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
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