( )

= =( )( )* +

& '( )

=( )( )*

=

+

= = %

2

pBut velocity of individual waves

is same as el

making up the wave packet

ect

V

ron's physical v

(not physical

e

)!

i y

loc t

w cc

k v= = >

vgt

x

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Wave Packets & Uncertainty Principles

• Distance ΔX between adjacent minima = (X2)node - (X1)node

• Define X1=0 then phase diff from X1 X2 = π (similarly for t1t2)w

Node at y = 0 = 2A cos ( ), Examine x or t behavior2 2

in x: Need to combine many waves of diff. to make small pulse

k

x= , for small x 0 k & Vik

.

ce

kt x

xk x !

!" " # $ "

" "%

$

&

$" " =

#"

"

In t : Need to combine many to make small pulse

Verca

waves of diff

= , for small 0 & Vice V e

ca r

.

a d

t

w ttn

t

'

'

!

!

'

" " =

" " # $ " &

"

#"

$

2 cos( ) cos( )2 2

k wy A x t kx wt

! !" #$ %= & &' () *+ ,- .

Amplitude Modulation

We added two Sinusoidal waves

What canwelearnfromthissimplemodel?

x1 x2

Signal Transmission and Bandwidth Theory

• Short duration pulses are used to transmit digital info– Over phone line as brief tone pulses– Over satellite link as brief radio pulses– Over optical fiber as brief laser light pulses

• Ragardless of type of wave or medium, any wave pulsemust obey the fundamental relation

» ΔωΔt ≅ π• Range of frequencies that can be transmitted are called

bandwidth of the medium• Shortest possible pulse that can be transmitted thru a

medium is Δtmin ≅ π/Δω• Higher bandwidths transmits shorter pulses & allows high data rate

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Wave Packets & Uncertainty Principles of Subatomic Physics

in space x: since

usual

2 hk = , p =

approximate relationly one writes

In time t : since =2 ,

.

.

. / 2

. / 2

k x

w f E hft

p x h

p x

!

! !

"

#

!

"$ $ =

$ $

%

%

% ==

$ $ =

$ $ & !

usually approximate reone write lations

. / 2

. / 2

E t h

E t

% $ $ =

$ $ & !

What do these inequalities mean physically?

Know the Error of Thy Ways: Measurement Error Δ• Measurements are made by observing something : length, time, momentum,

energy• All measurements have some (limited) precision`…no matter the instrument used• Examples:

– How long is a desk ? L = (5 ± 0.1) m = L ± ΔL (depends on ruler used)– How long was this lecture ? T = (50 ± 1)minutes = T ± ΔT (depends on the accuracy of

your watch)– How much does Prof. Sharma weigh ? M = (1000 ± 700) kg = m ± Δm

• Is this a correct measure of my weight ?– Correct (because of large error reported) but imprecise– My correct weight is covered by the (large) error in observation

Length Measure Voltage (or time) Measure

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• r

Measurement Error : x ± Δx• Measurement errors are unavoidable since the measurement procedure is an experimental one• True value of an measurable quantity is an abstract concept• In a set of repeated measurements with random errors, the distribution of measurements resembles a Gaussian distribution characterized by the parameter σ or Δ characterizing the widthof the distribution

Measurement error large Measurement error smaller

Interpreting Measurements with random Error : ΔTrue value

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Where in the World is Carmen San Diego?• Carmen San Diego hidden inside a big box of length L• Suppose you can’t see thru the (blue) box, what is you best estimate of her location inside box (she could be anywhere inside the box)

xX=0 X=L

Your best unbiased measure would be x = L/2 ± L/2

There is no perfect measurement, there are always measurement error

Wave Packets & Matter Waves

What is the Wave Length of this wave packet? λ−Δλ < λ < λ+Δλ De Broglie wavelength λ = h/p Momentum Uncertainty: p-Δp < p < p+Δp Similarly for frequency ω or f ω−Δω < ω < ω+Δω Planck’s condition E= hf = hω/2 E-ΔE < E < E + ΔE

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Back to Heisenberg’s Uncertainty Principle & Δ

• Δx. Δp ≥ h/4π ⇒– If the measurement of the position of a particle is made with a precision Δx

and a SIMULTANEOUS measurement of its momentum px in the X direction ,then the product of the two uncertainties (measurement errors) can never besmaller than ≅h/4π irrespective of how precise the measurement tools

• ΔE. Δt ≥ h/4π ⇒– If the measurement of the energy E of a particle is made with a

precision ΔE and it took time Δt to make that measurement, then theproduct of the two uncertainties (measurement errors) can never besmaller than ≅h/4π irrespective of how precise the measurement tools

These rules arise from the way we constructed the Wave packets describing Matter “pilot” waves

Perhaps these rulesAre bogus, can we verifythis with some physicalpicture ??

