
Atom och molekylfysik FYSC11  HT 2015
Date of the reexam:
Teachers
Lecturers:
Course information and updated scheduleFor the schedule, please refer to Physics 3 page. Course information, including reading suggestions (updated 26 August 2015)
Compulsory handin problems
First handin problem sheet, due on 10/9, noon
Exams from earlier semesters
CompendiumCompendium, covering QM intro and hydrogen atom
ExamThe written exam will take place on Monday, 26 October 2015, from 2 to 7 p.m. 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 literatureThe most straightforward way of finding an ebook from Springer is to log in with your StiL ID and to then directly go to Springer ebooks.
Lectures
Tuesday, 1 September 2015
Goals of the first week: We managed with quite a lot of the list above, although not all. The explicit construction of the spin states is of interest to see the reasoning behind how phases are chosen, but otherwise not essential. You can read about it in the compendium. In constrast, the commutator relations for the spin are highly important, and we will work on them tomorrow.
Reading for tomorrow:
Task for tomorrow:
Wednesday, 2 September 2015
Goals of today: Today we practised Dirac notation quite a lot, which implied that we otherwise made it up to the position and momentum basis of the quantum mechanical state space. We derived the very important commutator relations for the components of spin. We'll go to translation, momentum, and time evolution tomorrow. If we manage we'll also get started on the hydrogen atom, which is a prime example for orbital angular momentum. The reading instructions for tomorrow have changed, since we didn't make it that far, and so has the task for tomorrow.
Reading for tomorrow:
Task for tomorrow:
Thursday, 3 September 2015
Goals of today:
Handin problem sheet 1:
Reading for tomorrow:
Task for tomorrow:
Friday, 4 September 2015
Goals of today:
Reading for tomorrow:
Task until Monday:
Monday, 7 September 2015
Goals of today:
Reading for tomorrow:
Task until Tuesday:
Tuesday, 8 September 2015
Goals of today:
Reading for tomorrow:
Task until tomorrow:
Wednesday, 9 September 2015
Goals of today:
Reading for tomorrow: Here is the powerpoint from the lecture.
Task until tomorrow:
Thursday, 10 September 2015
Goals of today: Here is the powerpoint from the lecture.
Reading for tomorrow:
Handin problem sheet 2:
Tuesday, 15 September 2015
Goals of today: In the second half of the lecture we introduced the permutation , symmetrizer , and antisymmetrizeroperator. Using this we discussed the symmetry of wavefunctions and the required symmetry of bosons and fermions leading. Pauli's exclusion principle for bosons was also briefly discussed. At the end we derived the spin states of a two electron system and their eigenvalues of S^{2} and S_{z}.
Reading for tomorrow: B. H. Bransden and C. J. Joachain 7 (7.1  7.4)
Task until tomorrow: b) Calculate the following eigenvalues: S^{2}  > S_{z}  > S_{z}χ_{+} S_{z}χ_{} c) Imagine that you have a 3 electrons in a 3 state system with k'>, k''>, k'''>. Consider the following states: k'>k''>k'''> k'>k'''>k''> k''>k'>k'''> k''>k'''>k'> k'''>k'>k''> k'''>k''>k'> Show and discuss why the individual states shown above are forbidden. What linear combination of the states above is allowed?
Wednesday, 16 September 2015Today we discussed the independent particle model and the central field approximation to the He atom. The energy levels and the wave functions obtained from this very simple model were discussed. We did not manage to included the electronelectron repulsion in the energy of the ground state of He using perturbation theory and the variation principle, so I will start with this tomorrow.
Reading: B. H. Bransden and C. J. Joachain 7 (today we focused on 7.3  7.5)
Handin problem sheet 3:
Task until tomorrow: 2^{1}P 3^{3}S 4^{3}D 4^{3}S
Thursday, 17 September 2015Today we included the electronelectron repulsion in the energy of the ground state of He using perturbation theory and the variation principle. When we added perturbation theory and the variation principle to the independent particle model I tried to show you the basic mathematical tricks you need to do. We discussed perturbation theory and the variation principle directly leads to the central field approximation postulated last time.In the last part of the lecture we discussed the central field approximation for many electron systems. We also introduced Slater determinants and showed how they can be used to construct correctly symmetrized wavefunctions using the single electron solutions from the central field approximation.
