1. Clicker Questions Lecture Slides Professor John Price, Spring 2019
Physics 2130 Foundations of Modern Physics Quantum Mechanics Part III: The Schrödinger Equation in Two and Three Dimensions
Clicker Questions
Lecture Slides
Professor John Price, Spring 2019

2. The Schrödinger Equation in 2D and 3D
TZD Chapter 8
Schrödinger equation in 2D and 3D
Particle in a 2D box
2D central force problem
3D central force problem
Hydrogen atom
Electron Spin (TZD 9.1, 9.2)
Many Particles (TZD 10.1 – 10.5)
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3. Plane waves in 2 dimensionsy
Plane waves in 2 dimensions
𝜓 𝑥,𝑦 =𝐴 𝑒 𝑖( 𝑘 𝑥 𝑥+ 𝑘 𝑦 𝑦) x
Suppose 𝑘 𝑥 =0, 𝑘 𝑦 =𝑞 , where q is positive and real.
Which way does the wave propagate?
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 75 Physics 2130, Spring 2019

4. Plane waves in 2 dimensionsy
Plane waves in 2 dimensions
𝜓 𝑥,𝑦 =𝐴 𝑒 𝑖( 𝑘 𝑥 𝑥+ 𝑘 𝑦 𝑦) x
Suppose 𝑘 𝑥 =0, 𝑘 𝑦 =−𝑞 , where q is positive and real.
Which way does the wave propagate?
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 76 Physics 2130, Spring 2019

5. Plane waves in 2 dimensionsy
Plane waves in 2 dimensions
𝜓 𝑥,𝑦 =𝐴 𝑒 𝑖( 𝑘 𝑥 𝑥+ 𝑘 𝑦 𝑦) x
Suppose 𝑘 𝑥 =−𝑞, 𝑘 𝑦 =𝑞 , where q is positive and real.
Which way does the wave propagate?
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 77 Physics 2130, Spring 2019

6. Plane waves in 2 dimensionsy
Plane waves in 2 dimensions
𝜓 𝑥,𝑦 =𝐴 𝑒 𝑖( 𝑘 𝑥 𝑥+ 𝑘 𝑦 𝑦) x
Suppose 𝑘 𝑥 =−𝑞, 𝑘 𝑦 =𝑞 , where q is positive and real.
What is the direction of the vector 𝑘 ?
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 78 Physics 2130, Spring 2019

7. Suppose 𝜓 𝑥,𝑦 =𝐴 𝑒 𝑖( 𝑘 𝑥 𝑥+ 𝑘 𝑦 𝑦) =𝐴𝑒 𝑖 𝑘 ∙ 𝑟 .
What is 𝜕𝜓 𝜕𝑥 ?
𝑘 𝑥 𝑒 𝑖 𝑘 𝑥 𝑥
𝑖 𝑘 ∙ 𝑟 𝑒 𝑖 𝑘 ∙ 𝑟
𝑖 𝑘 𝜓
𝑖 𝑘 𝑥 𝜓
𝑖𝑥 𝑒 𝑖 𝑘 𝑥 𝑥
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 79 Physics 2130, Spring 2019

8. Suppose 𝜓 𝑥,𝑦 =𝐴 𝑒 𝑖( 𝑘 𝑥 𝑥+ 𝑘 𝑦 𝑦) =𝐴𝑒 𝑖 𝑘 ∙ 𝑟 .
What is 𝜕 2 𝜓 𝜕 𝑥 2 ?
𝑒 𝑖 𝑘 𝑥 𝑥
𝑖( 𝑘 ∙ 𝑟 )( 𝑘 ∙ 𝑟 ) 𝑒 𝑖 𝑘 ∙ 𝑟
𝑘 2𝜓
−𝑖 𝑘 𝑥 2 𝜓
− 𝑘 𝑥 2 𝜓
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 80 Physics 2130, Spring 2019

