1. PHYS541000 Quantum Mechanics(I)(II)
The first hour: p1-p12
The second hour: p13-p24
The third hour: p25-p32 and more
Chung-Yu Mou

2. Unique subjects in physics
-- quantum mechanics and cosmology
Cosmology：
closest subject to God、can
understand how God creates the world
Quantum Mechanics：
Describing the world by using the way
that electrons can pass two slits
at the same time

7. Quantum mechanics – Purpose of this course
The new way that was developed at the
beginning of the 20th century to interpret
& predict behaviors of microscopic objects
such as atoms, electrons, ..
Purpose of this course
Learn to calculate and think in
the quantum mechanical way

8. Conceptual problem will not be fully
discussed in this course.
In fact, as Richard Feynman remarked:
I think that it is fair to say that no one
understands the quantum theory…

9. Two tracks of discoveries
Spectroscopy
(discrete)
Theory of H
Atom (Bohr)
Matrix Mechanics
(Heisenberg)
Photoelectric
Effect (Einstein)
Matter wave
(De Broglie)
Wave Mechanics
(Schrodinger)
Equivalent
(shown by Schrodinger)
(based on the general formalism of Dirac)

10. Third Formulation Richard Feynman Path Integral
Equivalent to the above two formulations
but may be more general

11. Track 2: particle-wave controversy
Huygens: light is wave
Newton: light is particle
Hook: wave
Maxwell: wave
Young: wave
Planck, Einstein, and Compton: particle ..

12. The Maxwell’s theory of light
Light intensity (I) ~ |E|2 B

13. Wave-like properties of light
Hygen’s principle

14. Interference
Diffraction

15. Particle-like properties of light Pierre Gassendi and Newton
Reflection:
elastic collision of particles
Refraction：

16. Radiation and black-body radiation: all objects radiate electromagnetic waves

17. Thermodynamics of Radiation

18. Black Body Radiation
If Aλ=1, Eλ= f(λ,T) is universal => black body radiation
Ideal black body ＝ cavity

20. Max Planck
Expt & Planck

21. h = Planck constant =6.626×10-34 joule-sec
Planck’s consideration is based on statistical mechanics not dynamics
h = Planck constant
=6.626×10-34 joule-sec

22. Solid State Version : C versus T
Quantum region

23. Photoelectric effect:
metal

24. Wave-like interpretation of radiationλ
electron
Electric field experienced by electrons λ x

25. Expectation from wave properties
Electric field experienced by electrons
electron x
The larger E is, the higher the intensity is. It is easier to
shake off electrons with larger current. This is
independent of wavelength/frequency.
Any frequency of light can yield photoelectrons.
Need sufficient time to accumulate energy to shake off
electrons.

26. Photoelectric effect experiment

27. Einstein: E=nhν is dynamical!

28. Compton effect: λ

29. Compton：experimental results are consequences of particle’s collision
Light quanta collides with electron elastically:
After collision, the change of momentum for light quanta changes its wavelength.

30. Viewed as collision of particles

31. Is light particle or wave?
1924 de Broglie joined the debate
Is the fundamental particle –electron wave or particle?
Ph.D. thesis of de Broglie：electrons are also wave-like

32. Need more clear and direct evidence !
What is particle? Wave?
particle：
(1) One grain by one grain, discontinous
(2) trajectory
Both satisfy momentum and energy conservation
Wave：
(1)interference
(2)diffraction
Need more clear and direct evidence !

33. Expectation for electrons being particles:
expectation for particles
electron source

34. Expectation for electrons being waves:

35. Electronic version of Young’s experiment
electron source

36. Interference of waves Destructive interference
Constructive interfrence

37. Quantitative analysis
dsinθ = (m+1/2)λ destructive inteference
dsinθ= mλ constructive interference
dsinθ = m θ d

39. The Feynman Lectures on Physics (III) p. 1-4~1-5
…This experiment has never been done in just this way. The trouble is that the apparatus would have to be made on an impossible small scale … We are doing a “thought experiment”…
Reference:
Davisson and Germer: diffraction of electrons
wavelength: 0.165nm(1.65 Å, 50eV)

