Transcript De Broglie wavelength or “matter waves”

Wave Nature of Matter
 Light/photons have both wave & particle
behaviors.
 Waves – diffraction & interference,
Polarization.
 Acts like Particles – photoelectric effect, E =
hf.
de Broglie/Matter waves 1924
If light behaves as a particle, then particles should
behave like waves. Right?
Particles also have l, related to their momentum.
Where m = rest mass of the particle
Derive Eq Using E = mc2.
what is the wavelength of matter
 E = hf
 E = mc2 = mv2.
 hf = mv2. but f = v/l and v2.
 hv/l = mv2.
Cancel v.
h/l = mv
 h/l = p
 l = h/p
mv = p.
1: Find the l of an electron accelerated
through a p.d. of 30-V.
 Find the e- velocity
qV = ½ mv2.
v = 3.2 x 106 m/s
Calculate l.
l = h/p
2.3 x 10-10 m.
Handy Equation
KE e- = 1/2 mv2 = p2/2m
For e- accelerated through pd eV = KE = p2/2m
De Broglie wavelength or “matter waves”
are not physical.
They are not EM or mechanical waves
but determine the probability of finding a
particle in a particular place.
Evidence
Electrons
through look
2 slits like?
What
doesdiffracting
this pattern
Electron diffraction
Davisson-Germer experiment:
similar to xray diffraction
They know the e- speed thus know the deBroglie l
Maximum intensity from wave diffraction
pattern
Results of Davisson-Germer experiment:
Proof of deBroglie
Maxima observed
For e-. Diffraction
pattern.
Can calc l using position
of min & max.
l agrees with deBroglie l
from equation.
2. A 70kg person is running 5 m/s. Find l. How does
the l compare with the l on the EM spectrum?
3. Find l for an e- moving at 107 m/s. How does the
l compare with the l on the EM spectrum?
Hwk Read Hamper 243 – 246 IB Set
Electron in a Box
Bohr Model of Atom
Electrons jump “oscillate” up & down to
different energy levels absorbing or releasing
photons.
Bohr explains H well, not effective for larger
atoms.
Electron in a Box
The atomic orbits of Bohr can better be visualized as
e- oscillating in a box closed at both ends.
Picture that the de Broglie waves for e- are standing
waves.
This helps explain why energy is quantized.
If e- viewed as standing waves the
orbit model works better.
2L = l
2L/2 = l
2L/3 = l
Since p = h/l:
E
Orbit n=1
ground
Planck
= n2h2
8mL2.
Circular Diameter
Mass e-
De Broglie & e- in a box
The de Broglie l of e- are the
l‘s of the standing l allowed by the box;

since λ = 2L/n where n is an integer
energy is quantized;
If e- are standing waves. Only l’s that fit certain
orbits are possible.
Fit a standing wave into a circular orbit
Circumference = 2r = nl
deBroglie’s equation
for the electron:
l = h/mv
You get the equation for
quantized angular momentum:
mvr = nh/2
l’s that don’t fit circumference undergoes
destruction interference & cannot exist.
IB Prb Electron in a Box
Schrodinger Model
Schrodinger used deBroglie’s wave hypothesis
to develop wave equations to describe matter
waves. Electrons have undefined positions but
do have probability regions he called “electron
clouds”. The probability of finding an e- in a
given region is described by a wave function .
Schrodinger’s model works for all atoms.
Electron cloud
The structure of atoms
http://www.youtube.com/watch?v=-
YYBCNQnYNM&feature=related
Heisenberg Uncertainty.
 1927 Cannot make simultaneous measurements of
position & momentum on particle with accuracy.
 The act of making the measurement changes
something.
 The more certain we are of 1 aspect, the less certain
we are of the other.
 The total uncertainty will always be equal to or
greater than a value:

Dx = Uncertainty in position
Dp =Uncertainty in momentum
If you know the momentum exactly, then you
have no knowledge about position.
Another aspect to uncertainty is:
DEDt ≥ h/4P.
E = energy J.
t = time (s)
If a mass remains in a state for a long time,
it can have a well defined E.
Example Problem
The velocity of an electron is 1 x 106
m/s ± 0.01 x 106 m/s. What is the
maximum precision in its position?
5.8 x 10-9 m.
http://www.youtube.com/watch
?v=hZ8p7fIMo2k
Heisenberg.
Mechanical universe.
The End for now.
Minute Physics Heisenberg
http://www.youtube.com/watch?v=7vc-
Uvp3vwg
http://www.youtube.com/watch?v=hZ8p7fIMo2k
http://www.youtube.com/watch?v=groBKtfZfsA
HL stuff.
Constructive interference of e- waves scattered
from two atoms occurs when d sin = m l (m = 1,
 = 50o, solve for l)
The angle depends
on the voltage used to
accelerate the
electrons!
Positions of max/min
were similar to xray
diffraction
KE of electron = 1/2 mv2 = eV = p2/2m
= the same l that was found via the diffraction equation
Confirms the wave nature of electrons!
39.3
Probability and uncertainty
QM: a particle’s position and velocity
cannot be precisely determined
Single-slit diffraction:
l << a
1 = angle between central max. and first minimum
if 1 is very small, 1 = l / a (RADIANS!)
Interpret this result in terms of particles:
tan 1 = py / px
So 1 = py / px
There is uncertainty in py = Dpy
py / px = l / a
Dpy a > h
Can we fix this by making the slit width = a smaller?
No, because making the slit smaller makes central max wider
narrow slit,
Wide slit,
py could be
py is well
anything
defined (~0)
Slit width a is an uncertainty in position, now called Dx
h = h/2
y = 1/x
The longer the lifetime Dt of a state, the smaller
its spread in energy DE.
A state with a “poorlydefined” energy
A state with a “well-defined”
energy
Two-slit interference
With light…
Electrons diffracting through 2 slits
39.4 Electron microscope
Microscope resolution ~ 2 x wavelength
Better resolution because e- wavelengths << optical photons
Scanning electron microscope:
• e- beam sweeps across a specimen
• e- are knocked off and collected
• Specimen can be thick
• Image appears much more 3-D than a
regular microscope
SEM image
TEM image of a bacterium
In reality, wave functions are localized:
combinations of 2 or more sin & cos functions
Two waves with different wave numbers k = 2  l
p
h
l
 k
A wave packet: particle & wave properties
(x, y,z) 

 A(k)e

ikx
dk
(x, y,z) 

 A(k)e
ikx
dk

Does a wave packet represent a stationary state?
A stationary state

•
•
•
Has a definite energy (meaning, no uncertainty,
only 1 value of E)
* is independent of time
* = |(x,y,z)|2