9. Suppose 𝜓 𝑥,𝑦 =𝐴 𝑒 𝑖( 𝑘 𝑥 𝑥+ 𝑘 𝑦 𝑦) =𝐴𝑒 𝑖 𝑘 ∙ 𝑟 .
What is 𝜕 2 𝜓 𝜕 𝑦 2 ?
𝑒 𝑖 𝑘 𝑦 𝑦
𝑖( 𝑘 ∙ 𝑟 )( 𝑘 ∙ 𝑟 ) 𝑒 𝑖 𝑘 ∙ 𝑟
𝑘 2𝜓
−𝑖 𝑘 𝑦 2 𝜓
− 𝑘 𝑦 2 𝜓
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 81 Physics 2130, Spring 2019

10. Substitute it in and get… 𝐸𝑋𝑌=− ℏ 2 2𝑀 ( 𝑋 ′′ +𝑌′′)
Suppose 𝜓 𝑥,𝑦 =𝑋 𝑥 𝑌(𝑦)
and we know that
𝐸𝜓=− ℏ 2 2𝑀 𝜕 2 𝜕 𝑥 𝜕 2 𝜕 𝑦 2 𝜓
Substitute it in and get…
𝐸𝑋𝑌=− ℏ 2 2𝑀 ( 𝑋 ′′ +𝑌′′)
𝐸𝑋𝑌=− ℏ 2 2𝑀 ( 𝑋 ′′ 𝑌+ 𝑋 ′ 𝑌+𝑋𝑌′+𝑋𝑌′′)
𝐸𝑋𝑌=− ℏ 2 2𝑀 ( 𝑋 ′ 𝑌 ′ +𝑋 𝑌 ′′ + 𝑋 ′′ 𝑌)
𝐸𝑋𝑌=− ℏ 2 2𝑀 ( 𝑋 ′′ 𝑌+𝑋𝑌′′)
𝐸𝑋𝑌=− ℏ 2 2𝑀 ( 𝑋 ′ + 𝑋 ′′ + 𝑌 ′ +𝑌′′)
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 82 Physics 2130, Fall 2019

11. 𝑓 𝑥 +𝑔 𝑦 =𝑎 where a is a constant ≠0.
Suppose
𝑓 𝑥 +𝑔 𝑦 =𝑎 where a is a constant ≠0.
Which cases below are possible solutions?
(b and c are other constants.)
𝑓 𝑥 =𝑏, 𝑔 𝑦 =𝑐, 𝑎=𝑏+𝑐
𝑓 𝑥 =0, 𝑔 𝑦 =0
𝑓 𝑥 =0, 𝑔 𝑦 =𝑎
𝑓 𝑥 =𝑎, 𝑔 𝑦 =0
𝑓 𝑥 =𝑏, 𝑔 𝑦 =𝑐, 𝑎=𝑏+𝑐+𝑑 (𝑑 another constant ≠0)
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 83 Physics 2130, Spring 2019

12. and we know it is zero on the boundary. Then it follows that:
Suppose 𝜓 𝑥,𝑦 =𝑋 𝑥 𝑌(𝑦)
and we know it is zero on the boundary. Then it follows that:
𝑋 0 =𝑋 𝑎 =0 𝐚𝐧𝐝 𝑌 0 =𝑌 𝑎 =0
𝑋 𝑎 =0 and 𝑌 0 =0
𝑋 0 =𝑋 𝑎 =0 𝐨𝐫 𝑌 0 =𝑌 𝑎 =0
𝑋 0 =0 and 𝑌 𝑎 =0
𝑋 𝑎 =0 and 𝑌 𝑎 =0 a
a x
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 84 Physics 2130, Spring 2019

13. A) A stationary-state probability density of the 2D square well
What is this?
A) A stationary-state probability density of the 2D square well
B) The real part of a stationary-state wavefunction of the 2D square well
C) The real part of a time-dependent wavefunction of the 2D square well at t=0.
D) B or C
E) 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 85 Physics 2130, Spring 2019

14. What is this? A) nx=3, ny=3 B) nx=4, ny=4 C) nx=3, ny=4 D) nx=4, ny=3
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 86 Physics 2130, Spring 2019

