This evidence suggests pairwise associations of the histones in chromatin
but says nothing of details, such as
whether the F2A1 and F3 pair, which
occurs as an (F2Al)2(F3)2 tetramer
in solution, also occurs as a tetramer
in chromatin. The most direct evid
Electrolysis Revision
Electrolysis is the passing of a direct electric current through an ionic substance
that is either molten or dissolved in a suitable solvent, producing chemical
reactions at the electrodes and separation of materials.
The main compon
IB DIPLOMA PROGRAMME
PROGRAMME DU DIPLME DU BI
PROGRAMA DEL DIPLOMA DEL BI
N04/4/PHYSI/SPM/ENG/TZ0/XX
88046504
PHYSICS
STANDARD LEVEL
PAPER 1
Friday 5 November 2004 (afternoon)
45 minutes
INSTRUCTIONS TO CANDIDATES
Do not open this examination paper unt
Gordano: @ositioning for internationai Eananeion
Jochen Wirtz
As it looks to the future, a successful Asian retailer of casual apparel must decide whether to
maintain its existing positioning strategy. Management wonders what is stars will he critical to
778
Part2
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MANY-ELECTRON
MANY
ELECTRON
ATOMS
OUTLINE
Two-electron atom (He)
Spin and spatial part of the wavefunction
Singlet and triplet states
Many-electron atoms
The central field approximation
Spin, the Pauli exclusion principle and Slater determinants
Elec
METHODS
O SO
OF
APPROXIMATION
Time-independent perturbation theory
(non degenerate)
(non-degenerate)
Degenerate perturbation theory
Variational method
Time-dependent
Time dependent perturbation
pert rbation theory
theor
OUTLINE
VARIATIONAL METHOD
Why and
6CCP3221
QUANTUM MECHANICS II
Prof. Carla Molteni
Ph i Department,
Physics
D
t
t Kings
Ki College
C ll
London
L d
Strand London WC2R 2LS
Office S7.23 carla.molteni@kcl.ac.uk
LECTURES
(3h lectures + 1h problem class)
Monday
Tuesday
16:00 -18:00
16:00 -18
METHODS
O SO
OF
APPROXIMATION
Time-independent perturbation theory
(non degenerate)
(non-degenerate)
Degenerate perturbation theory
Variational method
Time-dependent
Time dependent perturbation
pert rbation theory
theor
OUTLINE
VARIATIONAL METHOD
Why and
METHODS
O SO
OF
APPROXIMATION
THE SCHROEDINGER EQUATION
i
H
t
Erwin Schroedinger (1926)
The g
general theoryy of q
quantum mechanics is now almost
complete. The underlying physical laws necessary for the
mathematical theory of a large part of physics a
SPIN
&
IDENTICAL
PARTICLES
C S
SPIN
SUMMARY
Orbital angular momentum
Spin angular momentum
Spin one-half systems
The Pauli spin matrices
Combining angular momenta
Combining spins
Singlet & triplet states
Combining
g spin
p and orbital angular
g
momen
METHODS
O SO
OF
APPROXIMATION
Time-independent perturbation theory
(non degenerate)
(non-degenerate)
Degenerate perturbation theory
Variational
V i i l method
h d
Time-dependent
Time dependent perturbation theory
TIME-DEPENDENT PERTURBATION THEORY
OUTLINE
THE REAL
HYDROGENIC ATOM
OUTLINE
Fine
Fi structure
t t off the
th hydrogenic
h d
i atom
t
Relativistic correction to the kinetic energy
Spin-orbit coupling
The Darwin term
Combining
g the effects
Fine structure and spectral lines
Allowed transitions
Zee
MOLECULES
RIGID ROTATOR diatomic molecule
Find the rotational energy levels of a diatomic molecule made by
two atoms of mass m1 and m2 at internuclear separation r
Rigid rotator Hamiltonian
2
2
L
L
=
=
H
2 r 2
2I
I = r 2 : moment of inertia
1 1
1
: reduc
Solutions to Problem Set 5
Physics 342
by: Callum Quigley
1
Spherical Potential Well Bound States
(i) For bound states we must have E < U0 . Defining u(r) = r(r), then for the ` = 0 states
we have
2m
2m
k 2 u(r), r < a
~2 E u(r)
00
u (r) = 2 (E U (r)u(
Solutions to Problem Set 3
Physics 342
Q 1,2 by: Rhys Povey
Q 3,4 by: Callum Quigley
1
Earths angular momentum
The angular momentum of the Earth is dependent on which reference frame you are using
and can be calculated using
2
L=I =I
.
