Class #23 of 30
Tennis racket demo
Euler’s equations



Motion of a free body
No external torques
Response to small disturbances
 Tennis racket theorem
 Nutation

Chandler wobble
Inertia tensor redux



Inertia tensor of a cube
Inertia tensor of a rectangle
Lamina theorem
1 :02
Inertia Tensor
I xx
L  I yx
I zx
I xx
I yy
I zy
I xz  x
I yz  y
I zz  z
1
0
0
I 0
0
2
0
0
3
L  I
If choose principal axes,
Inertia tensor is
diagonal
L  (11 , 22 , 33 )
2 :02
Inertia Tensor – U-solve it
Calculate elements of an
inertia tensor for a
rectangular lamina.
B
A
I
2
2

(
y

z
)dxdydz

  ( xy)dxdydz
  (xz)dxdydz
  ( yx)dxdydz
...
etc.
...
...
2
2

(
x

y
)dxdydz
3

Vectors expressed
in rotating frames

r
r
S
  ri eˆi
i
For fixed e-sub-I
dri
r   eˆi
S0
i dt
For rotating e-sub-I
dri
deˆi
r   eˆi   ri
S0
dt
i dt
i
Because eˆi is any arbitrary FIXED
vector in frame S 0, it transforms
deˆi
Imagine same axes
in S as
   eˆi
dt
(x,y,z) expressed in
two frames S0
dri
r   eˆi   ri   eˆi
S0
stationary) and S (fixed
i dt
i
to earth).
 ri   eˆi     ri eˆi    r 
i
r
i
S0
dri
  eˆi    r  r    r
S
i dt
4
:60
Euler’s Equation
r

S0
S
 L   L
L
L
 r   r
S0
S
 external  L   L
S
1  11  (2  3 )23
 2  22  (3  1 )31
3  33  (1  2 )12
11  (2  3 )23
22  (3  1 )31
33  (1  2 )12
5 :60
Small
perturbations

L
11  (2  3 )23
22  (3  1 )31
33  (1  2 )12
Let 0  ( ,  , 0 )

  0
1 
2 
(2  3 )
1
(3  1 )
2
3  0
0
0 
11  (2  3 )0
22  (3  1 )0
33  (1  2 )  0:60
6
Small perturbations -II
1 
(2  3 )
1
20
11  (2  3 )23
(3  1 )
22  (3  1 )31  2 
01
2
33  (1  2 )12
3  0  3  0
 (3  2 ) (3  1 ) 2 
1   
 0  1
2
 1

 (3  1 ) (3  2 ) 2 
2   
 0  2
1
 2

7 :60
Class #23 Windup - I
Euler
 external  L    L
S
Without external torques
11  (2  3 )23
22  (3  1 )31
33  (1  2 )12
•Rotations about the “middle-valued” principal axis
are unstable
8 :60
Home stretch
#12 – Supplement H, Taylor 9.11,
10.7, 10.18
23
11/12
Ch. 10 Euler’s equations – “tennis
racket theorem”
24
11/14
Exam review + Ch. 11 Coupled
Oscillators and Normal modes
25
11/19
Test #3
Central force and accelerated frames
26
11/21
Coupled Oscillators / Nonlinear
Mechanics and Chaos
11/26
27
11/26
Ch. 12 Nonlinear Mechanics and
Chaos
12/5
11/28
THANKSGIVING
28
12/3
Ch. 13 Hamiltonian Mechanics
29
12/5
Test #4
30
12/10
Review problems for Final
11/14
11/21
This exam was moved to 11/19
from 11/14
Inertia Tensor / Normal Coords /
Chaos
-
9 :02
Class #23 Windup - II
Thursday bring problems you want to
see worked
Homework is due day of exam (next
Tuesday) – A few more problems will
be added to assignment
Will go over old homeworks and your
q’s in class Thursday
10 :60
Class #24 of 30
Exam -- Tuesday



Additional HW problems posted Friday
(also due Tuesday).
Bring Index Card #3.
Office hours on Monday 3:30-6:00
Topics

Central Force






Kepler’s Laws
Gravitational PE and Force
Small Oscillations about equilibrium
Reduced Mass
Momentum conservation
Pseudopotential
11 :02
Class #24 of 30

Central Force
 Polar form of orbital equations



Elliptical orbits
Hyperbolic orbits
Parabolic / Circular Orbits
 Scattering

Accelerated Reference Frames
 Effective Gravity
 Centrifugal force
 Coriolis force
12 :02
Problem Review







