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

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) Clicker a. start / stop b, hide/ unhide c. ppt-1page fwd d. ppt 1 pg back e. select correct answer

Plane waves in 2 dimensions y 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

Plane waves in 2 dimensions y 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

Plane waves in 2 dimensions y 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

Plane waves in 2 dimensions y 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

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

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

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

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 + 𝜕 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

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

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 0 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

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

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

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

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

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

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

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

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

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

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

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

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

𝛻 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

Eψ 𝑟,𝜃,𝜑 = − ℏ 2 2𝑀 1 𝑟 𝜕 2 𝜕 𝑟 2 𝑟ψ + 1 𝑟 2 sin 𝜃 𝜕 𝜕𝜃 sin 𝜃 𝜕ψ 𝜕𝜃 + 1 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

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 + 1 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

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 + 1 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

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 + 1 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 𝛻 1 2 + 𝛻 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

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 𝛻 1 2 + 𝛻 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

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 𝛻 1 2 + 𝛻 2 2 𝜓+ −𝑘 𝑒 2 𝑟 1 + −𝑘 𝑒 2 𝑟 2 + 𝑘 𝑒 2 𝑟 1 − 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 116 Physics 2130, Spring 2019

Is this the TISE for helium? 𝜓=𝜓( 𝑟 1 , 𝑟 2 ) E𝜓=− ℏ 2 2𝑚 𝛻 1 2 + 𝛻 2 2 𝜓+ −𝑘 𝑒 2 𝑟 1 + −𝑘 𝑒 2 𝑟 2 + 𝑘 𝑒 2 𝑟 1 − 𝑟 2 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

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

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

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

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