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Wave Nature of Matter. Light/photons have both wave & particle behaviors. Waves – diffraction & interference, Polarization. Particle – photoelectric effect, E = h f. de Broglie/Matter waves. If light behaves as a particle, then particles should behave like waves. Right?
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Wave Nature of Matter • Light/photons have both wave & particle behaviors. • Waves – diffraction & interference, Polarization. • Particle – photoelectric effect, E = hf.
de Broglie/Matter waves 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. • E = hf • E = mc2 = mv2. • hf = mv2. but f = v/l and v2 = vv. • hv/l = mvv. • Cancel v. • h/l = mv mv = p. • h/l = p • l = h/p
Ex: 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.
De Broglie wavelength or “matter waves” are not EM or mechanical waves but determine the probability of finding a particle in a particular place.
What does this look like? Electrons diffracting through 2 slits
Electron diffraction Davisson-Germer experiment: similar to xray diffraction They know the e- speed thus know the deBroglie
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.
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.
2L = l 2L/2 = l If e- viewed as standing waves the orbit model works better. 2L/3 = l Since p = h/l: E = n2h2 8mL2. Orbit n=1 ground Planck Circular Diameter Mass e-
De Broglie & e- in a box • The de Broglie l of e- are thel‘s of the standing l allowed by the box; • since λ = 2L/n where n is an integerenergy 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 = 2r = n deBroglie’s equation for the electron: = 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.
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.
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:
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.
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
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
KE of an electron = 1/2 mv2 = eV = p2/2m = the same 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: << a 1 = angle between central max. and first minimum if 1 is very small, 1 = / a (RADIANS!)
py a > h 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?
No, because making the slit smaller makes central max wider narrow slit, py could be anything Wide slit, py is well defined (~0)
h = h/2 Slit width a is an uncertainty in position, now called x y = 1/x
The longer the lifetime t of a state, the smaller its spread in energy E. A state with a “poorly-defined” energy A state with a “well-defined” energy
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
In reality, wave functions are localized: combinations of 2 or more sin & cos functions Two waves with different wave numbers k = 2
