Lect21 - PHYS 172: Modern Mechanics Lecture 21 – Angular...

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Unformatted text preview: PHYS 172: Modern Mechanics Lecture 21 – Angular Momentum Quantization Fall 2011 1 Read 11.8 – 11.11 Announcements Exam 3 will be held on Wednesday, Nov. 16, 8-9 pm in Elliott Hall. Material through chapters 8, 9, 10, and 11 will be covered. There WILL be a lecture on Nov. 15 and 17. No lectures, labs, recitations, or homework the week of Thanksgiving. A practice exam 3 has been put in the Exams folder. 2 Predicting Position with Rotation A light string is wrapped around disk of radius R and moment of inertia I that can freely spin around its fixed axis. The string is pulled with force F during time Δt. Assume that the disk was initially at rest (ωi=0) 1) What will be the angular speed ωf ? I θ R m F=mg Solution: ! ! !Ltot = " net !t !! ! ! ! I! f " I! i = I! f = R # F $ %t ! dLtot ! = ! net dt I! f = RF "t RF "t !f = I 3 Predicting Position with Rotation I θ A light string is wrapped around disk of radius R and moment of inertia I that can freely spin around its fixed axis. The string is pulled with force F during time Δt. Assume that the disk was initially at rest (ωi=0) 1)  What will be the angular speed ωf ? 2)  How far (Δx) will the end of string move? Solution: R m F=mg ! dLtot ! RF "t = ! net ! f = dt I See also examples in Section 11.8 ! aver !" = # aver !t #$ " #t ω changes linearly with time: ! aver = !" = !i + ! f 2 #f 2 !t = !x = R!" = = !f 2 2 RF ( !t ) 2I 2 F ( R!t ) 2I 4 Angular momentum quantization Many elementary particles behave as if they posses intrinsic rotational angular momentum Electron can have translational (orbital around nucleus), and intrinsic rotational angular momenta [J s] = ⎡kg m2s-1 ⎤ ⎣ ⎦ Strange but true: Angular momentum is quantized Angular momentum quantum = h= h = 1.05 ×10−34 J s 2π Whenever you measure a vector component of angular momentum you get either half-integer or integer multiple of h Orbital angular momentum comes in integer multiples, but intrinsic spin of Fermions (building blocks) is ½ unit of h 5 Orbital Angular Momentum Where is the orbital angular momentum in a hydrogen orbital? + px i py Electron "current" circles around the atom. = |L=1, Lz=1> Quantized because these are 3D standing electron waves around the nucleus. See Atom in a Box 6 www.daugerresearch.com Bohr s Atomic Model ! LA,trans ,electron = mrkqe2 Niels Bohr 1913: IDEA: Electron can only take orbits where its translational angular momentum is integer multiple of h Allowed radii: !2 r = N2 2 kqe me ! = 1.05 "10!34 J # s N = 1, 2, 3,… This implies that only certain values of LA,trans,electron are allowed: ! LA,trans ,electron = N " where N=1,2,3,… NOTE: Because K and U are functions of r and v, energy levels are quantized also. 7 Bohr Model Consider an electron in circular orbit A about a proton. What are the possible values of LA,trans,electron? Assume circular motion: ￿ ￿! 2 2 kqe mv 2 pν mv ￿￿ ￿ = F ! = ￿Fe ￿ e= 2 r r r r Thus, kqe2 ! v= mr ! LA,trans ,electron = mvr = mrkqe2 If any orbital radius r is allowed, LA,trans,electron can be anything. However, only certain values of r are allowed . . . 8 The Bohr model: allowed radii and energies See derivation on page 444-446 This is 2K Use EN = K+U and E ￿ ￿ ￿ mν 2 2 kqe ￿￿ ￿ =2 ￿ Fe ￿ = r r k= Bohr model energy levels: ￿2 2 ⎛ 1 e2 ⎞ 44 1 m ⎜ 4π￿o ⎟ee m ⎝ 4πε 0 r ⎠ E= =N − − 2N h 2N 22￿22 13.6 eV =− , 2 N N=1,2,3,... 9 1 4π￿o The Bohr model: and photon emission 13.6 eV EN = − N2 10 Particle spin Rotational angular momentum Electron, muon, neutrino have spin 1/2 : mesurements of a component of their angular momentum yields ±½ħ Quarks have spin ½ Protons and neutrons (three quarks) have spin ½ Mesons: (quark+antiquark) have spin 0 or 1 Macroscopic objects: quantization of L is too small to notice! Two lowest energy electrons in any atom have total angular momentum 0 Fermions: spin ½, Pauli exclusion principle Bosons: integer spin Cooper pairs: superconductivity Rotational energies of molecules are quantized Quantum mechanics: Lx, Ly, Lz can only be integer or half-integer multiple of ħ Quantized values of L2 = l (l + 1) h2 where l is integer or half-integer 11 Gyroscopic Stability Edmund Scientifics In 1917, the Chandler Company of Indianapolis, Indiana, created the "Chandler gyroscope, a toy gyroscope with a pull string and pedestal. It has been in continuous production ever since and is considered a classic American toy. -- Wikipedia 12 Best Trick in the Book dp ￿ ￿ = Fnet dt p = | p| p ￿ ˆ Vectors have direction and magnitude. Vector Notation and the Momentum Principle: dp ￿ dp ˆ d | p| = | p| + p ˆ dt dt dt Use the chain rule ￿ ￿ = F⊥ + F|| ￿ F⊥ causes changes ￿ in the direction of p ￿ F|| causes changes in the magnitude of p ￿ 13 Best Trick Not in the Book ￿ dL = ￿net τ dt Vectors have direction and magnitude. ￿ = |τ |τ τ ˆ Vector Notation and the Angular Momentum Principle: ￿ ˆ dL dL ˆ d| L| = |L| +L dt dt dt Use the chain rule = ￿⊥ + ￿|| τ τ ￿⊥ causes changes τ ￿ in the direction of L ￿|| causes changes τ ￿ in the magnitude of L 14 Gyroscopes Precession Precession and nutation 15 ! FN ! ! Lrot Gyroscopes CLICKER: What is the direction of M ! R ! Lrot A ω ! Mg B ! dLrot ! = ! cm dt !! ! ! cm = R " F ! Lrot , A or B? CLICKER: What is the direction of A) Left B) Right !! ! ! cm = R " FN ! ! cm = RFN = RMg For rotating vector: ! dLrot = !Lrot = RMg dt CLICKER: What is the direction of A) Down C) into the page B) Up D) out of the page RMg RMg != = Lrot I" ! R? ! ! cm ? 16 i>clicker Ω= RMg Iω A B In which of the two gyroscopes is the disk spinning faster? 17 Precession phenomena (see book) Magnetic Resonance Imaging (MRI) Precession of spin axes in astronomy Tidal torques NMR - nuclear magnetic resonance Independently discovered (1946) Nobel Price (1952) Felix Bloch Edward Mills Purcell 1912-1997 1905-1983 B.S.E.E. from Purdue electrical engineering NMRI = MRI 18 ...
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This note was uploaded on 12/07/2011 for the course PHYS 172 taught by Professor ? during the Fall '08 term at Purdue University-West Lafayette.

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