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Atom- och molekylfysik FYSC11 - HT 2015

 

Date of the re-exam:

The re-exam will take place on Saturday, 30 January 2016, from 8.00 to 13.00 in Matteannexet 8A-B-C.

If you have any questions please come and ask!


Teachers

Lecturers:
Joachim Schnadt, Division of Synchrotron Radiation Research, joachim.schnadt@sljus.lu.se, office K516
Jan Knudsen, Division of Synchrotron Radiation Research, jan.knudsen@sljus.lu.se, office K518

Lab supervisors:
Hampus Nilsson, Department of Astronomy and Theoretical Physics, hampus.nilsson@astro.lu.se, office B207B in the Astronomy building
Shabnam Oghbaie, Division of Synchrotron Radiation Research, shabnam.ohhbaie@sljus.lu.se, office K509
Payam Shayesteh, Division of Synchrotron Radiation Research, payam.shayesteh@sljus.lu.se, office K528

Problem solving class assistant:
Bart Oostenrijk, Division of Synchrotron Radiation Research, bart.oostenrijk@sljus.lu.se, office K528

Course information and updated schedule

For the schedule, please refer to Physics 3 page.

Course information, including reading suggestions (updated 26 August 2015)


Compulsory hand-in problems

First hand-in problem sheet, due on 10/9, noon
Second hand-in problem sheet, due on 21/9, noon
Third hand-in problem sheet, due on 21/9, noon
Fourth hand-in problem sheet, due on 15/10, noon
Fifth hand-in problem sheet, due on 22/10, noon

 

Exams from earlier semesters

 

Compendium

Compendium, covering QM intro and hydrogen atom


Exam

The 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:

  • 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

The most straightforward way of finding an e-book from Springer is to log in with your StiL ID and to then directly go to Springer e-books.

  • 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


Lectures

Tuesday, 1 September 2015

Goals of the first week:
Introduction to and use of Dirac notation (makes the physicist's life easier)
Spin angular momentum of the electron: a rather straightforward example of angular momentum
Some quantum mechanical formalism
Excursion into translation and time evolution to see where fundamental relationships in quantum mechanics come from (canonical commutator relationships, replacement of linear momentum p by the first derivative with respect to position in wave function notation, Schrödinger equation)
Orbital angular momentum, using the hydrogen atom as an example


Goals of today:
Reminder of quantum mechanical concepts from FYSA21
Intro to Dirac notation
Introduction to spin angular momentum: the Stern-Gerlach experiment
Postulation of spin states
Kets, bras, and quantum mechanical operations
Explicit construction of the spin states
Commutator relations for the spin

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:
Up to page 29 of the compendium.

Task for tomorrow:
Solve the tasks on pages 13 to 15 of the slides from the lectures.


Wednesday, 2 September 2015

Goals of today:
Practice Dirac notation: tasks from yesterday
Finish off from yesterday: Commutator relations for the spin
Continuous bases: the position basis
Translation and momentum: the commutator relations for position and momentum
Time evolution: the Schrödinger equation

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:
In the compendium: Read up to page 43.

Task for tomorrow:
Show (for a spin-1/2 system such as a ground-state silver atom) that [S2,Si]=0, where i=x,y,z. This relationship is quite central since it shows that one can use both the length and z-component as labels for a spin-1/2 quantum state. Show also that S2=3/4 ℏ21, where 1 is the unit operator (it is often left out in writing).


Thursday, 3 September 2015

Goals of today:
Continuous bases: the position basis
Commuting observable
Translation and momentum: the commutator relations for position and momentum
Time evolution: the Schrödinger equation
Getting started with the hydrogen atom

Hand-in problem sheet 1:
The first hand-in problem sheet is due on 10 September at 12 o'clock noon. Please note the changed date! You are most welcome to hand in earlier, which will give you time for the nuclear physics hand-in problems.

Reading for tomorrow:
In the compendium: Read up to page 60.

Task for tomorrow:
Show that T(dx')=1-iKdx' fulfills the rules required of an infinitesimal translation operator.


Friday, 4 September 2015

Goals of today:
The replacement recipe for the momentum operator in wave function notation
Time evolution: the Schrödinger equation
Getting started with the hydrogen atom

Reading for tomorrow:
In the compendium: Read up to page 60.

Task until Monday:
None.


Monday, 7 September 2015

Goals of today:
Hydrogenic atoms and their central potential
Solution of the Schrödinger equation: separation of variables and orbital angular momentum
Orbital angular momentum: Commutator relationships and ladder operators
The solutions to the angular Schrödinger equation: the spherical harmonics

Reading for tomorrow:
The remaining part of hte compendium.

Task until Tuesday:
All tasks on pp. 60 to 62 of the compendium.