KKR9-4 Holy Earth
KKR9-6 Eccentric comet
KKR9-8 Escape to the moon
Taylor 8-9 Small oscillations
Taylor 9-1/9-2 Buoyant
doughnuts
Taylor 9-9 Tilted Plum-line
Taylor 9-10 Spin the bucket
13 :02
Gravity and Electrostatics
Gravity
Universal
Constant
Force Law
Gauss’s Law
Potential
G  6.673(75)  10 11
Nm 2
kg 2
F  G
Electrostatics
1
4 0
 6.987551...  109
Nm 2
Coulomb 2
m1m2
rˆ
2
r
F 
1
4 0
1
 Fg  4 G  mass
m
 E
mm
U  G 1 2
r
U 
1
0
1
4 0
q1q2
rˆ
2
r
 ch arg e
q1q2
r
14 :08
Kepler’s 1st, 2nd and 3rd laws
2 3
4

r
2
 
GM
 
  r 2  const.
1st Law – Planets move in ellipses
with sun at one focus
Third law demonstrated previously
relates period to semi-minor radius
2nd law is direct consequence of
momentum conservation
“Equal areas are swept out in
equal times”
True for ALL central forces
So  r  const.
dA 1
1
 bh  ( r )(r )
dt 2
2
b1
"
 const.
2
'
(1610)
h1
h2
b2
15 :37
E, L and Eccentricity
r ( ) 
2
Gm1m2  (1   cos  )
Gm1m2 

E
2
 E
 1
2

2
2

2
2
 Gm1m2 
2


1
1
 1
0   1
 0
The physics is in E and L. Epsilon is purely a
geometrical factor.
Epsilon equation applies to ALL conic sections
(hyperbolae, ellipses, parabolas).
16 :30
Central Force
2
c
r ( ) 
; c
(1   cos  )

rmin
c

;   Gm1m2
1 


E
2
rmax
2
2

2

  1
c

;
1 
b
 1  2
a
<- Completely general for inverse
square forces … All types of
orbits.
<- “Gamma” makes it specific for
gravity.
Key constants are, (E and L), OR (c
and L) or (L and epsilon) or (c
and epsilon)
<- Specific for elliptical orbits.
17 :30
Planetary Scattering Angle
Epsilon
2
r ( ) 
1
1.001
1.02
1.05
1.1
1.3
2
5
20
50
100
Gm1m2  (1   cos  )
r (max )    1   cos max  0
r ( )
 
  360  2arccos 1

rimpact
r ( )
-1/eps 2*arccos Omega
-1.00
-1.00
-0.98
-0.95
-0.91
-0.77
-0.50
-0.20
-0.05
-0.02
-0.01
360.0
354.9
337.3
324.5
310.8
280.6
240.0
203.1
185.7
182.3
181.2
0.0
5.1
22.7
35.5
49.2
79.4
120.0
156.9
174.3
177.7
178.8
Sketch for epsilon=2
2max
18 :37
Reduced two-body problem
r2
r1
mmoon
mEarth
rrelative


19
:15
Equivalent 1-D problem
m1m2
1
2
2 2
L rel   (r  r  )  G
2
r
m1m2
r :  r  G 2   r 2
r
2
m1m2
 r  G 2  3
r
r
Total Radial Force
Ftotal
U pseudo
Relative
Lagrangian
Radial equation
 
 r
2
2
m1m2
 G

2
r
2 r
  Ftotal dr  U pseudo
20 :30
Class #24 Windup
mr  Fexternal  2mr    m(  r )  
U pseudo  U actual 
2
2 r
2
for any Central force.
21 :60
Parameter
Sim.
Time
Comment
(s)
Lightly damped, undriven
15
More damping, undriven
(dies in 5 secs)
15
Lightly damped, state
space
15
Same damping, driven
15
Same damping, state space 15
Different drive frequency
15
Drive freq~res. Freq.
15
Drive freq>res. Freq.
15
Drive freq>>res. Freq.
15
Ordinary driven
15
Ordinary driven
more damping
15
Long Transient
22.5
Long Transient, State space 100
Period doubling
45
Period tripling
45
Different Attractors diff IC's
45
Period tripling, state space
45
Different Attractors state space 45
Period doubling
45
Period quadrupling
45
Period quadrupling (state)
100
Period octupling
45
True chaos
45
Chaos in state space
100
F0
F
Damping Gamma Ovrd Theta-0
Hz
3
Hz
1
3
1
1
3
3
3
3
1
1
1
1
1
1
1
0.5
2
5
10
0.67
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0.67
0.67
0.67
0.67
0.67
0.67
0.67
0.67
0.67
0.67
0.67
0.67
0.67
0.67
Nt-s/m Nt-s^2/m
0.4
0
State
1
Degrees
90
0,1,2
0
0
1
90
0
1
1
1
1
1
1
1
1
0
220
220
150
150
150
150
0.3
1
1
1
1
1
1
1
0
90
1
0
0
0
0
0
0
1
0
1
0
0
0
0
0
3.14
3.14
3.14
3.14
3.14
3.14
3.14
3.14
3.14
3.14
3.14
3.14
3.14
3.14
0.3
1.06
1.06
1.073
1.077
1.077
1.077
1.077
1.078
1.081
1.081
1.0826
1.105
1.105
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-90
0
-90
-90
-90
-90
-90
-90
-90
0
0
1
0
0
0
0
0
0
0
2
0
0
2
22 :60
Class #26 of 30
Nonlinear Systems and Chaos