Reading: B. H. Bransden and C. J. Joachain 8
Task until tomorrow:
Friday, 18 September 2015Today we first looked at the task given yesterday. We used Slater determinants (or the sum of them) to find the correctly symmetrized wavefunctions for the 2^{3}S state of Helium. Then we continued with filling of electrons in energy levels (aufbau principle), electronic configurations, and their possible terms. Finally, we discussed the energy levels of different terms of identical electronic configurations within the LS coupling.
Reading: B. H. Bransden and C. J. Joachain 8
Task until Monday the 5th of October:
LabsPlease find the signup sheet here. If there are any problems, please contact Joachim immediately. Monday, 28 September 2015Today a short summary of the course until now was given.
Task until Monday 5th of October:
Monday, 5 October 2015Today we first finished the tasks given at Friday the 18th of September.
Handin problem sheet 3:
Monday, 12 October 2015Due to the closure of the University today, the lecture from 10 to 12 a.m. obviously had to be cancelled. Please find below what we intended to do and read by yourselves. We will offer an additional meeting for your questions on these topics Thursday the 15th of October 15.15  16.30 in room XXX. Reading:1. Hydrogen atom: Lamb shift. Reading e.g. Demtröder (2010), section 5.7.3 or Bransden & Joachain (2003), section 5.2. 2. jj coupling (e.g. pp. 413  416 in Bransden & Joachain (2003) or Demtröder (2010) pp. 228  231. Tasks until Thursday the 15th of October: 3. Questions 4a and 4b about the Ce atom in the exam October 2014 (Exam October 2014). Setching of the microstate table and finding the term with the lowest energy (Hint: Hunds rule and the Landé interval rule). Energy of the other terms according to the same rules. 4. Consider the electron configuration ns^{1}n's^{1}. a) What are the possible terms assuming LS coupling? b) What are the possible terms written in the form (j_{1}, j_{2})J assuming JJ coupling? 5. Now consider two electrons in a sorbital but with the same n. The electronic configuration of this state ns^{2}. a) What are the possible terms assuming LS coupling? b) What are the possible terms assuming JJ coupling? 6. Return to the example of an np^{2} electronic configuration. a) Make the microstate table for this electronic configuration and show that the only ^{1}S, ^{3}P, and ^{1}D terms are allowed. b) What is the symmetry of the spin and spatial part of the wavefunction of each of these terms? Is this consistent with your knowledge of the symmetry requirements of Fermions? c) Use the microstate table to argue why the 3D term is forbidden. d) Use your answer of b together with the symmetry requirement for Fermions to argue why the 3D term is forbidden. 7. Consider the heavy Pb atom. a) What is the electron configuration of Pb? b) Find the allowed terms assuming JJ coupling. c) Compare with the situation if LS coupling was valid. 8. Assuming that jj coupling holds, list the possible terms (j_{1}, j_{2})_{J} of an np nd electronic configuration.
Tuesday, 13 October 2015We started the discussion of molecular structure. Using the i) Bornoppenheimer approximation (or adiabatic approximation), ii) the independent electron model (or orbital approximation), iii) and the linear combination of atomic orbitals (LCAO method) we calculated the wavefunctions and energies of the H_{2}^{+} ion. Using these molecular orbitals we looked at simple diatomic molecules like H_{2}^{+}, H_{2}, He_{2}^{+}, He_{2}. Finally, we had a quick look at the molecular orbitals of larger diatomic molecules such as O_{2} and N_{2}.
