Quantum Foundations and Quantum Information: Problems Week 2
SCA
QF & QI 2014
Questions on physics
Problem 2.1: Double Slit Experiment, General Questions
In the double slit experiment how would the pattern change (qualitatively, draw the result) if the sy
QF & QI: Postulates of QM.
1
W4.3b
State
[1] McIntyre: Postulate 1) The state, including all you can know about it, is represented mathematically by
a normalized ket | .
[2] Shankar, Postulate I: The state of a system is represented by a vector | in Hilbe
Some calculus exercises to keep you sharp
SOLUTIONS
1: Some Basic Derivatives
What are the following derivatives? for this problem you may enter your answers on this sheet. (These are the
key results we will use).
1.
dex
dx
2.
d2 ex
dx2
=
3.
deix
dx
= iei
QFQI 2014
.
Quantum Mechanics: Problems Lecture 15
15.1 Density matrices
The quantum state of an object is represented in the most general manner through the density matrix, = | |.
The motivation for this form was to express the mathematical form of a mix
Ch. 3 Solutions
Assignment Week 6
Ch 3: 12, 13, 14
Ch 4: 1,2,3,4
3.12 The eigenstates and eigenvalues of H are, by inspection:
1
0 , E2
E = E1 , E2 ; E1
0
1
We need to find the eigenvectors of A:
0 a
A
a 0
Now diagonalize:
a
2
= 0 2 ( a ) = 0
Ch. 5 Solutions
Ch 5: 4,6,7,8,16,20,22,24,27,33
5.4 Assume the real solutions
E ( x ) = Asin kx + B cos kx
The boundary conditions yield
E ( a ) :Asin ka + B cos ka = 0 B cos ka = Asin ka
E ( a ) :Asin ka + B cos ka = 02Asin ka = 0or2B cos ka = 0
This
Ch. 2 Solutions
2.4 The general matrix is
a b
A
c d
The matrix elements are
+ A+ =
(1
a b 1
=
0
c d 0
(1
a
=a
0
c
+ A =
(1
a b 0
=
0
c d 1
(1
b
=b
0
d
A+ =
(
a b 1
=
0 1
c d 0
(
a
=c
0 1
c
A =
(
a b 0
=
0 1
c d 1
EPGY P055
Light and Heat
Some calculus exercises to keep you sharp
1: Some Basic Derivatives
What are the following derivatives?
1.
dex
dx
2.
dex
dx
3.
d2 ex
dx2
=
4.
de
dx
=
5.
d2 eix
dx2
6.
d cos x
dx
7.
d2 cos x
dx2
8.
d sin x
dx
9.
d2 sin x
dx2
=
2
=
Lecture 6B: Dirac formalism of Quantum Mechanics
Bra and ket vectors.
The following appendices are to provide a quick overview of some areas of mathematics that you may not
have been exposed to. The optional appendices are included for those who may want
Summer College Academy
Quantum Foundations and Quantum Information
Postulate I: Quantum States
This section combines the rst chapter of Baym and the rst chapter of Scarani. Begin weigh the classical notions of
polarization (Baym) up to representing them a
QF & QI: State vector
Administrative, technical issues.
Questions?
Classical polarization of light.
Quantum model of polarization.
Mathematical structure of states.
Born probability rule.
Bases
W3.1
QF & QI: State vector
W3.1
Maxwells EM Theory
2
Summer College Academy 2014
Q.F & Q.I.
Lecture 12B: Existence of Joint Probability Distributions -signicance
The basic notions of probability have been reviewed before and here it will be assumed you know the denition of
expectation value and correlation.
QF & QI: State vectors: Polarization
Administrative, technical issues.
Questions?
Fun with polarizers.
State vectors for polarization
Bases
W3.2
QF & QI: State vectors: Polarization
W3.2
Obtaining probabilities from the state vector
Recall that for i
10
Quantum Mechanics Lecture 11c Addendum: Mathematical details
Basic Experiments with Polarizers
We can verify the properties of the wavefunction and operators for a photon in some polarization state by examining
some simple experiments. Recall Malus law
QF & QI: State vectors: Polarization, details
Administrative, technical issues.
Questions?
Solution to previous days question
A little about circular polarization
Bases, transferring between.
Superpositions
Intrinsic randomness.
Polaroids as proje