Most important concepts
 Sensitive Dependence on Initial conditions
 Attractors

Other concepts










State-space orbits
Non-linear diff. eq.
Driven oscillations
Second Harmonic Generation
Subharmonics
Period-doubling cascade
Bifurcation plot
Poincare diagram
Feigenbaum number
Universality
23 :02
Outline
Origins and Definitions of chaos
State Space
Behavior of a driven damped
pendulum (DDP)
Non-linear behavior of a DDP




Attractor
Period doubling
Sensitive dependence
Bifurcation Plot
24 :02
Definition of chaos
The dynamical evolution that is aperiodic and sensitively
dependent on initial conditions. In dissipative dynamical systems
this involves trajectories that move on a strange attractor, a
fractal subspace of the phase space. This term takes advantage
of the colloquial meaning of chaos as random, unpredictable,
and disorderly behavior, but the phenomena given the technical
name chaos have an intrinsic feature of determinism and some
characteristics of order.
Colloquial meaning – disorder, randomness, unpredictability
Technical meaning – Fundamental unpredictability and apparent
randomness from a system that is deterministic. Some real
randomness may be included in real systems, but a model of the
system without ANY added randomness should display the same
behavior.
25 :02
Weather and climate prediction

Is important

Can we go:
a)
b)
c)

Will we be hurt by a:
a)
b)
c)

For a hike?
Get married outdoors?
Start a war?
Tornado?
Hurricane?
Lightning bolt?
How much more fossil fuel can we burn
before we:
a)
b)
c)
Fry?
Drown?
Starve?
26 :02
Early work by
Edward N. Lorenz

1960’s at MIT
 Early computer models of the atmosphere



Were very simple
 (Computers were stupid)
Were very helpful
Results were not reproducible!!
 Lorenz ultimately noticed that


For 7-10 days of prediction, all of his models reproduced
very well.
After 7-10 days, the same model could be run twice and
give the same result – BUT!!
 Changes that he thought were trivial
 (e.g. changing the density of air by rounding it out at
the 3th decimal place, or slightly modifying the initial
conditions at beginning of model)
 Produced COMPLETELY DIFFERENT results
 This came to be called “The Butterfly effect”
27 :02
State Space
x  sin t ;
d
y  x  sin t  cos t
dt
x  sin 2t ; 
d
y  x  sin 2t  2 cos 2t
dt
x  sin t  0.35sin 2t ;
y  x  cos t  0.7 cos 2t

28 :02
Viscous Drag III – Stokes Law

Fdrag
D
u xˆ


Fdrag  b r

Fdrag   3 D u xˆ
Form-factor k becomes 3
“D” is diameter of sphere
Viscous drag on walls of
sphere is responsible for
retarding force.
George Stokes [1819-1903] 
(Navier-Stokes equations/ Stokes’ theorem)
29
:45
Damped Driven Pendulum (DDP)



