Physics 2D Lecture SlidesfWeek of May 4,2009Part 2

(Oleg Shpyrko)Sunil Sinha

UCSD Physics

de Broglie’s matter-wave theory

Louis de Broglie

Bohr’s Explanation of Hydrogen like atoms• Bohr’s Semiclassical theory explained some spectroscopic dataBohr s Semiclassical theory explained some spectroscopic data

Nobel Prize : 1922• The “hotch-potch” of classical & quantum attributes left many

(Einstein) unconvinced– “appeared to me to be a miracle – and appears to me to be a miracle today

...... One ought to be ashamed of the successes of the theory”

• Problems with Bohr’s theory:– Failed to predict INTENSITY of spectral lines – Limited success in predicting spectra of Multi-electron atoms (He)ted success p ed ct g spect a o u t e ect o ato s ( e)– Failed to provide “time evolution ” of system from some initial state– Overemphasized Particle nature of matter-could not explain the wave-particle

duality of light y g– No general scheme applicable to non-periodic motion in subatomic systems

• Without fundamental insight …raised the question : Why was Bohr successful?successful?

Prince Louis de Broglie & Matter Waves • Key to Bohr atom was Angular momentum quantizationKey to Bohr atom was Angular momentum quantization• Why this Quantization: mvr = |L| = nh/2π ?• Invoking symmetry in nature, Louis de Broglie conjectured:g y y g j

Because photons have waveBecause photons have wave and particle like nature particles may have wave likeparticles may have wave like properties !!

Electrons have accompanying “pilot” waveaccompanying pilot wave (not EM) which guide particles thru spacetime

A PhD Thesis Fit For a Prince

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

frequency f = E/h– frequency f = E/h

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

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

t i t l i th ti ti diti fstring to explain the quantization condition of Bohr orbits

Matter Waves : How big, how small 1.Wavelength of baseball, m=140g, v=27m/s

3434h 6.63 10 . =

p (.14 )(27 / )1.75 10h J s

mv kg m smλ

−−×

= = ×=

size of nucleus

Baseball "looks"

like a particle baseballλ <<<

⇒

⇒

2. Wavelength of electr2

on K=120eV (assume NR)pK= 2p mK⇒ =

-31 19

K 22m

= 2(9.11 10 )(120 )(1.6 10 )

p mK

eV −

⇒ =

× ×

1

-24

30

4

=5.91 10 . /6.63 10 . 11 12 0

Kg m sJ sh mλ

−−

×

×= == ×24

Size

5.91 10 . /

of at

1

o

1.12 0e

e

kg m sm

p

λ

λ −= =×

⇒

= ×

m !!

Models of Vibrations on a Loop: Model of e in atom

Modes of vibration when a integral # of λ fit into

Fractional # of waves in a loop can not persist due to destructive interference# of λ fit into

loop( Standing waves)vibrations continue

Indefinitely

De Broglie’s Explanation of Bohr’s Quantization

Standing waves in H atom:Constructive interference whenn = 2 r

h hλ π

since h=p

......( )hm

NR

nhv

λ =

n = 32nh r

mvπ⇒

⇒

=

h

Angular momentum n mvr⇒ =h

Quantization conditio !nThis is too intense ! Must verify such “loony tunes” with experiment

Reminder: Light as a Wave : Bragg Scattering ExptRange of X-ray wavelengths scatterRange of X ray wavelengths scatterOff a crystal sampleX-rays constructively interfere from y yCertain planes producing bright spots

Interference Path diff=2dsinϑ = nλ

Verification of Matter Waves: Davisson & Germer Expt

If electrons have associated wave like properties expect interferenceIf electrons have associated wave like properties expect interference pattern when incident on a layer of atoms (reflection diffraction grating) with inter-atomic separation d such that

path diff AB= dsinϕ = nλ

Atomic lattice as diffraction grating

Layer of Nickel atoms

Electrons Diffract in Crystal, just like X-rays

iff iDiffraction pattern produced by 600eVelectrons incident onelectrons incident on a Al foil target

Notice the waxing and waning of scattered electron Intensity.

What to expect if l t h delectron had no wave

like attribute

Davisson-Germer Experiment: 54 eV electron Beam

nten

sity max Max scatter angle

atte

red

In

Polar Plot Cartesian plot

Sca Polar Plot Cartesian plot

Polar graphs of DG expt with different electron accelerating potentialPolar graphs of DG expt with different electron accelerating potentialwhen incident on same crystal (d = const)

Peak at Φ=50o

when V = 54 Vwhen Vacc 54 V

Analyzing Davisson-Germer Expt with de Broglie idea

de Broglie for electron accelerated thru V =54Vλ acc

22

de Broglie for electron accelerated thru V =54V

1 2 ; 2 2

2p eVmv K eV vm m

eVp mv mm

λ

• = = = ⇒ = ==

If you believe de Broglie

h=2p 2

predicth hmv eV

hmeV

λ λ= = ==

10acc (de Br

2

V = 54 Volts 1.6 og

p 2

F lie)E ptal d

7 10 or

mv eVmm

meV

mλ −×⇒ =

ata from Da isson Germer Obser ation:Exptal d

nickelm

-10

ax

ata from Davisson-Germer Observation:=2.15 10 (from Bragg Scattering)

