Lecture 20 Angular Momentum Quantization (mostly) Chapter

Transcription

1 PHYS 172H: Modern Mechanics Fall 2012 Lecture 20 Angular Momentum Quantization (mostly) Chapter EVENING EXAM III - 8:00-9:30 PM, TUE NOV 13, Room 112

2 r dl dt tot I R r =τ net θ Predicting Position with Rotation 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? F Solution: r r Δ Ltot = τ netδt r r r r r Iω Iω = Iω = R F Δt f i f Iω = RFΔt ω f f = RFΔt I Because the angle, call it φ, in the cross product is 90 o and sin φ = 1. Note that θhere is NOT the cross-product angle

3 Predicting Position with Rotation r dl dt tot I r =τ R net θ ω f = 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? F RFΔt I See also examples in chapter 11.8 Solution: Δ θ = ω aver Δt ω changes linearly with time: ωi + ωf ωf ωaver = = 2 2 ω f RF Δt Δ θ = Δ t = 2 2I Δ x= RΔ θ = ( ) 2 ( Δ ) 2 F R t 2I ω aver Δθ Δ t This is the rotational analog of accelerated linear displacement And this string motion is in the linear world, once we multiply the angular displacement by R

4 Predicting Position with Rotation r dl dt tot I r =τ R net θ ω f = 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? F RFΔt I See also examples in chapter 11.8 Alternative form of Solution: ω changes linearly with time: RF = τ = I α θ= ½ α (Δ t) 2 where α =τ /I ( Δ ) 2. ω f RF t Δ θ = Δ t = 2 2I Δ x= RΔ θ = uniform angular acceleration ( Δ ) 2 F R t 2I

5 Angular momentum quantization Elementary particles may behave as if they have rotational angular momentum Electron may have translational (orbital around nucleus), and intrinsic rotational angular momenta Angular momentum is quantized 2-1 [ J s] kg m s = Angular momentum quantum = h h = = 2π J s 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

6 Angular momentum quantization Elementary particles may behave as if they have rotational angular momentum Orbital angular momentum comes in integer multiples, but intrinsic spin of Fermions (building blocks) is ½ unit of h Electron intrinsic rotational angular momentum is ½ h Also for quarks, and for protons and neutrons. Force carriers have integer spin, 1 for E&M, Weak, and Strong Nuclear forces (photons, W and Z bosons, gluons) h h 2 for gravity (gravitons)

7 The Bohr model of the hydrogen atom Niels Bohr p r 1913: electron can only take orbits where its translational angular momentum is integer multiple of h r r r r r L = p = Nh trans, C =r p = r m v Equate electric force to F c. p 2 /mr = mv 2 /r = ke 2 /r 2 (where k = 1/4πε o = 9E9) (N h /r) 2 /mr = k (e/r) 2 cancel r 2 terms, solve for r, find r = (N h / e) 2 / k m

8 The Bohr model: allowed radii and energies See derivation on page Allowed Bohr radii for electron orbits: h r = N N N πε em 0 ( m) k The (positive) K is half as big as the (negative) electrostatic U. The total binding energy is K+U = -1/2 k e 2 / r. But r is quantized, so Bohr model energy levels: E N e 4 em 4πε. 0 r 13.6 ev 2 2 2, N=1,2,3,... = = 2N h N

9 The Bohr model: and photon emission 13.6 ev E N = 2 N

10 Refined quantum mechanical model Probability density of the electron for the ground and two excited states of Hydrogen. N = Translational angular momentum of 1 st level (N=1) is 0 of 2 nd level (N=2) is 0 or ħ (z component) etc. As we ll see, the N=2 state with one unit of h-bar can have 3 possible angular momentum projections along any chosen axis.

11 Particle spin angular momentum Electron, muon, tau lepton, three flavors of neutrino all 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 an atom have orbital angular momentum 0 One has spin up, the other has spin down (+- ½ along any chosen axis) (requires Helium or higher, Hydrogen has only one electron total) 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 h

12 Particle spin angular momentum Bosons: integer spin = Force carrying particles Name comes from fact that bosons obey Bose-Einstein statistics, and prefer to all crowd into the same quantum state -- for example photons in radio waves or in laser beams 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 Graviton has spin 2 (and Gravity is a Tensor theory unlike the other three forces.) Higgs particle (if it exists) has spin 0 News flash: we may well have recently seen the Higgs at the LHC

13 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 L l( l 1) = + h where l is integer or half-integer 2 2 For a given spin, J, the projections of J along ANY chosen axis can only be J, J-1, J-2,. -J [all these integers are understood multiplied by h-bar] There are thus 2J+1 possible projections. These form multiplets and are deeply involved in the structure of atomic electron orbitals, and many other quantum structures. 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 ll get a sense of in this course.