Damping
 Pendulum immersed in fluid
F (t )


with Newtonian viscosity
 Damping proportional to
velocity (and angular velocity)
Driving
 Constant amplitude drive bar
 Connected to pendulum via
torsion spring
 Torque on Pendulum is
bv
mg
  FL  k (   )
mL   bL   mgL sin   LF (t )
2
2
30 :02
Damped Driven Pendulum (DDP)
mL2  bL2  mgL sin   LF (t )
b
g
F (t )
     sin  
m
L
mL
2
2 L F (t )
  2  0 sin   0
g mL
2
2 F (t )
  2  0 sin   0
mg
  2  02 sin   02 cos t
F (t )

mg
f 0  0 / 2  1 Hz
  0.667 Hz
2  3.14 N  s / m
Force

pendulum weight
31 :02
Conditions for chaos
Dissipative Chaos
1. Requires a differential equation with 3 or
more independent variables.
2. Requires a non-linear coupling between at
least two of the variables.
3. Requires a dissipative term (that will use up
energy).
Non-dissipative chaos
Not in this course
32 :60
Sensitive dependence on initial conditions
33 :60
Sensitive dependence on initial conditions
34 :60
Worked problem
Sketch a state-space plot for the magnetic
pendulum

Indicate the attractors and repellers

Show some representative trajectories
First explore the trajectories beginning with thetadot=0

Then explore trajectories that begin with theta in
some state near an attractor but through proper
choice of theta-dot move to the other attractor.


Sketch the basins of attraction
35 :60
Class #26 Windup
Homework

Is on Chapters 9 and 10 – Read them!
Is partly posted now … more will be added
tomorrow.

Is due BEFORE Thanksgiving (Wednesday … or under
my door Thursday).

36 :60
Class #27 Notes
Homework

Is due before you leave
Problem 10-25 has been upgraded
to extra-credit.

Probs 10-21 and 10-24 are CORE
PROBLEMS. Make sure you
understand these.


When you return –
Will spend another lecture on tops,
tensors and Euler’s theorem before
the exam.

37 :60
Class #27 of 30
Nonlinear Systems and Chaos

Most important concepts
 Sensitive Dependence on Initial conditions
 Attractors

Other concepts











State-space orbits
Non-linear diff. eq.
Driven oscillations
Second Harmonic Generation
Subharmonics
Period-doubling cascade
Bifurcation plot
Poincare diagram
Mappings
Feigenbaum number
Universality
38 :02
Chaos on the ski-slope
7 “Ideal skiers” follow the fall-line and end up very different places
39 :60
Insensitive dependence on initial
conditions
Plot A :  0  90
Plot A  B : ( Log of  )
  0.1
Plot B : 0  175
40 :60
Sensitive dependence on initial conditions
Plot A :  0  90
Plot A  B : ( Log of  )
  1.105
Plot B : 0  90.0001
41 :60
Resampled pendulum data
Gamma=0.3
Gamma=1.0826
Gamma=1.077
Gamma=1.105
42 :60
Bifurcation plot
100
0.3
50
0
50
100
43 :60
Bifurcation plot and universality
( n 1   n ) 
1

( n   n 1 )
  4.6692016


For ANY chaotic
system, the period
doubling route to
chaos takes a similar
form
The intervals of
“critical parameter”
required to create a
new bifurcation get
ever shorter by a ratio
called the
Feigenbaum #.
44 :60
In and out of chaos
45 :60
Poincare plot
46 :60
Poincare plot

Poincare plot is set of allowed states at any
time t.
 States far from these points converge on
these points after transients die out
 Because it has fractal dimension, the
Poincare plot is called a “strange
attractor”
47 :60
State-space of flows
48 :60
Cooking with state-space






Dissipative system
 The net volume of
possible states in
phase space ->0
Bounded behavior
 The range of possible
states is bounded
The evolution of the
dynamic system “stirs”
phase space.
The set of possible states
gets infinitely long and
with zero area.
It becomes fractal
A cut through it is a
“Cantor Set”
49 :60
Mapping vs. Flow
Gamma=1.0826





Gamma=1.105


A Flow is a continuous system
A flow moves from one state to another
by a differential equation
Our DDP is a flow
A mapping is a discrete system.
State n-> State n+1 according to a
difference equation
Evaluating a flow at discrete times turns
it into a mapping
Mappings are much easier to analyze.
50
Logistic map
xt 1  rxt (1  xt ) defined on interval x  0  1 ,
for r  1.
For xt  1,
xt 1
rxt . Typical exp onentia l growth.
for xt
1, growth is lim ited
“Interesting” values of r are.
R=2.8 (interesting because it’s boring)
R=3.2 (Well into period doubling_
R= 3.4 (Period quadrupling)
R=3.7 (Chaos)
R=3.84 (Period tripling)
51 :02
Class #28 of 31
Inertia tensor


Kinetic energy from inertia tensor.
Eigenvalues w/ MATLAB
Tops and Free bodies using Euler
equations



