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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
⎜ 4πo ⎟ee m ⎝ 4πε 0 r ⎠
E=
=N − −
2N h
2N 2222 13.6 eV
=−
,
2
N N=1,2,3,...
9 1
4πo 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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