Wave Nature of Matter
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Transcript of Wave Nature of Matter
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Wave Nature of Matter
Light/photons have both wave & particle behaviors.
Waves – diffraction & interference, Polarization.
Acts like Particles – photoelectric effect, E = hf.
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de Broglie/Matter waves 1924
If light behaves as a particle, then particles should behave like waves. Right?
Particles also have , related to their momentum.
Where m = rest mass of the particle
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Derive Eq Using E = mc2.what is the wavelength of matter
E = hf E = mc2 = mv2. hf = mv2. but f = v/ and v2. hv/ = mv2.Cancel v.h/ = mv mv = p. h/ = p = h/p
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1: Find the of an electron accelerated through a p.d. of 30-V.
Find the e- velocityqV = ½ mv2.v = 3.2 x 106 m/s
Calculate . = h/p2.3 x 10-10 m.
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Handy Equation
KE e- = 1/2 mv2 = p2/2m
For e- accelerated through pd eV = KE = p2/2m
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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.
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Evidence
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Electrons diffracting through 2 slitsWhat does this pattern look like?
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Electron diffraction
Davisson-Germer experiment: similar to xray diffraction
They know the e- speed thus know the deBroglie
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Maximum intensity from wave diffraction pattern
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Maxima observedFor e-. Diffraction pattern.
Can calc using position of min & max.
agrees with deBroglie from equation.
Results of Davisson-Germer experiment:Proof of deBroglie
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2. A 70kg person is running 5 m/s. Find . How does the compare with the on the EM spectrum?
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3. Find for an e- moving at 107 m/s. How does the compare with the on the EM spectrum?
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Hwk Read Hamper 243 – 246 IB Set
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Electron in a Box
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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.
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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.
Electron in a Box
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If e- viewed as standing waves the orbit model works better.
2L =
2L/2 =
2L/3 =
Since p = h/:
E = n2h2
8mL2.
Orbit n=1 ground Planck
Circular Diameter
Mass e-
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De Broglie & e- in a box
The de Broglie of e- are the‘s of the standing allowed by the box;
since λ = 2L/n where n is an integerenergy is quantized;
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If e- are standing waves. Only ’s that fit certain orbits are possible.
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Fit a standing wave into a circular orbit
Circumference = 2r = n
deBroglie’s equation for the electron:
= h/mv
You get the equation for quantized angular momentum:
mvr = nh/2
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’s that don’t fit circumference undergoes destruction interference & cannot exist.
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IB Prb Electron in a Box
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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.
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Electron cloud
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http://www.youtube.com/watch?v=-YYBCNQnYNM&feature=related
The structure of atoms
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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:
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x = Uncertainty in positionp =Uncertainty in momentum
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If you know the momentum exactly, then you have no knowledge about position.
Another aspect to uncertainty is:
Et ≥ h/4.
E = energy J. t = time (s)
If a mass remains in a state for a long time, it can have a well defined E.
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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.
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http://www.youtube.com/watch?v=hZ8p7fIMo2k
Heisenberg.
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Mechanical universe.
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The End for now.Minute Physics Heisenberg
http://www.youtube.com/watch?v=7vc-Uvp3vwg
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http://www.youtube.com/watch?v=hZ8p7fIMo2k
http://www.youtube.com/watch?v=groBKtfZfsA
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HL stuff.
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Constructive interference of e- waves scattered from two atoms occurs when d sin = m (m = 1, = 50o, solve for )
The angle depends on the voltage used to accelerate the electrons!Positions of max/min were similar to xray diffraction
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KE of electron = 1/2 mv2 = eV = p2/2m
= the same that was found via the diffraction equation
Confirms the wave nature of electrons!
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39.3 Probability and uncertainty
QM: a particle’s position and velocity cannot be precisely determined
Single-slit diffraction: << a 1 = angle between central max. and first minimumif 1 is very small, 1 = / a (RADIANS!)
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Interpret this result in terms of particles:
tan1 = py / px So 1 = py / px py / px = / a
There is uncertainty in py = py
Can we fix this by making the slit width = a smaller?
py a > h
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No, because making the slit smaller makes central max wider
Wide slit, py is well defined (~0)
narrow slit, py could be anything
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h = h/2
Slit width a is an uncertainty in position, now called x
y = 1/x
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The longer the lifetime t of a state, the smaller its spread in energy E.
A state with a “well-defined” energy
A state with a “poorly-defined” energy
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Two-slit interference
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With light…
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Electrons diffracting through 2 slits
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39.4 Electron microscope
Better resolution because e- wavelengths << optical photons
Microscope resolution ~ 2 x wavelength
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
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SEM image
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TEM image of a bacterium
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Two waves with different wave numbers k = 2
In reality, wave functions are localized: combinations of 2 or more sin & cos functions
ph
k
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A wave packet: particle & wave properties
(x,y,z) A(k)e ikx
dk
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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
(x,y,z) A(k)e ikx
dk
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