Precession
Lamina theorem
Free-body wobble
52 :02
Rest of course
11/28
THANKSGIVING
28
12/3
Tops, Gyros and Rotations
29
12/5
Ch. 13 Hamiltonian and
Quantum Mechanics or
Chaos
30
12/10
Test #4
31
12/12
Review for Final
FINAL
EXAM
12/16
MONDAY – 9AM-12NOON
#14 –Supplement /
Taylor
-
Inertia Tensor / Tops /
Euler equations /Chaos
12/10
WORKMAN 310
53 :60
Angular Momentum and Kinetic Energy
p
L
We derived the moment of inertia
 mv  m(  r ) tensor from the fundamental
definitions of L, by working out
 r  m(  r ) the double cross-product
Do the same for T (kinetic energy)
1
1
T  mv v  (  r ) mv
2
2
1
T   (r  mv ) (vector identity )
2
1
T  L
2
LI 
1 T
T  I 
2
54 :02
L 28-2 Angular Momentum and Kinetic
Energy
1) A square plate of side L and mass M is rotated
about a diagonal.
2) In the coordinate system with the origin at lower
left corner of the square, the inertia tensor is.
LI 
3


 1 4 0 
1

2 
0  
ML  3

I
 10 0  
1


3  4
2 
 0 0 2 
0




1 T
T  I 
2
Calculate L and T .
55 :02
Symmetrical top
Euler equation
 3  33  (1  2 )12
1  2 and 3  0 
33  0  3  const
56 :02
dL
  rF
dt
dL
 RCM mg sin  ˆ
dt
L
 
L sin 
Precession
Ignore in limit
2
L  Lz  L   L 
2
2
2
 p  3
Lz   33 
2
2
d
1 dL RCM mg sin  RCM mg
 p 


dt
L sin  dt
L sin 
33
57 :02
Lamina Theorem
I yy
I zz
I xx  I yy   ( y 2  z 2 )    ( x 2  z 2 )   dxdydz
I xx
I zz   ( x 2  y 2 )  dxdydz
I zz  I xx  I yy for laminar objects ( z  0)
I zz
I yy
I xx
I yy  I xx  I zz for laminar objects ( y  0)
58 :60
Euler’s equations for symmetrical bodies
1
2
I yy
For Disk I xx  I yy  MR  
4
I zz
1
2
I xx I zz  I xx  I yy  MR  2
2
Note 3  0 even for non-laminar
symmetrical tops AND even for 1 , 2  0
11  (2  3 )23 1  (  2 )23 1  23
22  (3  1 )31 2  (2   )31  2  31
33  (1  2 )12 23  (   )12 3  0
59 :60
Euler’s equations for symmetrical bodies
p
L
3 ẑ
1  23
1  321
2  31 
2
2  3 2
3  0

2
p

2
3
Precession frequency=rotation frequency
for symmetrical lamina
60 :60
Euler’s equations for symmetrical bodies
p
L
p
3 ẑ
3 ẑ
L
3  1
3  1
61 :60
L28-1 – Chandler Wobble
I zz
1) The earth is an ovoid
thinner at the poles than
I xx the equator.
I yy
2a
2) For a general ovoid,
2b
11  (1  3 )23
12  (3  1 )31
33  0
 (3  1 ) 2 2 
1   
3  1
2
 1

1
I xx  M (a 2  b 2 )
5
3) For Earth, what are
I yy and I zz , and  p ?
a  b  a
a 6400 km

20 km
2
b  2 b
2
62 :60
L 28-2 Angular Momentum and Kinetic
Energy
1) A square plate of side L and mass M is rotated
about a diagonal.
2) In the coordinate system with the origin at lower
left corner of the square, the inertia tensor is.
LI 
3


 1 4 0 
1

2 
0  
ML  3

I
 10 0  
1


3  4
2 
 0 0 2 
0




1 T
T  I 
2
Calculate L and T .
63 :02
Lecture 28 windup
p 
RCM mg
33
LI 
1 T
T  I 
2
for top
I zz  I xx  I yy for la min a
1
I xx  M (a 2  b 2 ) for ellipsoid
5
64 :02
Angular Momentum and Kinetic Energy
L

1) A complex arbitrary system is
subject to multi-axis rotation.
2) The inertia tensor is
3) A 3-axis rotation is
 5.0 
 
   8.2 
 3.0 
 
applied
LI 
15  6  1
A   6 10  5 
 1  5 20 
1 T
T   I  Calculate L and T .
2
65 :02
Physics Concepts
Classical Mechanics









