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Atom- och molekylfysik FYSC11 - HT 2014



The re-exam will take place on 7 February 2015 from 8 a.m. to 1 p.m. at Sparta. You need to register if you want to sit the exam. Please make sure to check out the homepage a day or two before the exam to make sure that the room has not been changed.

Course information and updated schedule

For the schedule, please refer to Physics 3 page.

Course information, including reading suggestions (updated 28 August 2014)

Compulsory hand-in problems

First hand-in problem sheet, due on 11/9, noon

Second hand-in problem sheet, due on 15/9, noon

Third hand-in problem sheet, due on 22/9, 10 a.m.

Fourth hand-in problem sheet, due on 6/10, 12 a.m.

Fifth hand-in problem sheet.


Exams from earlier semesters



Compendium, covering QM intro and hydrogen atom


The written exam will take place on Monday, 27 October.

As a preparation for the exam, you should look at the following things:

  • hand-in problems,
  • labs,
  • practice questions below,
  • reading according to the List of subjects and reading suggestions (you *don't* have to read all the books, choose a book that suits you - but remember that you (in principle) have to cover all topics),
  • homework according to this page.

In the exam, there will be exercises in which you will be asked to describe a topic (experiment, outline of a theory, physical models, etc.). There will also be exercises where you will have to calculate, similar to what you have done in the hand-in problems, and there will be exercises in which you will have to interpret physical data. No detailed derivation will be required (e.g. of the solution to the Schrödinger equation of the hydrogen atom), but you should know the starting point (e.g. Schrödinger equation), ansatz (e.g. separation of wave function into a radial and an angular part), and principal solution (e.g. that the solution to the angular Schrödinger equation of the hydrogen atom are the spherical harmonics and that the hydrogen energy is -13.6 eV / n2).

Suitable literature

  • W. Demtröder, Atoms, Molecules and Photons, Springer, 2006, available as e-book from the University Library
  • S. Andersson, F. Bruhn, J. Hedman, L. Karlsson, S. Lunell, K. Nilson, & J. Wall, Atom- och molekylfysik, Uppsala universitet
  • H. Haken & H. C. Wolf, The Physics of Atoms and Quanta, Springer, 2005 (or earlier editions), available as e-book from the University Library
  • B. H. Bransden and C. J. Joachain, Physics of Atoms and Molecules, 2nd edition, Prentice Hall , Harlow, 2003
  • A. Thorne, U. Litzén, & S. Johansson, Spectrophysics

Compulsory reading, exercises for following lecture, material from lectures

Monday, 1 September 2014

We had a quick look at the quantum mechanical concepts that you already know (see compendium). We also started to have a look at the hydrogen atom from a classical perspective. We'll continue with this tomorrow, turn to experimental observations, and a quick discussion of the Bohr model. After that it is time to get started with angular momentum, which is one of the main topics of this first week.

Reading for tomorrow: chapter 2 in the compendium.

Task for tomorrow: Using the infinite nucleus mass approximation, calculate the wavelengths (in nm) and energies (in eV) of the first four lines of the Lyman, Balmer, and Paschen series for hydrogen and He+.

Tuesday, 2 September 2014

We went through some of the aspects of the classical two-body problem. In particular, we discussed the angular momentum constants of motion and the centrifugal barrier. We then looked at the predictions of the semi-classical Bohr model of the hydrogen atom and how it related to absorption and emission of radiation. We identified the wavelength regimes of the Lyman, Balmers, and Paschen series. Moseley's law was discussed briefly. After that we discussed the setup of the Stern-Gerlach experiment and its outcome, followed by a consideration of sequential Stern-Gerlach experiments. This discuss forms the basis for providing a description of spin in terms of quantum mechanical states.

Questions and slides from lecture 2.

Reading for tomorrow: pages 21 to 39 in the compendium.

Task for tomorrow: answer the question and do the task on page 33 of the compendium.

Wednesday, 3 September 2014

We discussed the Stern-Gerlach experiment once more, this time in terms of measurement probabilities. Then we practiced using the following questions and tasks: Questions for lecture 3.

Reading for tomorrow: pages 40 to 50 in the compendium.

Task for tomorrow: Finish the tasks in the Questions and tasks from lecture 3 file.

Thursday, 4 September 2014

The first hand-in problem sheet was given out. You'll find it here.

We discussed the solutions of the Questions and tasks from lecture 3.

Then we compared discrete and continuous bases of state space. A particularly important relationship is the closure relation, which expresses the completeness of a basis of state space. Then we considered commutating observable, for which we found that they have simultaneous eigenstates. A maximum set of commuting observables gives the necessary means to completely label any eigenstate on the quantum mechanical vector space on which these observables act.

Reading for tomorrow: pages 50 to 57 in the compendium.