T
Earth spin angula
Solutions to Problem Set 1
Physics 342
Q 1,2 by: Rhys Povey
Q 3,4 by: Callum Quigley
1
A Commutator Quantity
We want to determine the value of [pi , i j k [Lj , xk ]. First, lets use Lj = j a b xa pb to obtain
[Lj , xk ] = j a b [xa pb , xk ] ,
Now,
[xa p
Solutions to Problem Set 4
Physics 342
by: Callum Quigley
1
Rotations and Angular Momentum
(i)
In addition to the actions of J~2 and Jz on the |j, mi states, its helpful to recall the actions
of J = Jx iJy :
p
J |j, mi = ~ (j m)(j m + 1)|j, m 1i.
(1.1)
Th
Problem Set 3
Physics 342
Due February 4
Some abbreviations: S - Shankar.
1. Estimate the angular momentum of the earth in units of ~.
2. After discussions in office hours, it seems that clarifying the notion of a representation
would be helpful. Lets try
Problem Set 2 Solutions
Physics 342
Problem 1 by Zimu Khakhaleva-Li
Problems 2 - 4 by Callum Quigley
1(i) Trace is invariant under cyclic permutations.
= Tr(U XU ) = Tr(XU U ) = Tr(X) = 0.
Tr(X)
= (U XU ) = U X U = U XU = X.
X
is still a traceless Herm
Problem Set 2
Physics 342
Due January 28
Some abbreviations: S - Shankar.
1. At this point, you might be asking yourself: are SU (2) and SO(3) really different? You
might also wonder: why do we care about the difference? To answer these questions, let
us
Problem Set 5
Physics 342
Due February 18
Some abbreviations: S - Shankar.
1. Lets consider a spherical potential well.
(i) Determine the energy levels for a particle with angular momentum l = 0 in the spherical
potential well:
U (r) = 0 r < a,
U (r) = U0
Problem Set 1
Physics 342
Due January 21
Some abbreviations: S - Shankar.
1. As a warm up, compute the quantity [pi , ijk [Lj , xk ] with a sum over repeated indices
implied.
2. Prove the two basic relations for the symbol we used in lecture. Namely,
iab
Problem Set 4
Physics 342
Due February 11
Some abbreviations: S - Shankar.
1. Consider a particle with angular momentum j in the state |j, mi which, as usual, is an
eigenstate of J~2 and Jz :
J~2 |j, mi = j(j + 1)~2 |j, mi,
Jz |j, mi = m~|j, mi.
(i) Compu
The University of Chicago, Department of Physics
Page 1 of 8
Problem set 8 solutions [50]
Michael A. Fedderke
Standard disclaimer: while every eort is made to ensure the correctness of these
solutions, if you nd anything that you suspect is an error, plea
The University of Chicago, Department of Physics
Page 1 of 6
Problem set 7 solutions
Michael A. Fedderke
Standard disclaimer: while every eort is made to ensure the correctness of these
solutions, if you nd anything that you suspect is an error, please em
The University of Chicago, Department of Physics
Page 1 of 10
Problem set 4 solutions [40]
Michael A. Fedderke
Standard disclaimer: while every eort is made to ensure the correctness of these
solutions, if you nd anything that you suspect is an error, ple
The University of Chicago, Department of Physics
Page 1 of 9
Problem set 5 solutions [40]
Michael A. Fedderke
Standard disclaimer: while every eort is made to ensure the correctness of these
solutions, if you nd anything that you suspect is an error, plea
The University of Chicago, Department of Physics
Page 1 of 3
Problem set 2 solutions
Michael A. Fedderke
Standard disclaimer: while every eort is made to ensure the correctness of these
solutions, if you nd anything that you suspect is an error, please em