Wave-particle duality
Physics 123
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Lecture XII
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Concepts
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De Broigle waves
Energy levels
Quantum numbers
Emission and absorption spectra
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Lecture XII
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Wave – Particle duality
• If light exhibits both wave and particle properties
then particles (e.g. electrons) must also exhibit
wave properties – e.g. interference.
• Matter (de Broglie) waves
l=h/p
p=mv
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Interference of electrons
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•
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Send electron beam (a lot of electrons) on crystal structure
Interference pattern is determined by l=h/p
Double slits distance d~1nm
Interference pattern
– Maxima (more e):
– Minima (no e):
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d sinq = m l
d sinq = (m+½ ) l
Lecture XII
m=0,1,2,3,….
4
Matter waves
• Particle position in space cannot be predicted with
infinite precision
• Heisenberg uncertainty principle x  p  h / 2
t  E  h / 2
• (Wave function Y of matter wave)2 dV=probability to
find particle in volume dV.
• But while probability is a real number, wave function is
a complex number. It has a phase.
• When two matter waves meet we add wave functions,
not probabilities! Interference can be observed (phase is
important
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Particle in a box
• Infinite potential well
• Particle mass m in a box
length L  standing wave
• Similar to guitar string
• Wave function - string
• We do not know with
certainty where in the box the
particle is
• More chances to find the
particle at a cress
• No chance at a knot
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Yn = A sin( 2x / ln )
Pn = A sin (2x / ln )
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2
6
Particle in a box
• Infinite potential well
• Boundary condition:
Y(0)=0;
Y(L)=0;
• Solve for wavelength:
Yn = A sin( 2x / ln )
2L / ln = n
ln = 2 L / n,
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n = 1,2,3...
Wavelength is quantized!
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Particle in a box
• Mass m
• Length L
• Possible wave lengths
ln=2L/n
• De Broigle waves
pn=h/ln
pn=hn/2L
• Possible kinetic energy states
2
2
2
pn
h n
En =
=
2m 8mL2
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Energy is quantized!
Energy levels – spectrum.
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Electron in a box
• Mass mec2=0.5MeV
• Length L=0.62 nm
E4 = 16eV
h2n2
h 2c 2
2
En =
=
n
2
2 2
8me L 8me c L
1240eV  nm 2
1
2
En = (
)
n
0.62nm
8  0.5 106 eV
E3 = 9eV
E2 = 4eV
En = 1eV  n 2
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E1 = 1eV
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Energy transitions
• These are kinetic energy
elevels, PE=0
• What happens when e jumps
from n=4 to n=3 level?
• KEe=16 eV  KEe=9 eV
• Where did 7 eV of energy go?
• 7 eV photon is emitted
• This photon was not
“sitting inside the electron”.
• It is born in this energy
transition
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Lecture XII
E4 = 16eV
E = 7eV
e-
E3 = 9eV
E2 = 4eV
E1 = 1eV
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Energy transitions
• What if e is on n=3 level and 7eV
photon comes by?
E4 = 16eV
e-
• e will gulp this photon and jump to n=4
level.
• Photon is not hiding inside e,
E = 7eV
It is absorbed.
• What if white light goes through this
system?
• Photons of 7 eV energy will be taken
out
• As will be photons of
– 5 eV, 3 eV
– 15, 12, 8 eV
• Absorption spectrum – dark spectral lines
• Note that 8.5 eV photon will pass by without
any interaction!
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Lecture XII
e-
E3 = 9eV
E2 = 4eV
E1 = 1eV
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Hydrogen atom
• Positively charged
nucleus inside, negatively
charged electrons around
• Electron is attracted to
nucleus
• Electron is trapped in a
potential well created by
nucleus (“a box”)
1
U
r
• Energy levels in atom
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Standing electron waves in
Hydrogen atom
• Standing waves:
• 2rn=nl
l=h/mv
mvrn=nh/2
• Angular momentum L=mvrn
is quantized
L=nh/2
• n – orbital quantum number
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Hydrogen atom
• Energy levels in H
13.6
En =  2 eV
n
• Electron from level
n goes to level n’
• Energy of emitted
photon
13.6 13.6
E = En  En ' =  2  2 eV
n
n'
1
1
E = 13.6( 2  2 )eV
n' n
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Absorption and emission spectra
UV
hc
Visible light
1
1
= 13.6( 2  2 )eV
l
n' n
1
IR
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Lyman series
n’=1
Balmer series
n’=2
Paschen series
n’=3
Rydberg constant R=1/91.2nm
1
1
= R( 2  2 )
l
n' n The first one to be discovered
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