Tuesday, 8 September 2015

Goals of today:
Solutions to the tasks in the compendium, pp 59-62. Derivation and discussion of the solutions to the angular Schrödinger equation.

Reading for tomorrow:
The remaining part of hte compendium.

Task until tomorrow:
Finish those tasks on pp. 60 to 62 of the compendium that you haven't done yet.


Wednesday, 9 September 2015

Goals of today:
Derivation and discussion of the solutions to the radial Schrödinger equation of the hydrogen atom.
Experimental observations. Get started with corrections to the Bohr formula for the energy levels of the hydrogen atom.
The last points requires a consideration of the addition of angular momenta.

Reading for tomorrow:
About the relativistic corrections to the energy of the hydrogen atom, e.g. in Bransden/Joachain or in Demtröder.

Here is the powerpoint from the lecture.

Task until tomorrow:
For the hydrogen atom in a state with principal quantum number n=2 show that the maximum of the radial probability density is located at a0n2 (which agrees with Bohr's value for the electron radius).


Thursday, 10 September 2015

Goals of today:
Yesterday we didn't make it to the experimental observations and the relativistic corrections to the energy of the hydrogen atom, so that's the goal for today.

Here is the powerpoint from the lecture.

Reading for tomorrow:
About the Lamb shift and hyperfine structure in the spectrum of the hydrogen atom and the Zeeman effect for hydrogen in e.g. Bransden/Joachain or Demtröder.

Hand-in problem sheet 2:
The second hand-in problem sheet is due on 21 September at 12 o'clock noon.


Tuesday, 15 September 2015

Goals of today:
Finish off the H atom: optical transitions and dipole selection rules, Lamb shift, hyperfine structure, and, if we have the time, the Zeeman effect. Get started with atoms with more than one electron. Pauli principle. Helium atom.

In the second half of the lecture we introduced the permutation- , symmetrizer -, and antisymmetrizer-operator. 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 S2 and Sz.

Reading for tomorrow:
Demtröder 6.

B. H. Bransden and C. J. Joachain 7 (7.1 - 7.4)

Task until tomorrow:
a) Show explicitly that the optical transition 1s to 3d in the hydrogen atom is not allowed, i.e. evaluate the relevant integral. Assume that the polarisation is in the z direction, i.e. you can replace the dipole operator -er by -ez. It is enough to show that the transition is not allowed for one of the d states.

b) Calculate the following eigenvalues:

S2|- - >

Sz|- - >

Szχ+

Szχ-

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 2015

Today 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 electron-electron 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:
Demtröder 6.1

B. H. Bransden and C. J. Joachain 7 (today we focused on 7.3 - 7.5)

Hand-in problem sheet 3:
The third hand-in problem sheet is due on 21 September at 12 o'clock noon.


Task until tomorrow:
a) Write the correctly summetrized zero-order wavefunctions for the following states in the He atom (one of the electrons will be in the 1s state):

21P

33S

43D

43S


Thursday, 17 September 2015

Today we included the electron-electron 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:
Demtröder 6

B. H. Bransden and C. J. Joachain 8

Task until tomorrow:
Look at the 23S state of Helium. This state could be described by the spatial functions u100 and u200 and the spin can the electrons can be up or down. Use Slater determinants (or a sum of Slater determinants) to construct the 3 correctly symmetrized wavefunctions (problem 8.2 in B. H. Bransden and C. J. Joachain)


Friday, 18 September 2015

Today 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 23S 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 L-S coupling.

Reading:
Demtröder 6.2 - 6.5

B. H. Bransden and C. J. Joachain 8

Task until Monday the 5th of October:
What are the possible fine structure terms (2S+1LJ) of an electronic configuration with np3?

Labs

Please find the sign-up sheet here. If there are any problems, please contact Joachim immediately.

Monday, 28 September 2015

Today a short summary of the course until now was given.

Task until Monday 5th of October:
Question 4a and 4b about the Ce atom in the exam October 2014 (Exam October 2014)


Monday, 5 October 2015

Today we first finished the tasks given at Friday the 18th of September.

Hand-in problem sheet 3:
The fourth hand-in problem sheet is due on 15 October at 12 o'clock noon.


Monday, 12 October 2015

Due 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. j-j 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 ns1n's1.
a) What are the possible terms assuming L-S coupling?
b) What are the possible terms written in the form (j1, j2)J assuming J-J coupling?

5. Now consider two electrons in a s-orbital but with the same n. The electronic configuration of this state ns2.
a) What are the possible terms assuming L-S coupling?
b) What are the possible terms assuming J-J coupling?

6. Return to the example of an np2 electronic configuration.
a) Make the microstate table for this electronic configuration and show that the only 1S, 3P, and 1D 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 J-J coupling.
c) Compare with the situation if L-S coupling was valid.

