1. Clicker Questions Lecture Slides Professor John Price, Spring 2019
Physics 2130 Foundations of Modern Physics Quantum Mechanics Part I: Background
Clicker Questions
Lecture Slides
Professor John Price, Spring 2019

2. Background and Motivations for Quantum Mechanics
TZD:
3.1, 3.2, 3.3 Atoms – charge, size, mass
5.4 Atoms – stability
4.2 Black body radiation and Planck (1900)
4.7, de Broglie waves (1924)
5.1, 5.2, 5.3 Atomic spectra and the Balmer- Rydberg formula
5.5, 5.6, 5.7 Bohr’s great idea
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4. No, E&M does not apply because the motion is too fast - +
In hydrogen, an electron orbits a proton. As the electron orbits, does it emit radiation like a radio transmitter?
No, E&M does not apply because the motion is too fast
No, E&M does not apply because the system is too small
No, because **quantum magic**
Yes, it does
Clicker Slide 18 Physics 2130, Spring 2019

5. Why don’t planets emit radiation and spiral into the sun?
They do, but very, very slowly.
Because planets obey quantum mechanics, not classical mechanics.
Because gravitational forces work differently than electrical forces.
Because there are no photons with low enough frequency.
Because they are not charged.
Clicker Slide 19 Physics 2130, Spring 2019

6. The Hulse-Taylor binary pulsar PSR B1913+16
A binary neutron star system with an orbital period of 7.75 hours
17 Hz rotation frequency!
Observed time of periastron
GR Prediction
Orbital period: μs/year
Lifetime: 300 My

7. every point vibrates sinusoidally at the same frequency
Normal Modes
every point vibrates sinusoidally at the same frequency
every point goes through equilibrium at the same instant
there nodes (places that don’t move) and anti-nodes (maximum motion)
string normal modes
drum head normal modes (circular membrane)
tuning fork normal modes
Chladni figures (1787)

8. Electromagnetic normal modes of a cylindrical cavity

9. Continuum
E=0
A B C D E F
Which sketch is the spectrum possible energies of a quantum harmonic oscillator? 𝐴 𝐵 𝐶 𝐷 𝐸
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 19 Physics 2130 Spring 2019

10. Electron diffraction: Davisson-Germer experiment
wave character of electrons (and all matter) (1927)
diffraction of electrons off nickel crystal
estimate of e wavelength: 1 Angstrom for 150 eV
if detect which slit, diffraction pattern disappears
double-slit experiment with: e, p, n, molecules,…

11. Quantum interference of large organic molecules, Gerlich et al 2011
G1 and G2: 266 nm SiN transmission gratings
G2: 532 nm standing wave grating
S1, S2, S3: velocity selection slits, parabolic trajectory

12. Empirical Balmer-Rydberg formula
Lyman series: n’=1, n = 2,3,4… UV (1914)
Balmer series: n’=2, n = 3,4,5… visible (1885)
Paschen series: n’=3, n = 4,5,6… IR (1908)
410.3
486.1
Balmer’s formula
656.3 nm
434.0
R = nm-1
Rydberg constant
1 𝜆 =𝑅 1 𝑛′ 2 − 1 𝑛 2
where n’=1,2,3
and where n = n’+1, n’+2,…

13. Balmer-Rydberg formula as transitions between energy levels
Hydrogen energy levels
410.3
486.1
Balmer’s formula
656.3 nm
434.0
𝐸 𝛾 = ℎ𝑐 𝜆 =− 𝐸 𝑅 1 𝑛 2 − 1 𝑛′ 2
where n’=1,2,3
and where n = n’+1, n’+2

15. Which sketch is the spectrum possible energies of a hydrogen atom? 𝐴 𝐵
Continuum
E=0
A B C D E F
Which sketch is the spectrum possible energies of a hydrogen atom? 𝐴 𝐵 𝐶 𝐷 𝐸
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 20 Physics 2130 Spring 2019

16. Standing Waves on a Ring
Just like a standing wave
on a string, but now the
two ends of the string
are joined.
What are the restrictions on the wavelength?
r = 
r = n
r = n
2r = n
2r = /n r
2,3,10
n = 1, 2, 3, …
Clicker Slide 21 Physics Spring 2019

17. Which sketch could be the spectrum of a qubit? 𝐴 𝐵 𝐶 𝐷 𝐸
Continuum
E=0
A B C D E F
Which sketch could be the spectrum of a qubit? 𝐴 𝐵 𝐶 𝐷 𝐸
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 22 Physics 2130 Spring 2019

