Fall 2010 PHYS 172H: Modern Mechanics

Lecture 20 – Angular Momentum Quantization (mostly)

Chapter 11.8 – 11.11

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

CLICKER POLL Not for Credit

Predicting Position with Rotation

F=mg

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 (ωi=0) 1) What will be the angular speed ωf ?

θ

R

I

Solution:

m

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 ? 2)  How far (Δx) will the end of string move?

θ

R

I

See also examples in Chapter 11.8

Solution:

ω changes linearly with time:

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

F=mg

m

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 angular momentum in a hydrogen orbital?

px

+

i py

= |L=1, Lz=1>

Electron "current" circles around

the atom.

Quantized because these are 3D standing

electron waves around the nucleus. See Atom in a Box

www.daugerresearch.com

Bohr’s Atomic Model 2

, ,A trans electron eL mrkq=

Allowed radii: 2

22e e

r Nkq m

= 341.05 10

1,2,3,J s

N

−= × ⋅

=

…

This implies that only certain values of LA,trans,electron are allowed:

, , N=1,2,3 where ,A trans electronL N=

…

NOTE: Because K and U are functions of r and v, energy levels are quantized also.

Niels Bohr

1913: IDEA: Electron can only take orbits where its translational angular momentum is integer multiple of

Bohr Model

Consider an electron in circular orbit about a proton. What are the possible values of LA,trans,electron?

A

2, ,A trans electron eL mvr mrkq= =

2 22

2e e

ekq kqpv mvF v

r r r mr= ⇒ = ⇒ =

Thus,

If any orbital radius r is allowed, LA,trans,electron can be anything.

However, only certain values of r are allowed . . .

Assume circular motion:

The Bohr model: allowed radii and energies

Allowed Bohr radii for electron orbits:

Bohr model energy levels:

See derivation on page 444-446

Use EN = K+U and

This is 2K

The Bohr model: and photon emission

Gyroscopic Stability

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

Edmund Scientifics

Best Trick in the Book

Vector Notation and the Momentum Principle:

Vectors have direction and magnitude.

causes changes in the magnitude of

causes changes in the direction of

Use the chain rule

Best Trick Not in the Book

Vector Notation and the Angular Momentum Principle:

Vectors have direction and magnitude.

Use the chain rule

causes changes in the magnitude of

causes changes in the direction of

Gyroscopes Precession and nutation Precession

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.

Gyroscopes

A B

CLICKER: What is the direction of , A or B?

CLICKER: What is the direction of ? A) Left B) Right

CLICKER: What is the direction of ? A) Down C) into the page B) Up D) out of the page

M

For rotating vector:

The precession angular frequency

N

i>clicker

In which of the two gyroscopes does the disk spin faster?

A B

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 Slow precession

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

Felix Bloch 1905-1983

NMR - nuclear magnetic resonance

Edward Mills Purcell 1912-1997

Independently discovered (1946)

Nobel Prize (1952)

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.

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

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

Particle spin Rotational angular momentum

Macroscopic objects: quantization of L is too small to notice!

Rotational energies of molecules are quantized

Quantum mechanics: Lx, Ly, Lz 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.