Friday, 5 September 2014

Although not strictly relevant to this course, we went through what it entails that translation (induced by momentum) and position measurement do not commute: this leads to the uncertainty of position and momentum, which also can be expressed by stating that momentum and position do not commute. In quite much the same way, we look at time evolution (which is generated by the energy) and found the Schrödinger equation from some basic statements, namely that of probability conversation, combination of subsequent time evolutions, and on zero time evolution.

We then started to solve the Schrödinger equation for the hydrogen atom. The motion was separated into angular and radial variables. In the treatment of the radial variables orbital angular momentum plays an important roles, and we started looking at some more properties (besides those treated earlier an in the hand-in problem) by considering the tasks on page 59-61 of the compendium.

Reading for Monday: the remaining pages of the compendium.

Tasks for Monday: solve the exercises on pages 59 and 60 of the compendium.


Monday, 8 September 2014

We went on with solving the problems on pages 59 to 61 of the compendium. Then we considered the full solution to the angular Schrödinger equation of the hydrogen atom and found that the wave functions for the different l and ml quantum numbers can be found from applying the ladder operators to the the wave function for l and ml=l. These wave functions are the spherical harmonics and we discussed how they can be depicted. In addition we discussed the "real spherical harmonics" for the p and d orbitals.

Reading for tomorrow: Demtröder 7.1.2 and 7.2 (optical transitions: matrix elements and selection rules) or accordingly in other books.

Tasks for tomorrow: solve the remaining problems on page 61 of the compendium.


Tuesday, 9 September 2014

After solving the final problems on page 61 of the compendium, we went on with considering the radial Schrödinger equation of the hydrogen atom and its solutions, which can be calculated analytically after making a power series expansion ansatz. We had a look at the radial wave functions Rn,l(r) as well as the radial probability densities r2 Rn,l(r). Further, we compared the radial wave functions to those of the doubly ionised Li atom.

Slides and questions from the lectures on Monday and Tuesday

Reading for tomorrow: Demtröder 6.3 (alkali atoms) or corresponding passages in different book.

No task for tomorrow.


Wednesday, 10 September 2014

We discussed the parity of wave functions in general and of the hydrogen waves functions in general (the parity of the total hydrogen wave functions is (-1)l). We then looked at optical transitions (absorption and emission of electromagnetic radiation in/from the hydrogen atom) and derived the transition matrix element and therefore the probability for a transition in the electric dipole approximation. Further, we derived the parity selection rule (for a transition to be allowed the parity of the two involved wave functions needs to be opposite). Finally, we discussed the dipole selection rules, which give more stringent rules for which transitions are allowed. You should memorise these rules!

Questions on parity from today

Reading for tomorrow: You should get started with reading on Fine structure, e.g. sections 5.3 and 5.4 in Demtröder.

Task for tomorrow: Show explicitly that <2,0,0|-ez|1,0,0>=0, i.e. that transitions between the 1s and 2s state of hydrogen are not allowed.


Thursday, 11 September 2014

We looked at hydrogenic atoms, with the alkali atoms as the most important examples. The wave functions for the valence electrons in these atoms are described by energies and wave functions which are very similar to those of hydrogen, but the l-degeneracy of the states for a particular n is lifted. The reason is that the penetration into the filled shell of inner electrons is larger the smaller the l, we implies that wave function with lower l are stabilised in comparison to wave functions with larger l. Here we had an interesting discussion about what it means if we say that a system becomes more stable, which implies that the (total) energy of the systems becomes smaller. Further, we started discussing the fine structure in the spectrum of the hydrogen atom, starting with the corrections due to the relativistic mass increase and the Darwin term, which is due to the position of the electron being smeared out a little bit, which leads to a modification of the potential for l=0.

Slides from today's lecture

Reading for tomorrow: Fine structure in the hydrogen atom and Lamb shift, e.g. Demtröder sections 5.6 (you may leave out the Einstein-de Haas effect, but consider equations 5.52 to 5.53) and 5.8.3 (these are for the 2006 edition of Demtröder; for the 2010 edition the sections are 5.5 and 5.7.3).

Task for tomorrow: Evaluate the expectation value <δ(r)> which you find in the Darwin term hamiltonian.


Friday, 12 September 2014

We went on discussing the fine structure in the hydrogen atom and considered in particular the spin-orbit coupling term. For this we had to consider the spin of the hydrogen atom. Spin and orbital angular together form the total angular momentum, with corresponding quantum numbers j and mj. When all relativistic effects are taken into account, the fine structure does not depend on l, but only on j, although the independent terms do depend on l.


Monday, 15 September 2014

Two topics were discussed today: first the general theory of addition of two angular momenta, which leads to the so-called Clebsch-Gordon coefficients. Second, we discussed the Lamb shift, which is an important correction to the energies found after non-relativistic treatment with relativistic corrections. The Lamb-shift, which can only be explained using quantum electrodynamics, is largest for the lowest n and lowest l. It is comparable in size to the fine structure for the 1s1/2 state and still sizeable for the 2s1/2 and 2p1/2 states.

The third hand-in problem sheet was given out. You'll find it here.

Reading for tomorrow: Helium atom (e.g. Demtröder (2006) 6.1).

Task for tomorrow: For the hydrogen atom find an l and ml for wich the spin and orbital angular momenta are strictly parallel.


Tuesday, 16 September 2014

We finished off the discussion of the hydrogen atom by considering the interaction between the nuclear spin and the electron angular momentum, which leads to the occurrence of hyperfine structure in the hydrogen spectrum. After discussing the solutions to the first hand-in problem we started with the treatment of the helium atom, for which the electron-electron interaction makes an analytic solution impossible. The fact that the two electrons cannot be distinguished from each other in a measurement leads to conditions for the two-electron wave function and the Pauli principle.

The article "A critical compilation of experimental data on spectral lines and energy levels of hydrogen, deuterium, and tritium" by A.E. Kramida contains a very interesting section on the history of hydrogen research. Log in with your Stil-ID to access the article.

Slides and questions from today's and yesterday's lectures

Reading for tomorrow: Aufbau principle (e.g. Demtröder (2006) 6.2).

Task for tomorrow: You still have to find a solution for yesterday's task ...


Wednesday, 17 September 2014

After an hour of rehearsal on angular momentum operators, eigenvalue equations, and values for the angularn momentum quantum numbers we went on with the helium atom and considered the helium atom's spin wave functions.


Thursday, 18 September 2014

We almost finished the discussion of the helium atom and considered the values of the S and mS quantum numbers for the different spin wave functions. Then we made a rather poor approximation by introducing the electron-electron interaction as a perturbation of hamiltonian. Although a poor contribution, it reproduces the general increase of the energies due to the Coulomb interaction of the two electrons and the splitting due to the presence of the quantum mechanical exchange integral, which is a result of the Pauli principle.

Reading for tomorrow: LS-coupling (e.g. Demtröder (2006) 6.5).

Task for tomorrow: Evaluate all the S and mS quantum numbers for all spin wave functions of the helium atom.


Friday, 19 September 2014

We had a look at the spin quantum numbers S and mS for the different spin wave functions. The lower energy solution, which corresponds to an antisymmetric spatial wave function and thus to a symmetric spin wave function, is a triptlet, i.e. S=1 and mS=-1,0,1. The higher energy solution is a singlet with S=0 and mS=0.

We also discussed the LS-coupling scheme for a pd-configuration.

Reading for Monday: As a preparation for the labs re-read about the Zeeman effect and about the Fabry-Perot interferometer.


Monday, 22 September 2014

Topics of the day were the LS-coupling scheme for identical electrons (i.e. electrons with the same n and l quantum numbers) and the normal (S=0) and anomalous (S different from 0) Zeeman effect. Concerning LS-coupling, we used a counting scheme for finding the allowed terms for a p2 configuration.


Wednesday, 24 September 2014

No lecture, but you'll find the fourth hand-in problem sheet here.


Monday, 6 October 2014

Lars Engström gave a lecture on atomic lifetimes. You find his powerpoint material here.


Monday, 13 October 2014

We finished off a couple of topics related to angular momentum coupling, in particular we discussed the jj-coupling scheme.

For those students who have not secured 70% in average on the previous hand-in problems, here is a fifth hand-in problem sheet. It is recommended, but not compulsory, for everybody to solve it.


Tuesday and Wednesday, 14-15 October 2014

The Einstein coefficients were discussed as well as the three- and four-level schemes for lasers. The components of a laser were discussed as well as different types of laser (ruby, HeNe, solid state lasers, dye lasers). The ruby and HeNe laser were discussed in more detail.


Thursday, 16 October 2014

The main topic of the day was the potential for a diatomic molecule and (vibrational) optical transitions in molecules.


Monday, 20 October 2014

Today, we discussed the vibrational Raman effect, Raman spectra, and the Morse potential.


Tuesday, 21 October 2014

We discussed the rigid rotor for the case of a diatomic molecule as well as the application to infrared (rotational) spectra and rotational Raman spectra.


Monday, 27 October 2014: Written exam


You will find the exam here.


1 75 G
2 14 U
3 55 G
4 49 rest*
5 29 U
6 58 G
7 60 G
8 70 G
9 77 G
10 79 rest för VG**
11 44 U
12 47 U
13 60 G
14 59 G
15 43 U
16 84 VG
17 54 G
18 53 G
19 77 G
20 65 G
21 49 rest
22 74 G
23 55 G

VG: Väl godkänd/Pass with distinction, G: Godkänd/Pass, U: Underkänd/Fail
*: rest: possibility of obtaining a pass mark by oral exam, contact me if interested
**: rest för VG: possibility of obtaining a pass with distinction mark by oral exam, contact me if interested

Tentavisning: possibility of "examining" exam on 13 November 2014, 1.00 p.m., in lecture hall D. Results will be registered in Ladok first after that date.

Re-exam: A re-exam will be offered in January, date to be determined (probably in week 3).



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