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Time Dependent Perturbation Theory Homework In Spanish

Course Texts

  • Griffiths, D. J. Introduction to Quantum Mechanics. 2nd ed. Pearson, 2014. ISBN: 9789332535015. (Required)
  • Cohen-Tannoudji, C. Quantum Mechanics. Vol. 2. Wiley-VCH, 1991. ISBN: 9780471164357. (Strongly Recommended)
  • Shankar, R. Principles of Quantum Mechanics. Springer, 2013. ISBN: 9781461576754. (Strongly Recommended)
  • Sakurai, J. J. Modern Quantum Mechanics. Addison Wesley, 1993. ISBN: 9780201539295.
  • Feynman, R. The Feynman Lectures on Physics. Vol. 3. Paperback, 2003. ISBN: 9788131792131.
  • Ohanian, H. Principles of Quantum Mechanics. Prentice Hall, 1989. ISBN: 9780137127955.

Schedule and Reading List

The first third of the course covers review material from 8.05 Quantum Mechanics II, including wave mechanics, energy eigenstates, the variational principle, Stern Gerlach, spin 1/2, operators and spin states, vector spaces and operators, Dirac's bra-ket notation, x and p basis states, the uncertainty principle and compatible operators, the quantum harmonic oscillator, coherent states, two state systems, multiparticle states and tensor products, angular momentum, central potentials, and addition of angular momentum. The second two thirds of the class is summarized below.

Time-independent perturbation theory

Lecture Notes, Chapter 1

[Griffiths] Chapter 6

[Cohen-Tannoudji] Chapter XI(including Complements A-D)

[Cohen-Tannoudji] Chapter XII

  • Time-independent perturbation theory for degenerate states: Diagonalizing perturbations and lifting degeneracies
  • Time-independent perturbation theory for nondegenerate states: Energy and wavefunction perturbations through second order
  • Degeneracy reconsidered
  • Simple examples: Perturbing a two-state system, a simple harmonic oscillator, and a bead on a ring
  • The fine structure of hydrogen, revisited: Relativistic and spin-orbital effects
  • The hydrogen atom in a magnetic field, revisited: The Zeeman effect
  • The hydrogen atom in a electric field: The Stark effect
  • Van der Waals interaction between neutral atoms
The Semi-classical (or WKB) approximation

Lecture Notes, Chapter 1

[Griffiths] Chapter 8

  • Form of wave functions in classically allowed and classically forbidden regions
  • Handling turning points: Connection formulae
  • Tunnelling
  • Semiclassical approximation to bound state energies
The adiabatic approximation and Berry's phase

Lecture Notes, Chapter 2

[Griffiths] Chapter 10

  • The Born-Oppenheimer approximation and the rotation and vibration of molecules
  • The adiabatic theorem
  • Application to spin in a time-varying magnetic field
  • Berry's phase, and the Aharonov-Bohm effect revisited
  • Resonant adiabatic transitions and The Mikheyev-Smirnov-Wolfenstein solution to the solar neutrino problem
Time-dependent perturbation theory

Lecture Notes, Chapter 2

[Griffiths] Chapter 9

[Cohen-Tannoudji] Chapter XIII

  • General expression for transition probability; Adiabatic theorem revisited
  • Sinusoidal perturbations; Transition rate
  • Emission and absorption of light; Transition rate due to incoherent light; Fermi's Golden Rule
  • Spontaneous emission; Einstein's A and B coefficients; How excited states of atoms decay; Laser

Lecture Notes, Chapter 2

[Griffiths] Chapter 11

[Cohen-Tannoudji] Chapter VIII

  • Definition of cross-section \(\sigma\); and differential cross section \(\sigma/ \Omega\); General form of scattering solutions to the Schrodinger equation, the definition of scattering amplitude \(f\), and the relation of \(f\) to \(d\sigma/d\Omega\); Optical theorem
  • The Born approximation: Derivation of Born approximation to \(f\); Application to scattering from several spherically symmetric potentials, including Yukawa and Coulomb; Scattering from a charge distribution
  • Low energy scattering: The method of partial waves; Definition of phase shifts; Relation of scattering amplitude and cross section to phase shifts; Calculation of phase shifts; Behavior at low energies; Scattering length; Bound states at threshold; Ramsauer-Townsend effect; Resonances.
Density Operators

Lecture Notes, Chapter 3

[Sakurai] Chapter 3.4

[Cohen-Tannoudji] Complements EIII and FIV

  • Pure and mixed states
  • Spin-\(1/2\) density operators
  • Partial trace
  • Generalized measurements and quantum operations
  • Thermal states
  • Decoherence
Introduction to the quantum mechanics of identical particles

Lecture Notes, Chapter 4

[Griffiths] Chapter 5.1, 5.2

[Cohen-Tannoudji] Chapter XIV

  • N-particle systems: Identical particles are indistinguishable
  • Exchange operator, symmetrization and antisymmetrization
  • Exchange symmetry postulate: Bosons and fermions
  • Pauli exclusion principle: Slater determinants; Non-interacting fermions in a common potential well
  • Exchange force and a first look at hydrogen molecules and helium atoms
Degenerate Fermi systems

Lecture Notes, Chapter 4

[Griffiths] Chapter 5.3

[Cohen-Tannoudji] Chapter XI Complement F

  • Fermions in a box at zero temperature: Density of states; energy; degeneracy pressure
  • White dwarf stars: Equation of state at \(T = 0\); Chandrasekhar limit; neutron stars
  • Electrons in metals: Periodic potentials; Bloch waves; introduction to band structure; metals vs. insulators
Charged particles in a magnetic field

Supplementary notes

[Griffiths] Section 10.2.3 (Aharonov-Bohm effect)

[Cohen-Tannoudji] Chapter VI Complement E

  • Canonical quantization
  • The classical Hamiltonian for a particle in a static magnetic field
  • The Schrodinger equation for a charged particle in a magnetic field, via canonical quantization
  • Gauge invariance
  • Landau level wave functions. Counting the states in a Landau level
  • De Haas-Van Alphen effect
  • Integer Quantum Hall Effect: Introduction to the ordinary Hall effect; Quantum mechanical problem of a particle in crossed magnetic and electric fields; Calculation of Hall current due to a single filled Landau level; From this idealized calculation to real systems: The role of impurities.
  • The Aharonov-Bohm effect
Quantum Computing and quantum information

Lecture Notes, Chapter 5

  • Using many two-state systems as a quantum computer
  • Grover algorithm
  • Simon's algorithm


53755 Special course in cosmology / Kosmologian erikoiskurssi

Lecturer: Hannu Kurki-Suonio, office hour Mo 10-11, C328
Assistant: Anna-Stiina Suur-Uski, C329

Lectures: Mo 14-16 and Th 12-14 (Physicum A315)
Exercises: Fr 10-12 (D106 in period I, D114 in period II)
The grades were reported to the office on Dec 23rd, 2015.

The first lecture is on Thursday, Sep 3rd. The last lecture on Thursday, Dec 10th.

This is an advanced course on cosmological perturbation theory. Cosmological perturbation theory is the tool to study and understand the origin and evolution of structure (like galaxies and their clustering) in the universe, and lies at the heart of modern cosmology. If you plan to do research in cosmology, you should learn it.