15. ℏ 2 𝜋 2 2𝑀 𝑎 2 The energy of this state is…. multiplied by A) 9 B) 16
ℏ 2 𝜋 2 2𝑀 𝑎 2
multiplied by
A) 9
B) 16
C) 18
D) 24
E) 32
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 87 Physics 2130, Spring 2019

16. The degeneracy of this level is….
B) 2
C) 3
D) 4
E) 5
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 88 Physics 2130, Spring 2019

17. The degeneracy of this level is….ny
The degeneracy of this level is….
A) 1
B) 2
C) 3
D) 4
E) 5 nx
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 89 Physics 2130, Spring 2019

18. The degeneracy of the level with nx=1, ny=3 is…
B) 2
C) 3
D) 4
E) 5
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 90 Physics 2130, Spring 2019

19. In Cartesian coordinates, 𝛻 2 = 𝜕 2 𝜕 𝑥 2 + 𝜕 2 𝜕 𝑦 2
Find 𝛻 2 𝑓(𝑟) for 𝑓 𝑟 = 𝑟 2 = 𝑥 2 + 𝑦 2 . 𝑟 1 2 3 4
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 91 Physics 2130, Spring 2019

20. Suppose 𝛻 2 = 𝜕 2 𝜕 𝑟 2 + 1 𝑟 2 𝜕 2 𝜕 𝜑 2 in polar coordinates
Find 𝛻 2 𝑓(𝑟) for 𝑓 𝑟 = 𝑟 2 . 𝑟 1 2 3 4
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 92 Physics 2130, Spring 2019

21. Suppose 𝛻 2 = 𝜕 2 𝜕 𝑟 2 + 1 𝑟 2 𝜕 2 𝜕 𝜑 2 in polar coordinates Not Correct!
Find 𝛻 2 𝑓(𝑟) for 𝑓 𝑟 = 𝑟 2 . 𝑟 1 2 3 4
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 92 Physics 2130, Spring 2019

22. Suppose 𝛻 2 = 𝜕 2 𝜕 𝑟 2 + 1 𝑟 𝜕 𝜕𝑟 + 1 𝑟 2 𝜕 2 𝜕 𝜑 2 in polar coordinates
Find 𝛻 2 𝑓(𝑟) for 𝑓 𝑟 = 𝑟 2 . 𝑟 1 2 3 4
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 93 Physics 2130, Spring 2019

23. Suppose Φ 𝜑 = 𝑒 𝑖𝑚𝜑 and 𝜑 is the angle in polar coordinates.
What are the allowed values of m?
0, ±𝜋,±2𝜋,±3𝜋,…
0, ±1,±2,±3,…
0, 1, 2, 3,…
1, 2, 3,…
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 94 Physics 2130, Spring 2019

24. For the central force problem in 2D,
Φ 𝜑 = 𝑒 𝑖𝑚𝜑 where 𝜑 is the angle in polar coordinates and
m =0, ±1,±2,±3,…
Are the two states with m = +1 and m = -1 degenerate?
Yes No
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 95 Physics 2130, Spring 2019

25. 𝛻 2 𝑓= 𝜕 2 𝜕 𝑥 2 + 𝜕 2 𝜕 𝑦 2 + 𝜕 2 𝜕 𝑧 2 in Cartesian coordinates.
Find 𝛻 2 𝑓 for 𝑓(𝑟)= 𝑟 2 . 2 4 6 8 10
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 96 Physics 2130, Spring 2019