The Act of Observation (Compton Scattering)

Act of observation disturbsthe observed system

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Compton Scattering: Shining light to observe electron

Light (photon) scattering off an electronI watch the photon as it enters my eye

hgg

g

The act of Observation DISTURBSthe object being watched,

here the electron moves away fromwhere it was originally

λ=h/p= hc/E = c/f

Act of Watching: A Thought Experiment

Eye

Photons that go thru are restrictedto this region of lens

Observed Diffractionpattern

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Diffraction By a Circular Aperture (Lens)See Resnick, Halliday Walker 6th Ed (on S.Reserve), Ch 37, pages 898-900

Diffracted image of a point source of light thru a lens ( circular aperture of size d )

First minimum of diffraction pattern is located by

sin 1.22d

!" =

See previous picture for definitions of ϑ, λ, d

Resolving Power of Light Thru a Lens

Resolving power x 2sin

!

"# !

Image of 2 separate point sources formed by a converging lens of diameter d, ability to resolve them depends on λ & d because of the Inherent diffraction in image formation

Not resolved resolvedbarely resolvedΔX

d

ϑ Depends on d

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• Incident light (p,λ) scatters off electron• To be collected by lens γ must scatter thru angle α

• -ϑ ≤α≤ϑ• Due to Compton scatter, electron picks up momentum

•PX , PY

• After passing thru lens, photon diffracts, landssomewhere on screen, image (of electron) is fuzzy

• How fuzzy ? Optics says shortest distance between tworesolvable points is :

• Larger the lens radius, larger the ϑ⇒ better resolution

Putting it all together: act of Observing an electron

Eye

Photons that go thru are restrictedto this region of lens

ObservedDiffractionpattern

sin sin

electron momentum uncertainty is

~2hp sin

x

h hP! !

" "

!"

# $ $

% &

2sinx

!

"# =

2 sin.

2sin

. 2 /p

hp x h

x

! "" !

# $# $% % =& '& '( )

*

*)

% % +(!

"

Pseudo-Philosophical Aftermath of Uncertainty Principle

• Newtonian Physics & Deterministic physics topples over– Newton’s laws told you all you needed to know about trajectory of a

particle• Apply a force, watch the particle go !

– Know every thing ! X, v, p , F, a– Can predict exact trajectory of particle if you had perfect

device

• No so in the subatomic world !– Of small momenta, forces, energies– Cant predict anything exactly

• Can only predict probabilities– There is so much chance that the particle landed here or

there– Cant be sure !....cognizant of the errors of thy observationsPhilosophers went nuts !...what has happened to nature

Philosophers just talk, don’t do real life experiments!

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All Measurements Have Associated Errors

• If your measuring apparatus has an intrinsic inaccuracy(error) of amount Δp

• Then results of measurement of momentum p of anobject at rest can easily yield a range of valuesaccommodated by the measurement imprecision :– -Δp ≤ p ≤ Δp

• Similarly for all measurable quantities like x, t, Energy !

Matter Diffraction & Uncertainty Principle

Incident Electron beam In Y direction

x

Y

Prob

abili

ty

Momentum measurement beyondslit show particle not moving exactly in Y direction, develops a X componentOf motion ΔPX =h/(2π a)

X component PX of momentum

ΔPX

0

slit size: a

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Particle at Rest Between Two Walls• Object of mass M at rest between two walls originally at infinity• What happens to our perception of George as the walls are brought in ?

m

George’s Momentum p0 2 2

On average, measure <p> = 0

but there are quite large fluctuations!

Width of Distribution =

( ) ( ) ; ave ave

P

PL

P PP

!

! != "!"

L