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UCF Physics: AST 5765/4762:
(Advanced) Astronomical Data Analysis Lecture Notes
Prof. Joseph Harrington
Planetary Sciences Group
Department of Physics
University of Central Florida
Orlando, Florida 32816
jh@physics.ucf.edu
c 2004 2010 Joseph Harrington. A
Sz |smi = m~|smi.
(269)
Here, s = 0, 1/2, 1, 3/2, ., m = s, s + 1, ., s 1, s. 0 mesons have spin
s = 0, electrons have spin 1/2, photons have spin 1, and gravitons have spin 2.
Bosons have integer spin and fermions have half-integer spin.
Spin 1/2 is the
Figure 5:
The Carnot cycle.
Source:
astr.gsu.edu/hbase/thermo/carnot.html.
http:/hyperphysics.phy-
For an ideal gas, U = U (T ) is a function of temperature only. In reality, all
gases cool slightly during expansion, due to intermolecular attraction poten
Figure 2:
Convex (top) and concave (bottom) lenses.
the position of the image relative to the object for each
Source:
http:/physics20p1.blogspot.com/2011 12 01 archive.html
http:/images.tutorvista.com/content/refraction-light/.
Note
case.
and
dO < 0 if o
The current in the RLC AC-circuit reaches a maximum at the resonant frequency,
1
= 0 =
.
(121)
LC
Electromagnetic waves obey the wave equation (in 1-D),
2E
1 2E
= 2 2,
2
x
c t
2
1 2B
B
= 2 2.
2
x
c t
(122)
(123)
Thus, the solutions are
E = Emax cos(kx
The conserved probability current density is given by
~j = Re( ~ ).
im
(234)
The momentum space wavefunction (p, t) is related to the position space
wavefunction by Fourier transform,
Z
1
eipx/~ (x, t)dx,
(235)
(p, t) =
2~
Z
1
(x, t) =
eipx/~ (p, t)dp.
Here, the d~s implies that this is a path integral. For the same process, the
change in electrostatic potential:
V =
U
=
q0
Z
B
~ d~s.
E
(37)
A
~ x) = E
~ 0,
If the E-field is uniform, E(~
~
~ 0 d,
V = E
(38)
where d~ is the displacement vector.
An equipo
where is the magnetic susceptibility. < 0 for diagmagnetic materials, >
0 for paramagnetic materials, and 1 for ferromagnetic materials. The
magnetic permeability m is defined according to
m = 0 (1 + ).
(96)
Faradays Law:
I
~ d~s = E = dB .
E
dt
(97)
The
This yields solutions
r
n (x) =
2
nx
sin(
),
a
a
n = 1, 2, 3, .,
(246)
with corresponding energy spectrum
En =
n2 2 ~2
,
2ma2
n = 1, 2, 3, .
These solutions satisfy the orthogonality condition,
Z a
m
n dx = mn .
(247)
(248)
0
And,
r Z a
2
2x
cn =
)(x, 0)d
Geometry
Isolated sphere, charge Q, radius R
Parallel plate capacitor, area A, separation d
Capacitance
C = 40 R
C = 0 A
d
Table 2: Geometry and Capacitance.
The capacitance for two important geometric setups can be found in Table 2.
With a dielectric, >
Figure 3: Fixed (left) and free (right) boundary. Note the phase shift in the
first case. Source: http:/snvphysics.blogspot.com/2011/03/waves.html.
Figure 4: Waves move from less to more dense regions (left) and more to
less dense regions (right). Note th
When the battery is turned off:
I(t) =
E t/
e
.
R
(108)
The energy stored in the magnetic field of an inductor:
U=
1 2
LI .
2
(109)
The energy per unit volume in the B-field region of the inductor:
uB =
B2
.
20
(110)
The current and charge of an LC-circui
CP = T (
S
)P .
T
(200)
The Helmholtz function:
F = U T S.
(201)
For a process in which the end point temperatures are the same as the surroundings, the maximum work obtainable equals the decrease in F . F at a minimum
is the condition for thermodynamic e
3. The square of the orbital period T of a planet is directly proportional to
the cube of the semi-major axis a of its orbit.
In mathematical terms, the third law is
T 2 a3 .
(4)
This can be seen, for circular motion, from
mv 2
GM m
mr(2)2
=
.
= m 2 r =
2
Force is related to pressure and area by F = P dA, so
dW = P dV
(172)
for a reversible process. Thus, the total work done on a system is,
Z V2
P dV.
W =
(173)
V1
The adiabatic free expansion is not reversible, and there is no work done on the
system and n
The partition function:
Z=
X
gj e
Ej
BT
k
,
(213)
j
where the sum runs over all microstates and gj is the degeneracy of the jth
microstate. The probability of being found in state j is,
P r(j) =
gj e
Ej
BT
k
Z
.
(214)
The pressure is,
P = kB T ln Z.
(215)
Running the Carnot cycle backward gives a refrigerator with efficiency,
R =
QC
QC
TC
=
=
.
W
QH QC
TH TC
(188)
The efficiency of a heat pump is,
H =
QH
QH
TH
=
=
.
W
QH QC
TH TC
(189)
The Clausius equation:
I
dQ
0.
(190)
T0
Here, T0 is the temperature of
I M P L E M E N T A T I O N
Supporting
Math
in the
AVID Elective
Teacher Guide
Dedicated in Loving
Memory to my Father,
Walter Bugno
You were the first to instill in me the
importance of education, that it is not
laws, but education that truly frees
human
From here, we see the expressions for the operators x
and p,
x
= x,
p =
h
.
i
(224)
(225)
For a general operator Q,
Z
hQi =
h|Q|i.
d3 x Q
(226)
Note that this 3-dimensional formalism simplifies to the 1-dimensional case in
the obvious waysimply replace
where is the angle with respect to the direction of propagation. The total
power is,
|2
2 q 2 |~x
.
(328)
P =
3 c3
The Auger effect describes the process by which an electron from a higher
energy level drops into a lower energy level and kicks another el
Renormalization group theory: Its basis and formulation in statistical physics*
Michael E. Fisher
Institute for Physical Science and Technology, University of Maryland, College Park,
Maryland 20742
The nature and origins of renormalization group ideas in
Elementary Particle Physics Lecture Notes 2013-14
Bobby Samir Acharya
March 4, 2014
1
Resources
Books:
Halzen-Martin: Quarks and Leptons: An Introductory Course in Modern Elementary Particle
Physics
and
Kane: Modern Elementary Particle Physics
and
Thompso
APS DPB
Division of Physics of Beams
Edited by Sam Posen
Assistant editor: Ernie Malamud
Newsletter 2015
In this issue:
Chairs Report . 2
From the Editor. 2
Twenty-Fifth Anniversary of the Founding of the DPB . 2
APS-DPB Amendments to Bylaws . 3
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Quantum information in a nutshell ()
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