26. Eψ 𝑟,𝜃,𝜑 =
− ℏ 2 2𝑀 1 𝑟 𝜕 2 𝜕 𝑟 2 𝑟ψ + 1 𝑟 2 sin 𝜃 𝜕 𝜕𝜃 sin 𝜃 𝜕ψ 𝜕𝜃 𝑟 2 sin 2 𝜃 𝜕 2 ψ 𝜕 𝜑 2 +𝑈(𝑟)ψ
𝛻 2 𝑓= 1 𝑟 𝜕 2 𝜕 𝑟 2 (𝑟𝑓)+ 1 𝑟 2 sin 𝜃 𝜕 𝜕𝜃 ( sin 𝜃 𝜕𝑓 𝜕𝜃 )+ 1 𝑟 2 sin 2 ⁡𝜃 𝜕 2 𝑓 𝜕 𝜑 2
Find 𝛻 2 𝑓 for 𝑓(𝑟)= 𝑟 2 . 2 4 6 8 10
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 97 Physics 2130, Spring 2019

27. 1 sin 𝜃 𝑑 𝑑𝜃 sin 𝜃 𝑑Θ 𝑑𝜃 + 𝑙 𝑙+1 − 𝑚 2 sin 2 𝜃 Θ=0
Eψ 𝑟,𝜃,𝜑 =
− ℏ 2 2𝑀 1 𝑟 𝜕 2 𝜕 𝑟 2 𝑟ψ + 1 𝑟 2 sin 𝜃 𝜕 𝜕𝜃 sin 𝜃 𝜕ψ 𝜕𝜃 𝑟 2 sin 2 𝜃 𝜕 2 ψ 𝜕 𝜑 2 +𝑈(𝑟)ψ
ψ 𝑟,𝜃,𝜑 =𝑅(𝑟)Θ(𝜃)Φ(𝜑)
Φ ′′ 𝜑 =− 𝑚 2 Φ(𝜑)
1 sin 𝜃 𝑑 𝑑𝜃 sin 𝜃 𝑑Θ 𝑑𝜃 + 𝑙 𝑙+1 − 𝑚 2 sin 2 𝜃 Θ=0
𝐸𝑅 𝑟 =− ℏ 2 2𝑀 1 𝑟 𝜕 2 𝜕 𝑟 2 𝑟𝑅 + 𝑙 𝑙+1 ℏ 2 2𝑀 𝑟 2 +𝑈 𝑟 𝑅
What quantum number does Φ depend on? m 𝑙
m ,𝑙
𝑙, 𝑛 𝑛
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 98 Physics 2130, Spring 2019

28. 1 sin 𝜃 𝑑 𝑑𝜃 sin 𝜃 𝑑Θ 𝑑𝜃 + 𝑙 𝑙+1 − 𝑚 2 sin 2 𝜃 Θ=0
Eψ 𝑟,𝜃,𝜑 =
− ℏ 2 2𝑀 1 𝑟 𝜕 2 𝜕 𝑟 2 𝑟ψ + 1 𝑟 2 sin 𝜃 𝜕 𝜕𝜃 sin 𝜃 𝜕ψ 𝜕𝜃 𝑟 2 sin 2 𝜃 𝜕 2 ψ 𝜕 𝜑 2 +𝑈(𝑟)ψ
ψ 𝑟,𝜃,𝜑 =𝑅(𝑟)Θ(𝜃)Φ(𝜑)
Φ ′′ 𝜑 =− 𝑚 2 Φ(𝜑)
1 sin 𝜃 𝑑 𝑑𝜃 sin 𝜃 𝑑Θ 𝑑𝜃 + 𝑙 𝑙+1 − 𝑚 2 sin 2 𝜃 Θ=0
𝐸𝑅 𝑟 =− ℏ 2 2𝑀 1 𝑟 𝜕 2 𝜕 𝑟 2 𝑟𝑅 + 𝑙 𝑙+1 ℏ 2 2𝑀 𝑟 2 +𝑈 𝑟 𝑅
What quantum number does Θ depend on? m 𝑙
m ,𝑙
𝑙, 𝑛 𝑛
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 99 Physics 2130, Spring 2019