Reading:
Tasks until Wednesday the 14th of October: ψ_{el}> = N[c_{a}ψ_{1s}(r_{a})+c_{b}ψ_{1s}(r_{b})] Assume that the wavefunctions are real and show that: E_{el}=<ψ_{el}Hψ_{el}>=N^{2}[c_{a}^{2}H_{aa}+c_{b}^{2}H_{bb}+2c_{a}c_{b}H_{ab}] (1)
H_{aa}=<ψ_{1s}(r_{a})Hψ_{1s}(r_{a})> Require that ψ_{el}> is normalized and show that: N^{2}=[c_{a}^{2}+c_{b}^{2}+2c_{a}c_{b}S_{ab}]^{1} (2)
S_{ab}=<ψ_{1s}(r_{a})ψ_{1s}(r_{b})> Substitute (2) into (1) and take the derivative of E_{el} with respect to c_{a} and set this equal to zero to find the value of c_{a}. Hint: After taking the derivative with respect to c_{a} you should refind the expression for E_{el} (1). Show that : c_{a}H_{aa}+c_{b}H_{ab}E_{el}c_{a}E_{el}c_{b}S_{ab}=0 (3)
Analog by taking the derivative of E_{el} with respect to c_{b} and set this equal to zero you should get that: c_{b}H_{bb}+c_{a}H_{ab}E_{el}c_{b}E_{el}c_{a}S_{ab}=0 (4)
Write the secular equations (3) and (4) as a matrix times the vector of c_{a} and c_{b} and require that the matrix times this vector is equal to the zero vector. Set H_{bb}=H_{aa} and find the nontrivial solutions by requiring that the determinant of the matrix is equal to 0. Show that:
E_{+}=[H_{aa}H_{ab}]/[1S_{ab}] (5) or Substitute (5) and (6) back into (3) and (4) and show that:
c_{a}=c_{b} or
ψ_{}=(22S_{ab})^{1/2}[ψ_{1s}(r_{a})ψ_{1s}(r_{b})] (7) Sketch the wavefunctions (7) and (8) and explain for yourself why H_{2} is stable while He_{2} is unstable.
Wednesday, 14 October 2015Today we started by discussing the rigid rotor and molecular rotations. When we solved the Schrödinger equation for the rotation we found that its form is very similar to the spherical part of the Schrödinger equation for the hydrogen atoms. We took advantage of this similarity and used what we previously learned for the hydrogen atom when we solved the Scrödinger equation for the rigid rotor.In the second half of the lecture we looked at molecular vibrations in the Morse potential. We showed the solution to the Schrödinger equation for molecular vibrations in the Morse potential (without any proof). Finally, we discussed how the shape of the Morse potential (given by D_{e}, D_{0}, & alpha;) can be found in experiments. Finally, we started to discuss Raman spectroscopy. We will continue with this tomorrow.
Reading:
Thursday, 15 October 2015Today a classical description of vibrational Raman spectroscopy was given by looking at at the Taylor expansion of an electric dipole.In the second half of the lecture we looked at lineshapes and discussed experimental broadening, natural broadening, doppler broadening, and pressure broadening.
Reading:
Monday, 19 October 2015Today we started by comparing the jj and LS coupling for the electronic configuration of Pb. Please make sure that you are able find the Pauli allowed microstates for a electron configuration both with LS and jj coupling and deduce the term values from the microstates. It is also important that you understand the difference between nonequivalent electrons and equivalent electrons and how it affets the determination of possible terms. If you are unsure please ask! In the second half of the lecture today we looked at centrifugal distortion and the interaction between rotations and vibrations and we arrived at formulae 9.113b in Demtröder. Finally, we discussed vibrationalrotational transitions.
Reading:
Tasks until Tuesday the 20th of October:
Wednesday, 21 October 2015, 15.1517, DSalenHere we plan to go through the exam from 26 October 2012 (Exam October 2012). You will also get a chance to ask questions to the material we covered in the course.
Results of the exam of 26 October 2015:
VG: Väl godkänd/Pass with distinction, G: Godkänd/Pass, U: Underkänd/Fail ^{*}: rest VG: possibility of obtaining a pass with distinction mark by oral exam, contact me if interested Tentavisning: possibility of "examining" your exam on Wednesday, 18 November 2014, 3.00 p.m., in lecture hall F (room number K404).
Date of the reexam:
