
Atom och molekylfysik FYSC11  HT 2014
ReexamThe reexam 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 scheduleFor the schedule, please refer to Physics 3 page. Course information, including reading suggestions (updated 28 August 2014)
Compulsory handin problemsFirst handin problem sheet, due on 11/9, noon Second handin problem sheet, due on 15/9, noon Third handin problem sheet, due on 22/9, 10 a.m. Fourth handin problem sheet, due on 6/10, 12 a.m.
Exams from earlier semesters
CompendiumCompendium, covering QM intro and hydrogen atom
ExamThe written exam will take place on Monday, 27 October. As a preparation for the exam, you should look at the following things:
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 handin 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 / n^{2}).
Suitable literature
Compulsory reading, exercises for following lecture, material from lecturesMonday, 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 twobody problem. In particular, we discussed the angular momentum constants of motion and the centrifugal barrier. We then looked at the predictions of the semiclassical 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 SternGerlach experiment and its outcome, followed by a consideration of sequential SternGerlach 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 SternGerlach 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 handin 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 handin problem) by considering the tasks on page 5961 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 m_{l} quantum numbers can be found from applying the ladder operators to the the wave function for l and m_{l}=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 R_{n,l}(r) as well as the radial probability densities r^{2} R_{n,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 <2,0,0ez1,0,0>=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 ldegeneracy 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. Reading for tomorrow: Fine structure in the hydrogen atom and Lamb shift, e.g. Demtröder sections 5.6 (you may leave out the Einsteinde 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 spinorbit 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 m_{j}. 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 socalled ClebschGordon coefficients. Second, we discussed the Lamb shift, which is an important correction to the energies found after nonrelativistic treatment with relativistic corrections. The Lambshift, 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 1s_{1/2} state and still sizeable for the 2s_{1/2} and 2p_{1/2} states. The third handin 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 m_{l} 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 handin problem we started with the treatment of the helium atom, for which the electronelectron 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 twoelectron 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 StilID 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 m_{S} quantum numbers for the different spin wave functions. Then we made a rather poor approximation by introducing the electronelectron 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: LScoupling (e.g. Demtröder (2006) 6.5). Task for tomorrow: Evaluate all the S and m_{S} 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 m_{S} 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 m_{S}=1,0,1. The higher energy solution is a singlet with S=0 and m_{S}=0. We also discussed the LScoupling scheme for a pdconfiguration. Reading for Monday: As a preparation for the labs reread about the Zeeman effect and about the FabryPerot interferometer.
Monday, 22 September 2014 Topics of the day were the LScoupling 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 LScoupling, we used a counting scheme for finding the allowed terms for a p^{2} configuration.
Wednesday, 24 September 2014 No lecture, but you'll find the fourth handin 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 jjcoupling scheme. For those students who have not secured 70% in average on the previous handin problems, here is a fifth handin problem sheet. It is recommended, but not compulsory, for everybody to solve it.
Tuesday and Wednesday, 1415 October 2014 The Einstein coefficients were discussed as well as the three and fourlevel 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.
Results:
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. Reexam: A reexam will be offered in January, date to be determined (probably in week 3).