Study of how things move
Newton’s laws
Conservation laws
Solutions in different reference frames (including
rotating and accelerated reference frames)
Lagrangian formulation (and Hamiltonian form.)
Central force problems – orbital mechanics
Rigid body-motion
Oscillations lightly
Chaos
66 :04
Mathematical Methods
Vector Calculus





Differential equations of vector quantities
Partial differential equations
More tricks w/ cross product and dot product
Stokes Theorem
“Div, grad, curl and all that”
Matrices


Coordinate change / rotations
Diagonalization / eigenvalues / principal axes
Lagrangian formulation



Calculus of variations
“Functionals” and operators
Lagrange multipliers for constraints
General Mathematical competence
67 :06
Correlating Classical and Quantum Mechanics
Correspondence Principle
 In the limit of large quantum numbers,
quantum mechanics becomes classical
mechanics.
 First formulated by Niels Bohr, one of the
leading quantum theoreticians
We will illustrate with
 Particle in a box
 Simple harmonic oscillator
Equivalence principle is useful
 Prevents us from getting lost in “quantum
chaos”.
 Allows us to continue to use our classical
intuition as make small systems larger.
 Rule of thumb. System size>10 nm, use
classical mechanics.
68 :02
1-D free particle
Classical Lagrangian and Hamiltonian for free 1-D
particle L  T  V
H  T  V  Total Energy
for free particle,
Schroedinger’s
equation for
free particle
p2
p2
H 
E
  E
2m
2m



p  i
;Ei
x
t
2
 2

H   E 
i
2
2m x
t
  0 e  i ( kx t )
69 :02
Hydrogen Atom
Classical Lagrangian and Hamiltonian
1
1 q1q2
2
2 2
L rel   (r  r  ) 
2
4 0 r
2
pr 2
1 q1q2



2
2 2 r
4 0 r
H rel
2
pr 2
1 q1q2



2 2 r 2 4 0 r
Schroedinger’s equation for hydrogen
1 q1q2

H   E 
 
i
2
4 0 r
t
2
2
70 :02
Hydrogen Atom
P
2
r  op
1  

r
 r r 
2
2

1  
 
1  
2
Lop  
 sin 
 2
2
  sin   
 sin   
Schroedinger’s equation for hydrogen
q1q2

H   E 


i
2
2
2 r
4 0 r
t 71
Pr op 2
Lop 2
1
:02
Particle in a box
2

 V   E
2
2m x
2
n 
2
n
sin
x
L
L
n12  n22  n32 2
E
h
2
8mL
h  6.62 1034 j  sec
Let L  10 nm
m  16 1.66 1027 kg (Oxygen)
E1  2.06 1026 joules  107 eV
Thus ideal gas law doesn ' t need quantum mech.
72 :02
2
n
Particle in a box
n 
sin
x
L
L
2 2 n
*
 n  n  sin
x
N1, no match between
L
L
quantum and classical
probability
1 2
2E
mv  E  v 
2
m
C
1
Pclassical ( x) 
 const.
v( x)
2E
m
N51, Averaged quantum
probability approaches
classical constant
probability.
73 :02
Simple harmonic oscillator (SHO)
1 2 1 2
2E k 2
mv  kx  E  v 
 x
2
2
m m
C
1
Pclassical ( x) 
v( x)
2E k 2
 x
74 :02
m m
Expectation values
" Bra "   *
  " ket "
Bra-ket notation and Matrix formulation of QM
All wave functions may be written as linear combination
of eigenfunctions.
Thus effect of operator can be replaced by a matrix
showing effect of operator on each eigenfunction.
All QM operators (p, L, H) have real eigenvalues – They are
“Hermitian” operators
E  H 
*
functional formalism
E   H  matrix formalism
Exactly like
T
T  T I 
75 :02
Expectation values
" Bra "   *
  " ket "
Bra-ket notation and Matrix formulation of QM
All wave functions may be written as linear combination
of eigenfunctions.
Thus effect of operator can be replaced by a matrix
showing effect of operator on each eigenfunction.
All QM operators (p, L, H) have real eigenvalues – They are
“Hermitian” operators
E  * H 
functional formalism
E  *H  matrix formalism
Exactly like
T   I  (except  * is complex conjugate of  ) 76 :02
T
T
Spin Matrix
S z   Sz 
*
functional formalism
S z   Sz matrix formalism
*
1 0 0 
S z  0 0 0 
0 0 1
77 :02
Wind up
Classical mechanics is valid for
v  30, 000 km / s
r  10 nm
In other words … almost all of human experience
and endeavor.
Use it well!
78 :02