14 As you saw last week. 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 a circular motion, without speeding or slowing the tangential velocity. Precession Gyroscopes Precession and nutation

15 Gyroscopes Ω L r A rot F r N R r M L r rot r dlrot r L r =τ cm rot dt r r r ω B τ cm = R FN Mg r CLICKER: What is the direction of R r? A) Left B) Right r r r τ cm = R FN r τ = RF = RMg CLICKER: What is the direction of, A or B? cm N The precession angular frequency For rotating vector: r dl dt rot =ΩL rot = RMg CLICKER: What is the direction of? A) Down C) into the page B) Up D) out of the page RMg Ω= = L rot RMg Iω τ r cm

16 Ω= RMg Iω 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. Slow precession A Fast precession B Which of the two gyroscopes has its disk spinning faster?

17 Precession phenomena Magnetic Resonance Imaging (MRI) A magnetic field torques the magnetic dipole moment of an atom or a nucleus, trying to line it up with the field. The magnetic moment is tied to the particle s spin, which thus precesses exactly like a gyroscope. The precession frequency is specific to different molecules and atoms.

18 Precession phenomena Precession of spin axes in astronomy (moon and sun pulling on Earth s bulging midriff, or equatorial bulge) The Earth is somewhat oblate, due to centrifugal effects from the Earth s rotation. Also the Earth s spin is noticeably tipped away from perpendicular to the Earth/Moon/Sun orbits. The part of the bulge closest to the Moon is attracted more strongly by gravity. The part of the bulge farthest from the Moon is moving faster around the Earth-Moon center of mass, due to the Moon s orbit, and feels more centrifugal force. Both forces tend to line up the Earth s equator along the Earth-Moon line. Similar interactions due to Earth s orbit around Sun also try to line up the Earth s spin perpendicular to the orbital planes. Instead, the Earth s spin precesses just like a torqued gyroscope spin Moon F G F G There are also fictitious centrifugal (away) forces, smaller when closer to the Moon and larger when farther away. Net overall counterclockwise torque on the Earth.

19 Tidal torques The same stronger gravity of the Moon on the closest part of the Earth causes an additional hump pointing at the Moon. Also, the farthest part of the Earth feels greater centrifugal force away from the Earth- Moon center of mass (about which the system is rotating) so there is a hump facing away from the Moon. These humps affect the solid Earth, and more-so the fluid oceans. Hence, high tides occur approximately every twelve hours, as the Earth turns past the Earth-Moon line. The Earth spins faster than the Moon s orbit (one day vs. about 29 days) and the humps get somewhat ahead of the Earth-Moon line. This exerts a torque on the Moon s orbit (and an equal and opposite torque on the Earth s spin) and over Eons the Moon s orbital angular momentum has increased and the Earth s spin angular momentum has decreased accordingly. Growth bands in Devonian fossil shells show daily, monthly, and yearly patterns which reveal that there were about 400 days in a year in the Devonian Epoch. So the Earth is spinning slower now.

20 Tidal torques Growth bands in Devonian fossil shells show daily, monthly, and yearly patterns also reveal that the lunar month was shorter in the Devonian Epoch. So the Moon s orbital period is longer now. Paradoxically, the Moon now also takes longer to orbit the Earth. Because its increased orbital angular momentum goes with a larger Radius from the Earth. And the orbital period T ~ R 3/2 Derivation: mv 2 /R = F c = GMm/R 2 Equate gravity with centripetal force v 2 = GM/R mvr = L = m GMR Eventually the day and month could converge to the same rotation period, when enough energy and angular momentum from the Earth s spin are transferred to the Moon s orbit. The Moon already has a frozen-in tidal bulge that always points at the Earth, meaning that the Moon s rotation period is the same as its orbital period.

21 Tidal torques Gravity gradient stabilization of a satellite: The end closer to Earth feels stronger Gravity (1/r 2 variation) and the end farther feels weaker Gravity than at the center of mass. At the same time, the entire satellite is orbiting as a rigid body so the near and far ends are moving slower and faster, respectively, than orbiting mass points would move. In a rotating (non-inertial) coordinate system there are fictitious forces pointing away from the Earth, which vary as ω 2 r Gravity Centrifugal force Net force pulls near end towards Earth and vice versa

22 Precession phenomena 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 fortune to take introductory E&M from Prof. Purcell as an undergrad at Harvard.