40. Tonomura et al.
American Journal of Physics
57, 117(1989)
λ = 0.054Å (50kV), Va = 10V
a = 0.5μm, b = 5mm

43. Matter waves and mechanical properties
h = Planck constant
(6.626×10-34 joule-sec)
DeBroglie:
λ=h/p
Einstein:
E=hν=p2/2m

44. Bulk size
Nano size

45. Neutron
Reviews of Modern Physics 60, 1067 (1988)

46. Helium atoms
Physics Review Letters 66, 2689 (1991)

47. Macromolecules
C60

48. Nature 409, 304(2001)

49. Nature 401,680 (1999)
(1) Diffration grating is SiNx grating (period 100 nm) with
width 0.1 m.
(2) C60 is thermal ionized by a laser. The ions are then accelerated and directed towards a conversion electrode. The ejected electrons are subsequently counted by a Channeltron electron multiplier.

50. Other atoms:
Na, Phys. Rev. 66, 2693 (1991)

51. Bio-molecules
3D structure of tetraphenylporphyrin C44H30N4(TPP)

52. 3D structure of the fluorofullerene C60F48
L Hackermuller et al. Phys. Rev. Lett , (2003)

53. Interference of 106 Na atoms
Science 275, 637 (1997)

54. Conclusions
(i) Number of counting ~ probability of finding the particle
(ii) When there are many exclusive choices, 1,2,3,..
P (total probability) ≠ P1+P2+P3+ …
Particle-like behavior: P = P1+P2+P3+ …

55. How do we describe the results ?
Experience from electro-magentism:
(i) Light intensity (I) ~ |E|2
(ii) For multiple choices:
I (total intensity) ~ |E1+E2|2 ≠ |E1|2+|E2|2
A way out to escape the classical sum rule
of probability P (total probability) = P1+P2

56. Max Born’s probability interpretation
Ψ(x,y,z,t) =
wave function of matter wave (complex)
E(x,y,z,t) = Electric field
(i) Occurrence of events (particle’s appearance) = Probability density ~ |Ψ |2
(ii) For many alternative routes (choices), each alternative route is represented by Ψi
total probability ~ |Ψ1+ Ψ2+ …. |2
(principle of superposition)

57. Difference between classical and quantum:
|Ψ1+ Ψ2|2 – (|Ψ1|2+ |Ψ2|2)= Ψ1Ψ2 *+ Ψ1*Ψ2 = interference term r1 r2

58. Particle-wave duality
When the material is detected,
it is a whole particle that is being detected.
It is particle-like.
The matter wave commands the particle where
to go and arrive in different positions with
the associated probability.
pilot wave

59. New concepts classical: |Ψ1|2+ |Ψ2|2 quantum: |Ψ1+ Ψ2|2
The electron does not pass 1 or 2
“It can pass 1 and 2 at one time.”
electron source 1 2
Ref: The ghost in atom

60. Which-way experiment Electron source Nature 395, 33(1998)
PRL70,2359(1993)

61. Once which-way is known...
Particle-like behavior
Electron source

62. Which-way experiment 198Hg+ polarized Phys. Rev. Lett. 70,2359(1993)
Ground state: 6s2S1/2, Excited state: 6p2P1/2
degenerate: mJ
Photons: σ & π polarized
π : ΔmJ=0 two atoms are in the same state
σ: |Δ mJ|=1 two atoms are not in the same state

63. π polarized
σ polarized

64. Five quantum effects (to be covered)
Interference
Quantization
Tunneling effect
Spin
Entanglement

65. Entanglement/糾纏
(Quantum telepathy)

66. Entangled photon-pair experiment?
focal plane

67. Zeilinger, Rev. Mod. Phys. S288, (1999)
焦平面上Detector之距離
Dopfer, B., 1998
Zeilinger, Rev. Mod. Phys. S288, (1999)

68. Einstein: Spooky action at a distance!
“ghost-like behavior”
Due to the nature of probability,
the propagation of information
does not exceed speed of light.

69. I think that it is fair to say that no one
understands the quantum theory…

70. Description of plane wavesx λ t T
phase

71. Sinusoidal wave in higher dimension: being directional
phase:

72. Counting number of waves
Number of possible directions for a given ν