(observation from scattering intensity p

d =2.15 A

50 lo )todiff

m

θ =

×&

Diffraction Rule : d sin = n

oF r P

ff

φ λ

⇒pred

oict 1.67

rincipal Maxima (n=1); = agreement(2 15 A)(sin 50 )meas AExcellent

λλ

= && oF r P⇒

observrincipal Maxima (n=1); = agreement(2.15 A)(sin

=150 )

.65 Excellent

Aλ

λ &

Davisson Germer Experiment: Matter Waves !

h

Excellent Agreeme2

nt

predicthmeV

λ=

Excellent Agreement

Practical Application : Electron Microscope

Electron Microscopy

Electron Microscope : Excellent Resolving Power

Electron Micrograph Sh i B i hShowing BacteriophageViruses in E. Coli bacterium

The bacterium is ≅ 1μ size

Swine Flu Virus

Electron microscope image of the H1N1 virus, April 27, 2009, at the U.S. Centers for Disease Control and Prevention's headquarters in AtlantaCenters for Disease Control and Prevention s headquarters in Atlanta, Georgia (AP Photo/Center for Disease Control and Prevention, C. S. Goldsmith and A. Balish) The viruses are 80–120 nanometres in diameter.

1918 Swine Flu Virus

Negative stained transmission electron micrograph (TEM) showed recreated 1918 influenza virions that were collected from therecreated 1918 influenza virions that were collected from the supernatant of a 1918-infected Madin-Darby Canine Kidney (MDCK) cell culture 18 hours after infection.

The 1918 Spanish flu epidemic was caused by an influenza A (H1N1) virus, killing more than 500,000 people in the United States, and up to 50 million worldwide.

West Nile Virus extracted from a crow brain

TEM pictures of Carbon Nanotubes

TEM of 6 nm Au Nanoparticles(Shpyrko Group, UCSD)

Just What is Waving in Matter Waves ?• For waves in an ocean, it’s the Imagine Wave pulse moving alongFor waves in an ocean, it s the

water that “waves”• For sound waves, it’s the

molecules in medium

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

• For light it’s the E & B vectors • What’s waving for matter

space (unlike a pure harmonic wave)

waves ?– It’s the PROBABLILITY OF

FINDING THE PARTICLE that waves !waves !

– Particle can be represented by a wave packet in

• Space p• Time • Made by superposition of

many sinusoidal waves of different λ

• It’s a “pulse” of probability

What Wave Does Not Describe a Particle( )A k Φ 2 2k fπcos ( )y A kx tω= − +Φ

x t

y2 , 2k w fπ πλ

= =

• What wave form can be associated with particle’s pilot wave?

x,t

• 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 tω= − +Φ

as particle ……(like me and my shadow)

p p

In Matter:

Phase velocity of sinusoida l wave: (v ) v fλ=

Conflicts withSingle sinusoidal wave of infinite extent does not represent particle

2

h( ) =

E(b) f =

hap mv

mc

λγγ

=

=

Conflicts withRelativity Unphysical

extent does not represent particle localized in space

N d “ k ” l li d2 2

p

(b) f

v

h

!

E mc cf cp vm

h

vγλγ

= = = = >⇒

p yNeed “wave packets” localizedSpatially (x) and Temporally (t)

Wave Group or Wave Pulse • Wave Group/packet: Imagine Wave pulse moving alongWave Group/packet:

– Superposition of many sinusoidal waves with different wavelengths and frequencies

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

– Localized in space, time – Size designated by

• Δx or Δt

space (unlike a pure harmonic wave)

Δx or Δt– Wave groups travel with the

speed vg = v0 of particle

• Constructing Wave PacketsConstructing Wave Packets– Add waves of diff λ, – For each wave, pick

• AmplitudeWave packet represents particle prob

• Amplitude• Phase

– Constructive interference over the space time of particlethe space-time of particle

– Destructive interference elsewhere ! localized

Phase velocity

Group velocity

See animation of group/phase velocity at:

http://en.wikipedia.org/wiki/Group_velocity

Resulting wave's "displacement " y = y1 + y2 :

y = A cos(k1x −ω1t) + cos(k2x −ω2t)⎡⎣ ⎤⎦

Trignometry : cosA+cos B =2cos(A+B

2)cos(

A-B2

)

∴ y 2A cos(k2 − k1 x

ω2 −ω1 t)⎛⎜

⎞⎟ cos(

k2 + k1 xω2 +ω1 t)

⎛⎜

⎞⎟

⎡⎢

⎤⎥∴ y = 2A cos( 2 1

2x − 2 1

2t)

⎝⎜ ⎠⎟cos( 2 1

2x − 2 1

2t)

⎝⎜ ⎠⎟⎣⎢⎢ ⎦

⎥⎥

since k2 ≅ k1 ≅ kave , ω2 ≅ ω1 ≅ ω ave ,Δk k,Δω ω

⎛ ⎞⎡ ⎤∴ y = 2A cos(

Δk2

x −Δω2

t)⎛⎝⎜

⎞⎠⎟

cos(kx −ωt)⎡

⎣⎢

⎤

⎦⎥ ≡ y = A' cos(kx −ωt), A' oscillates in x,t

A ' 2A (Δk Δω

t)⎛ ⎞

d l t d lit d A ' = 2A cos(

2x −

2t)

⎝⎜ ⎠⎟= modulated amplitude

Phase Vel V avep

w=

gGroup Vel V

pavekwk

Δ=Δ

waveGroup

: Vel of envelope=g

kdwVdk

ΔOr packet