18. Which sketch is the spectrum of a free particle? 𝐴 𝐵 𝐶 𝐷 𝐸
Continuum
E=0
A B C D E F
Which sketch is the spectrum of a free particle? 𝐴 𝐵 𝐶 𝐷 𝐸
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 23 Physics 2130 Spring 2019

19. The Quantum Wavefunction
TZD:
6.4, Young’s double slit and interference
6.4, Probability interpretation and normalization
Why is the wave function complex?
Later chapters More general wave functions
6.6 Sinusoidal waves and phase velocity
6.7,6.8,6.9,6.10 Wave packets and group velocity
Complex numbers
Complex waves
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20. Young’s double-slit experiment
T. Young

21. Both slits open
Youtube animation

22. One slit open

23. One slit open

24. Quantum particles

25. Complex running wave: i x 𝑓 𝑥,𝑡 =𝑒 𝑖(𝑘𝑥−𝜔𝑡) 1 R-click to freeze
2,3,10 1 x
R-click to freeze

26. 𝑧= 4−𝑖 𝑒 (1+5𝑖) What is z*z? 17 17e i5 17e2 -i𝜋e
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 24 Physics 2130, Spring 2019

27. 𝑃 𝑟 , 𝑡 = 𝜓( 𝑟 ,𝑟) 2 = 𝜓 ∗ 𝜓 𝑃( 𝑟 ,𝑡) 𝑑 3 𝑟 =1
𝑃 𝑟 , 𝑡 = 𝜓( 𝑟 ,𝑟) 2 = 𝜓 ∗ 𝜓
𝑃( 𝑟 ,𝑡) 𝑑 3 𝑟 =1
What are the units of ψ ? m2
m-1/2
m2/3
m-3/2 m3
all space
Clicker Slide 25 Physics 2130, Spring 2019

28. 𝑃 𝑥, 𝑡 = 𝜓(𝑥,𝑟) 2 = 𝜓 ∗ 𝜓 𝑃 𝑥,𝑡 𝑑𝑥 =1 What are the units of ψ in 1-D?
𝑃 𝑥, 𝑡 = 𝜓(𝑥,𝑟) 2 = 𝜓 ∗ 𝜓
𝑃 𝑥,𝑡 𝑑𝑥 =1
What are the units of ψ in 1-D? m2
m-1/2
m2/3
m-3/2 m3
Clicker Slide 26 Physics 2130, Spring 2019

29. Beats T f1= 224 Hz f2= 225 Hz What is Tbeat? A) 1/224 s
B) ( )/2 s
C) 1 s
D) 0.5 s
E) 2.0 s
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c T
Clicker Slide 27 Physics 2130, Spring 2019
beat

30. Beats T Tbeat = 1 s fbeat = 1 Hz fbeat = |f1-f2| f1= 224 Hz f2= 225 Hz
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c T
Clicker Slide 27 Physics 2130, Spring 2019
beat

31. Wave Packets ∆𝑇 Superpose waves with ∆𝑓≈1/∆𝑇 or ∆𝑇∆𝑓≈1
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
More careful definition of  gives an exact result:
∆𝑇∆𝜔≥1/2
Wave packet in x instead of t:
∆𝑥∆𝑘≥1/2

32. ∆𝑇
Suppose I play a brief blast on a trumpet of the note A5 = 880 Hz. If the blast only lasts T = 0.1s, what frequencies f do I emit?
A) 880 Hz exactly
B) roughly Hz
C) roughly 879 – 881 Hz
D) roughly 870 – 890 Hz
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 28 Physics 2130 Spring 2019

33. What is the correct expression for i2 ? +1 -1 +i -i +65 i
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 29 Physics 2130, Fall 2019

34. What is the correct expression for i4 ? +1 -1 +i -i +65 i
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 30 Physics 2130, Fall 2019

35. What is the correct expression for i5 ? +1 -1 +i -i +65 i
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 31 Physics 2130, Fall 2019

36. What is the correct expression for: (6 + 7 i) · (2 – 3 i) ? +8 + 4 i
none of the above
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 32 Physics 2130, Fall 2019

37. What is this complex number -1 + i expressed in polar form (r,θ) ?
(1,π)
( 2 ,π/4)
( 2 ,3π/4)
(1,3π)
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 33 Physics 2130, Fall 2019