Prerequisites (recommended): Cosmology I+II, General Relativity. Knowledge of general relavity is essential for being able to understand the course. If you haven't taken Cosmology I+II, you could read Chapters 2, 3, 4 (luminosity distance not needed), 10 and 11 of my Cosmology lecture notes, available at the bottom of this page. (We'll be redoing Chapter 11 material at a deeper level in this course.) Alternatively, you can read the more recent Cosmology I+II lecture notes by Syksy Räsänen; note that he has different chapter numbering.
Tentative contents: Cosmological perturbation theory. Scalar, vector, and tensor perturbations. Gauges. Newtonian gauge and synchronous gauge. Adiabatic and isocurvature perturbations. Initial conditions. Primordial power spectra. Transfer functions. Generation and evolution of perturbations during inflation. Multi-field inflation. Observables. Dependence of cosmological parameters. CAMB and CosmoMC.
The course does not follow any textbook. Lecture notes (in English) will be made available.
Exams and grades: The grade is based entirely on the homework. There is no other way to pass the course than doing the homework in time. Instead of an exam there will an additional homework to problem set to be done after the lectures have ended.
Some literature:
[1] V.F. Mukhanov, H.A. Feldman, and R.H. Brandenberger: Theory of Cosmological Perturbations, Phys. Rep. 215, 203 (1992).
[2] A.R. Liddle and D.H. Lyth: Cosmological Inflation and Large-Scale Structure (Cambridge University Press 2000), Chapters 14 and 15. Check the errata!
[3] C.-P. Ma and E. Bertschinger: Cosmological Perturbation Theory in the Synchronous and Conformal Newtonian Gauges, ApJ 455, 7 (1995). You can get it from NASA ADS.
[4] A.R. Liddle and D.H.Lyth: The Cold Dark Matter Density Perturbation, Phys. Rep. 231, 1 (1993).
[5] C. Gordon: Adiabatic and entropy perturbations in cosmology, Ph.D. thesis, Univ. of Portsmouth, astro-ph/0112523.
[6] S. Dodelson: Modern Cosmology (Academic Press 2003). Errata. (In the reference library)


This course was last lectured in fall 2010. This year the course will be very similar, but there will be some additional material. An updated version of the 2010 lecture notes is available below. They will be further updated during the course.

Lecture notes:

Cosmological Perturbation Theory, part 1, 28.11.2015 version

In Chapter 21 I take some results from my 2007 CMB Physics course.

My old notes about the late-time evolution of the small scale perturbations (based on the book by S. Dodelson, Chapter 7) from my 2004 CMB Physics course are below. This material is now included in the above, but not yet the figures; so look at the figures here:
M1. Prelude (hand, 7 pages, 154 KB pdf)
M2. Large Scales (hand, 4 pages, 83 KB pdf)
M3. Small Scales (hand, 7 pages, 139 KB pdf)
M4. Transfer Function (hand, 2 pages, 44 KB pdf)

This year we did not have time to discuss tensor perturbation, but I attach my hand-written notes on them from 2007:
T1. Einstein Equations
T2. Evolution in a Matter-Dominated Universe
T3. Evolution in the Radiation-Dominated Universe
T4. Radiation+Matter Universe and the Transfer Function
T5. Power Spectrum
T6. The Effect of Late-Time Acceleration (Vacuum Energy)

Cosmological Perturbation Theory, part 2, 31.12.2015 version

Homework problem sets:

Homework 1 due Wed, Sep 16th
Homework 2 due Wed, Sep 23rd
Homework 3 due Wed, Oct 7th
Homework 4 due Wed, Oct 14th
Homework 5 due Wed, Oct 28th
Homework 6 due Wed, Nov 4th
Homework 7 due Wed, Nov 11th
Homework 8 due Wed, Nov 25th
Homework 9 due Wed, Nov 25th
Homework 10 due Wed, Dec 2nd
Homework 11 due Wed, Dec 9th
Homework 12 (the last one!) due Wed, Dec 16th
--> Return your solutions to Anna-Stiina by Wednesday evening. Put them in her mailbox, at the front end of the 3rd floor C corridor.

Codes to calculate primordial power spectra:


Cosmology I+II lecture notes:

Lecture 1
Lecture 1
Lecture 2
Lecture 3
Lecture 4
Lecture 5
Lecture 6
Lecture 7
Lecture 8
Lecture 9
Lecture 10
Lecture 11
Lecture 12

Last updated: December 31st, 2015.

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