1. Wave Nature of Matter
Light/photons have both wave & particle behaviors.
Waves – diffraction & interference, Polarization.
Acts like Particles – photoelectric effect, E = hf.

2. 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

3. 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 mv = p.
h/l = p
l = h/p

4. 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 m.

5. Handy Equation KE e- = 1/2 mv2 = p2/2m
For e- accelerated through pd eV = KE = p2/2m

6. De Broglie wavelength or “matter waves” are not physical
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.

7. Evidence

8. What does this pattern look like?
Electrons diffracting through 2 slits

9. Electron diffraction Davisson-Germer experiment:
similar to xray diffraction
They know the e- speed thus know the deBroglie 

11. Maximum intensity from wave diffraction pattern

12. 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.

13. 2. A 70kg person is running 5 m/s. Find l
2. A 70kg person is running 5 m/s. Find l. How does the l compare with the l on the EM spectrum?

14. 3. Find l for an e- moving at 107 m/s
3. Find l for an e- moving at 107 m/s. How does the l compare with the l on the EM spectrum?

15. Hwk Read Hamper 243 – 246 IB Set

16. Electron in a Box

17. Bohr explains H well, not effective for larger atoms.
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.

18. 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.

19. If e- viewed as standing waves the orbit model works better.
2L = l
2L/2 = l
2L/3 = l
Since p = h/l:
E = n2h2
8mL2.
Orbit n=1 ground
Planck
Circular Diameter
Mass e-

20. 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;

21. If e- are standing waves. Only l’s that fit certain orbits are possible.

22. Fit a standing wave into a circular orbit
Circumference = 2r = n
deBroglie’s equation for the electron:
 = h/mv
You get the equation for quantized angular momentum:
mvr = nh/2

23. l’s that don’t fit circumference undergoes destruction interference & cannot exist.

24. IB Prb Electron in a Box

25. 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.

26. Electron cloud

27. The structure of atoms

28. 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:

29. Dx = Uncertainty in position Dp =Uncertainty in momentum
Dx = Uncertainty in position Dp =Uncertainty in momentum

30. DEDt ≥ h/4P. E = energy J. t = time (s)
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.

31. 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.

32.
Heisenberg.

34. Mechanical universe.

35. The End for now. Minute Physics Heisenberg

37. HL stuff.

38. 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

39. KE of electron = 1/2 mv2 = eV = p2/2m
= the same  that was found via the diffraction equation
Confirms the wave nature of electrons!

41. 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!)

42. Interpret this result in terms of particles:
Lambda = h/px
tan1 = py / px
So 1 = py / px
py / px =  / a
There is uncertainty in py = py
py a > h
Can we fix this by making the slit width = a smaller?

43. No, because making the slit smaller makes central max wider
narrow slit,
py could be anything
Wide slit,
py is well defined (~0)

44. Slit width a is an uncertainty in position, now called x
y = 1/x

45. 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

46. Two-slit interference

47. With light…

48. Electrons diffracting through 2 slits

49. 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

50. SEM image

52. TEM image of a bacterium

53. In reality, wave functions are localized:
combinations of 2 or more sin & cos functions
Two waves with different wave numbers k = 2

54. A wave packet: particle & wave properties

55. Does a wave packet represent a stationary state?
Has a definite energy (meaning, no uncertainty, only 1 value of E)
* is independent of time
* = |(x,y,z)|2