29. 1 sin 𝜃 𝑑 𝑑𝜃 sin 𝜃 𝑑Θ 𝑑𝜃 + 𝑙 𝑙+1 − 𝑚 2 sin 2 𝜃 Θ=0
Eψ 𝑟,𝜃,𝜑 =
− ℏ 2 2𝑀 1 𝑟 𝜕 2 𝜕 𝑟 2 𝑟ψ + 1 𝑟 2 sin 𝜃 𝜕 𝜕𝜃 sin 𝜃 𝜕ψ 𝜕𝜃 𝑟 2 sin 2 𝜃 𝜕 2 ψ 𝜕 𝜑 2 +𝑈(𝑟)ψ
ψ 𝑟,𝜃,𝜑 =𝑅(𝑟)Θ(𝜃)Φ(𝜑)
Φ ′′ 𝜑 =− 𝑚 2 Φ(𝜑)
1 sin 𝜃 𝑑 𝑑𝜃 sin 𝜃 𝑑Θ 𝑑𝜃 + 𝑙 𝑙+1 − 𝑚 2 sin 2 𝜃 Θ=0
𝐸𝑅 𝑟 =− ℏ 2 2𝑀 1 𝑟 𝜕 2 𝜕 𝑟 2 𝑟𝑅 + 𝑙 𝑙+1 ℏ 2 2𝑀 𝑟 2 +𝑈 𝑟 𝑅
What quantum number does 𝑅 depend on? m 𝑙
m ,𝑙
𝑙, 𝑛 𝑛
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 100 Physics 2130, Spring 2019

30. z Suppose we plot r= Θ 𝑙,𝑚 (𝜃) Φ 𝑚 (𝜑) Which one is this? l m 2 0 2 ±1
2 0
2 ±1
2 ±2
1 0
1 ±1 z
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 101 Physics 2130, Spring 2019

31. Suppose we plot r= Θ 𝑙,𝑚 (𝜃) Φ 𝑚 (𝜑) Which one is this? l m 2 0
2 0
2 ±1
2 ±2
1 0
1 ±1
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 102 Physics 2130, Spring 2019

32. Suppose we plot r= Θ 𝑙,𝑚 (𝜃) Φ 𝑚 (𝜑) Which one is this? l m 2 0
2 0
2 ±1
2 ±2
1 0
1 ±1
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide103 Physics 2130, Spring 2019

33. Suppose we plot r= Θ 𝑙,𝑚 (𝜃) Φ 𝑚 (𝜑) Which one is this? l m 2 0
2 0
2 ±1
2 ±2
1 0
1 ±1
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 104 Physics 2130, Spring 2019

34. Suppose we plot r= Θ 𝑙,𝑚 (𝜃) Φ 𝑚 (𝜑) Which one is this? l m 2 0
2 0
2 ±1
2 ±2
1 0
1 ±1
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 105 Physics 2130, Spring 2019

35. These are plots of R(r). Which one is n=1, l=0? blue dark green red
blue-green
purple
yellow-green
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 106 Physics 2130, Spring 2019

36. These are plots of R(r). Which one is n=1, l=0? blue dark green red
blue-green
purple
yellow-green
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 106 Physics 2130, Spring 2019

37. These are plots of R(r). Which one is n=2, l=1? blue dark green red
blue-green
purple
yellow-green
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 107 Physics 2130, Spring 2019

38. These are plots of R(r). Which one is n=2, l=1? blue dark green red
blue-green
purple
yellow-green
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 107 Physics 2130, Spring 2019

39. These are plots of R(r). Which one is n=3, l=2? blue dark green red
blue-green
purple
yellow-green
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 108 Physics 2130, Spring 2019

40. These are plots of R(r). Which one is n=3, l=2? blue dark green red
blue-green
purple
yellow-green
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 108 Physics 2130, Spring 2019

41. These are plots of R(r). Which one is n=2, l=0? blue dark green red
blue-green
purple
yellow-green
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 109 Physics 2130, Spring 2019

42. These are plots of R(r). Which one is n=2, l=0? blue dark green red
blue-green
purple
yellow-green
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 109 Physics 2130, Spring 2019

43. These are plots of R(r). blue dark green red blue-green purple
yellow-green
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Physics 2130, Spring 2019

44. . Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Physics 2130, Spring 2019

45. In the ground state of hydrogen, where is the electron most likely to be found?
At the origin
At 𝑟= 𝑎 𝐵
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 110 Physics 2130, Spring 2019

46. In the ground state of hydrogen, at what radius is the electron most likely to be found?
At the origin
At 𝑟= 𝑎 𝐵
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 111 Physics 2130, Spring 2019

47. Radial “shells” . Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c

48. 2p states (n=2, l=1) m=0 m=±1 states
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
m=0
m=±1 states

49. 𝐿 = 𝑙(𝑙+1) ℏ
Suppose allowed values of Lz are −2ℏ, −ℏ,0,ℏ,2ℏ. What is the value of 𝐿 ?
2 ℏ 2ℏ
4 ℏ
5 ℏ
6 ℏ
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 112 Physics 2130, Spring 2019

50. Suppose 𝑆 = 3/4 ℏ What is the value of s? 1/2 1 3/2 5/2
1/2 1
3/2
5/2
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 113 Physics 2130, Spring 2019

51. Can we solve this in the form 𝜓= 𝜓 1 𝑟 1 𝜓 2 𝑟 2 ?
𝜓=𝜓( 𝑟 1 , 𝑟 2 )
E𝜓=− ℏ 2 2𝑚 𝛻 𝛻 2 2 𝜓
Can we solve this in the form
𝜓= 𝜓 1 𝑟 1 𝜓 2 𝑟 2 ?
YES NO
What do you mean, we?
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 114 Physics 2130, Spring 2019

52. E𝜓=− ℏ 2 2𝑚 𝛻 1 2 + 𝛻 2 2 𝜓+ −𝑘 𝑒 2 𝑟 1 + −𝑘 𝑒 2 𝑟 2 𝜓
𝜓=𝜓( 𝑟 1 , 𝑟 2 )
E𝜓=− ℏ 2 2𝑚 𝛻 𝛻 2 2 𝜓+ −𝑘 𝑒 2 𝑟 1 + −𝑘 𝑒 2 𝑟 2 𝜓
Can we solve this in the form
𝜓= 𝜓 1 𝑟 1 𝜓 2 𝑟 2 ?
YES NO
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 115 Physics 2130, Spring 2019

53. Can we solve this in the form 𝜓= 𝜓 1 𝑟 1 𝜓 2 𝑟 2 ?
𝜓=𝜓( 𝑟 1 , 𝑟 2 )
E𝜓=− ℏ 2 2𝑚 𝛻 𝛻 2 2 𝜓+ −𝑘 𝑒 2 𝑟 1 + −𝑘 𝑒 2 𝑟 2 + 𝑘 𝑒 𝑟 1 − 𝑟 𝜓
Can we solve this in the form
𝜓= 𝜓 1 𝑟 1 𝜓 2 𝑟 2 ?
YES NO
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 116 Physics 2130, Spring 2019

54. Is this the TISE for helium?
𝜓=𝜓( 𝑟 1 , 𝑟 2 )
E𝜓=− ℏ 2 2𝑚 𝛻 𝛻 2 2 𝜓+ −𝑘 𝑒 2 𝑟 1 + −𝑘 𝑒 2 𝑟 2 + 𝑘 𝑒 𝑟 1 − 𝑟 𝜓
Is this the TISE for helium?
YES NO
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 117 Physics 2130, Spring 2019

55. Stable atomic nuclei with spin>0
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c

56. If the three wavefunctions a, b, and c are all different, is this a correct product state for three non-interacting electrons?
A) Yes
B) No
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 118 Physics 2130, Spring 2019

57. If the three wavefunctions a, b, and c are all different, is this a correct product state for three non-interacting electrons?
A) Yes
B) No
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Clicker Slide 119 Physics 2130, Spring 2019

58. Antisymmetric product states can also be written as determinants:
. Discuss the form of the v transformation formula
. Why plus not minus
. Discuss that is always smaller than c
Physics 2130, Spring 2019