8. Assuming that j-j coupling holds, list the possible terms (j1, j2)J of an np nd electronic configuration.


Tuesday, 13 October 2015

We started the discussion of molecular structure. Using the i) Born-oppenheimer 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 H2+ ion. Using these molecular orbitals we looked at simple diatomic molecules like H2+, H2, He2+, He2. Finally, we had a quick look at the molecular orbitals of larger diatomic molecules such as O2 and N2.

Reading:
Demtröder (2010), section 9, 9.1, 9.1.1, 9.1.2, 9.2

Tasks until Wednesday the 14th of October:
Write the wavefunction as:

el> = N[caψ1s(ra)+cbψ1s(rb)]

Assume that the wavefunctions are real and show that:

Eel=<ψel|H|ψel>=N2[ca2Haa+cb2Hbb+2cacbHab] (1)

Haa=<ψ1s(ra)|H|ψ1s(ra)>
Hbb=<ψ1s(rb)|H|ψ1s(rb)>
Hab=<ψ1s(ra)|H|ψ1s(rb)>

Require that |ψel> is normalized and show that:

N2=[ca2+cb2+2cacbSab]-1 (2)

Sab=<ψ1s(ra)|ψ1s(rb)>

Substitute (2) into (1) and take the derivative of Eel with respect to ca and set this equal to zero to find the value of ca. Hint: After taking the derivative with respect to ca you should refind the expression for Eel (1). Show that :

caHaa+cbHab-Eelca-EelcbSab=0 (3)

Analog by taking the derivative of Eel with respect to cb and set this equal to zero you should get that:

cbHbb+caHab-Eelcb-EelcaSab=0 (4)

Write the secular equations (3) and (4) as a matrix times the vector of ca and cb and require that the matrix times this vector is equal to the zero vector. Set Hbb=Haa and find the non-trivial solutions by requiring that the determinant of the matrix is equal to 0. Show that:

E+=[Haa-Hab]/[1-Sab] (5) or
E-=[Haa+Hab]/[1+Sab] (6)

Substitute (5) and (6) back into (3) and (4) and show that:

ca=cb or
ca=-cb
Finally, you should show that the full solutions are:

ψ-=(2-2Sab)-1/21s(ra)-ψ1s(rb)] (7)
ψ+=(2+2Sab)-1/21s(ra)+ψ1s(rb)] (8)

Sketch the wavefunctions (7) and (8) and explain for yourself why H2 is stable while He2 is unstable.


Wednesday, 14 October 2015

Today 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 De, D0, & alpha;) can be found in experiments.
Finally, we started to discuss Raman spectroscopy. We will continue with this tomorrow.

Reading:
Demtröder (2010), section 9.5.2 (rigid rotor), 9.5.5 (vibrations of diatomic molecules), 11.6 Raman spectroscopy.


Thursday, 15 October 2015

Today 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:
Demtröder (2010), section 11.6, 11.6.1, (Raman), 7.4, 7.4.1, 7.4.2, 7.4.3 (Lineshapes)


Monday, 19 October 2015

Today 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 non-equivalent 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 vibrational-rotational transitions.

Reading:
Demtröder (2010), section 9.5.3 (centrifugal distortion), section 9.5.6 (Interaction between rotation and vibrations), section 9.6.2 (vibrational-rotational transitions).

Tasks until Tuesday the 20th of October:
1) Assume that the rotational constant is the same for v' and v''. Draw all the energy levels for v'=0, J'=0,1,2,3,4,5, and v''=1, J''=0,1,2,3,4,5 correctly spaced.
2) Indicate all allowed optical transitions. Name the with increasing energies as P(0), P(1)....and R(0), R(1)....
3) Question 5 from the exam October 2014.
4) In reality the rotational constant for v'=0 and v''=1 for H35Cl is different. Use tabel 9.7 in Demtröder to calculate the rotational constant for H35Cl for v'=0 and v''=1.
Please skip question 5 on the paper I handed out.


Wednesday, 21 October 2015, 15.15-17, D-Salen

Here 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:

NumberPercentMark
1 24 U
3 92 VG
4 76 G/rest VG*
5 34 U
6 83 VG
8 24 U
11 56 G
12 44 U
13 81 VG
15 43 U
16 53 G
17 77 G/rest VG*
18 81 VG
19 59 G
21 63 G
23 78 G/rest VG*
25 53 G
26 43 U
27 64 G
28 20 U
29 56 G
30 79 G/rest VG*
31 0 U

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 re-exam:

The re-exam will take place on Saturday, 30 January 2016, from 8.00 to 13.00 in Matteannexet 8A-B-C.

If you have any questions please come and ask!


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