PHYS 172H: Modern Mechanics Fall 2010

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1 PHYS 172H: Modern Mechanics Fall 2010 Lecture 20 Angular Momentum Quantization (mostly) Chapter

2 CLICKER POLL Not for Credit How many senses does the human body have? (Taste, smell...) A) 3 senses B) 4 senses C) 5 senses D) 6 senses E) More than 6

3 I R Predicting Position with Rotation θ A light string is wrapped around disk of radius R and moment of intertia 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 (ω i =0) 1) What will be the angular speed ω f? Solution: m F=mg

4 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 (ω i =0) 1) What will be the angular speed ω f? 2) How far (Δx) will the end of string move? Solution: R m F=mg ω changes linearly with time: This is the rotational analog of accelerated linear displacement See also examples in Chapter 11.8 And this string motion is in the linear world, once we multiply the angular displacement by R

5 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 Strange but true: Angular momentum is quantized Angular momentum quantum = Whenever you measure a vector component of angular momentum you get either half-integer or integer multiple of Orbital angular momentum comes in integer multiples, but intrinsic spin of Fermions (building blocks) is ½ unit of

Orbital Angular Momentum Where is the orbital

6 Orbital Angular Momentum Where is the orbital angular momentum in a hydrogen orbital? + Electron "current" circles around the atom. p x Quantized because these are 3D standing electron waves around the nucleus. i p y = L=1, L z =1> See Atom in a Box

Bohr s Atomic Model L A, trans, electron = mrkq 2 e Niels Bohr 1913: IDEA: Electron can only take orbits where its translational angular momentum is integer multiple of Allowed radii: r = 2 34 2 1.

7 Bohr s Atomic Model L A, trans, electron = mrkq 2 e Niels Bohr 1913: IDEA: Electron can only take orbits where its translational angular momentum is integer multiple of Allowed radii: r = N kq 2 e m e N = J s = 1, 2,3, This implies that only certain values of L A,trans,electron are allowed: L,, = N where N=1,2,3, A trans electron NOTE: Because K and U are functions of r and v, energy levels are quantized also.

8 Bohr Model A Consider an electron in circular orbit about a proton. What are the possible values of L A,trans,electron? Assume circular motion: F e Thus, pv kq mv kq = = = r r r mr L = mvr = mrkq A, trans, electron e e v 2 2 e If any orbital radius r is allowed, L A,trans,electron can be anything. However, only certain values of r are allowed...

9 The Bohr model: allowed radii and energies See derivation on page Allowed Bohr radii for electron orbits: Use E N = K+U and This is 2K Bohr model energy levels:

10 The Bohr model: and photon emission

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 Vectors have direction and magnitude. Vector Notation and the Momentum Principle: Use the chain rule causes changes in the direction of causes changes in the magnitude of

13 Best Trick Not in the Book Vectors have direction and magnitude. Vector Notation and the Angular Momentum Principle: Use the chain rule causes changes in the direction of causes changes in the magnitude of

gyroscope precesses here. Note that the angular impulse, being perpendicular to L, STEERS L but does not increase or decrease L.

14 Precession Gyroscopes Precession and nutation As you saw on Wednesday. Look straight down on the model at the left: the Torque and the Angular Impulse are COUNTERCLOCKWISE, and indeed that s how the gyroscope precesses here. Note that the angular impulse, being perpendicular to L, STEERS L but does not increase or decrease L. To understand better, it s useful to note: this is somewhat analogous to centripetal force STEERING momentum, in circular motion, without speeding or slowing the tangential velocity.

15 Gyroscopes M CLICKER: What is the direction of, A or B? A B CLICKER: What is the direction of? A) Left B) Right N The precession angular frequency For rotating vector: CLICKER: What is the direction of? A) Down C) into the page B) Up D) out of the page

16 i>clicker NOTE: if the shaft of the gyro isn t horizontal, R must be replaced by Rcos(θ) where θ is the shaft s angle above horizontal and is the complement of the angle between R and g. Fast precession A Slow precession B In which of the two gyroscopes does the disk spin faster?

discovered (1946) Nobel Prize (1952) Felix Bloch 1905-1983 Edward Mills Purcell 1912-1997 B.S.E.E. from Purdue electrical engineering V.

17 Precession phenomena (see book) Magnetic Resonance Imaging (MRI) Precession of spin axes in astronomy (moon pulling on Earth s bulging midriff as mentioned last time) Tidal torques NMR - nuclear magnetic resonance Independently discovered (1946) Nobel Prize (1952) Felix Bloch Edward Mills Purcell B.S.E.E. from Purdue electrical engineering V. Barnes had the good luck to take introductory E&M from Prof. Purcell as an undergrad at Harvard.

18 Particle spin angular momentum Electron, muon, neutrino have spin 1/2 : measurements of any component of their angular momentum yields ±½ħ (along any chosen axis.) Quarks have spin ½ Protons and neutrons (three quarks) have spin ½ Fermions: spin ½ are matter particles or building blocks They obey the Pauli exclusion principle Two identical Fermions cannot be in the same quantum state in the same place Two lowest energy electrons in any atom have orbital angular momentum 0 One has spin up, the other has spin down (+- ½ along any chosen axis) The different spin projections distinguish between the two electrons so that they are not identical for Pauli exclusion purposes. The two spin projections add up to 0, and when added to the above 0 orbital, the TOTAL ANGULAR MOMENTUM is 0

19 Particle spin angular momentum Bosons: integer spin = Force carrying particles Mesons: (quark+antiquark) have spin 0 or 1: composite, old nuclear force Gluons (elementary, the new strong color force) photons, gravitons, weak bosons W and Z All have spin 1. These are the force carriers of Quantum Field Theory Higgs particle (if it exists) has spin 0

20 Particle spin Rotational angular momentum Macroscopic objects: quantization of L is too small to notice! Rotational energies of molecules are quantized Quantum mechanics: L x, L y, L z can only be integer or half-integer multiple of ħ Quantized values of where l is integer or half-integer These rules are not at all intuitive. Rotation is a richly complicated phenomenon (as you have been told in lectures), and even more-so in the Quantum world. But the quantization of the PROJECTION of angular momentum along any chosen axis plays very directly into the structure of the microworld that we hope you get a sense of in this course.

Lecture 20 Angular Momentum Quantization (mostly) Chapter

PHYS 172H: Modern Mechanics Fall 2012 Lecture 20 Angular Momentum Quantization (mostly) Chapter 11.8 11.11 EVENING EXAM III - 8:00-9:30 PM, TUE NOV 13, Room 112 r dl dt tot I R r =τ net θ Predicting Position