Download Angular Momentum Quantization

PHYS 172H: Modern Mechanics
Fall 2010
Lecture 20 – Angular Momentum Quantization (mostly)
Chapter 11.8 – 11.11
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
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
(ωi=0)
1) What will be the angular speed ωf ?
I
θ
R
m
F=mg
Solution:
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
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
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
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
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
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.
Bohr Model
Consider an electron in circular orbit
A
about a proton. What are the possible
values of LA,trans,electron?
Assume circular motion:

pv
Fe =
r
Thus,
⇒
kqe2 mv 2
=
2
r
r
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 . . .
The Bohr model: allowed radii and energies
See derivation on page 444-446
Allowed Bohr radii for electron orbits:
This is 2K
Use EN = K+U and
Bohr model energy levels:
The Bohr model: and photon emission
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
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
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
Gyroscopes
Precession
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.
Gyroscopes
M
CLICKER: What is the direction of
, A or B?
A
B
N
CLICKER: What is the direction of
A) Left B) Right
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
?
?
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?
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
1912-1997
1905-1983
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.