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
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, 6.2 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 Clicker a. start / stop b, hide/ unhide c. ppt-1page fwd d. ppt 1 pg back e. select correct answer
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
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
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: -76.5 μs/year Lifetime: 300 My
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)
Electromagnetic normal modes of a cylindrical cavity
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
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,…
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
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 = 0.0110 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,…
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
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
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 2r = n 2r = /n r http://jilawww.colorado.edu/%7Emckagan/movies/atom1.avi 2,3,10 n = 1, 2, 3, … Clicker Slide 21 Physics 2130 Spring 2019
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
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
The Quantum Wavefunction TZD: 6.4, 6.5 Young’s double slit and interference 6.4, 6.5 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 Clicker a. start / stop b, hide/ unhide c. ppt-1page fwd d. ppt 1 pg back e. select correct answer
Young’s double-slit experiment T. Young 1773-1829
Both slits open Youtube animation
One slit open
One slit open
Quantum particles
Complex running wave: i x 𝑓 𝑥,𝑡 =𝑒 𝑖(𝑘𝑥−𝜔𝑡) 1 R-click to freeze http://jilawww.colorado.edu/%7Emckagan/movies/atom1.avi 2,3,10 1 x R-click to freeze
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
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
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
Beats T f1= 224 Hz f2= 225 Hz What is Tbeat? A) 1/224 s B) (224+225)/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
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
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
∆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 879.9 - 880.1 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
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
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
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
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
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
Which is the the complex conjugate of 3 – 2 i ? 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 34 Physics 2130, Fall 2019
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
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
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
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
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
Complex running wave: i x 𝑓 𝑥,𝑡 =𝑒 𝑖(𝑘𝑥−𝜔𝑡) 1 R-click to freeze http://jilawww.colorado.edu/%7Emckagan/movies/atom1.avi 2,3,10 1 x R-click to freeze
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
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
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
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
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
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