38. Which is the the complex conjugate of 3 – 2 i ? AB C D E
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 34 Physics 2130, Fall 2019

39. Suppose z = 3 – 2 i
Find 𝑧 2 5 7 9 11 13
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 35 Physics 2130, Fall 2019

40. 𝑒 𝑖3𝜋 = ? 1 -1 i -i -1-i Clicker Slide 36 Physics 2130, Fall 2019
𝑒 𝑖3𝜋 = ? 1 -1 i -i
-1-i
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 36 Physics 2130, Fall 2019

41. 𝑒 (1+𝑖𝜋) = ? -1 e ie -e -i𝜋 Clicker Slide 37 Physics 2130, Fall 2019
𝑒 (1+𝑖𝜋) = ? -1 e ie -e
-i𝜋
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 37 Physics 2130, Fall 2019

42. 𝑧= 4−𝑖 𝑒 (1+5𝑖) What is z*z? 17 17e i5 17e2 -i𝜋e
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 38 Physics 2130, Fall 2019

43. Running wave moving towards +x Running wave moving towards -x
Ψ 𝑥,𝑡 =𝐴𝑒 𝑖(𝑘𝑥−𝜔𝑡)
Ψ is a…
standing wave
Running wave moving towards +x
Running wave moving towards -x
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 39 Physics 2130, Fall 2019

44. Complex running wave: i x 𝑓 𝑥,𝑡 =𝑒 𝑖(𝑘𝑥−𝜔𝑡) 1 R-click to freeze
2,3,10 1 x
R-click to freeze

45. At fixed x, as t increases, Ψ …
Ψ 𝑥,𝑡 =𝐴𝑒 𝑖(𝑘𝑥−𝜔𝑡)
At fixed x, as t increases, Ψ …
moves towards the origin (in the complex plane)
moves away from the origin
orbits the origin in the CW direction
orbits the origin in the CCW direction
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 40 Physics 2130, Fall 2019

46. At fixed t, as x increases, Ψ …
Ψ 𝑥,𝑡 =𝐴𝑒 𝑖(𝑘𝑥−𝜔𝑡)
At fixed t, as x increases, Ψ …
moves towards the origin (in the complex plane)
moves away from the origin
orbits the origin in the clock-wise direction
orbits the origin in the counter-clock-wise direction
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 41 Physics 2130, Fall 2019

47. What is the correct formula for this wave animation?
(View in slide-show mode to see it move.)
Ψ 𝑥,𝑡 =𝐴𝑒 𝑖(+𝑘𝑥+𝜔𝑡)
Ψ 𝑥,𝑡 =𝐴𝑒 𝑖(−𝑘𝑥+𝜔𝑡)
Ψ 𝑥,𝑡 =𝐴𝑒 𝑖(+𝑘𝑥−𝜔𝑡)
Ψ 𝑥,𝑡 =𝐴𝑒 𝑖(−𝑘𝑥−𝜔𝑡)
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 42 Physics 2130, Fall 2019

48. Suppose Ψ 𝑥,𝑡 =𝐴𝑒 𝑖(𝑘𝑥−𝜔𝑡) . What is 𝜕Ψ 𝜕𝑥 ? −𝑘 𝑒 −𝑖𝜔𝑡 𝑖𝑘 𝑒 −𝑖𝑘𝑥 𝑖𝑘Ψ Ψ
(𝑖𝑘𝑥−𝜔𝑡)Ψ
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 43 Physics 2130, Fall 2019

49. Suppose Ψ 𝑥,𝑡 =𝐴𝑒 𝑖(𝑘𝑥−𝜔𝑡) . What is 𝜕Ψ 𝜕𝑡 ? −𝑘 𝑒 −𝑖𝜔𝑡 𝑖𝜔 𝑒 −𝑖𝜔𝑡 𝑖𝑘Ψ Ψ
−𝑖𝜔Ψ
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 44 Physics 2130, Fall 2019

50. Suppose Ψ 𝑥,𝑡 =𝐴𝑒 𝑖(𝑘𝑥−𝜔𝑡) . What is 𝜕 2 Ψ 𝜕 𝑥 2 ?
−2𝑘 𝑒 −𝑖𝜔𝑡
𝑘 2 𝑒 −𝑖𝑘𝑥
𝑖𝑘Ψ
− 𝑘 2 Ψ
(𝑖𝑘𝑥−𝜔𝑡) 2 Ψ
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 45 Physics 2130, Fall 2019