After completion of this course the student is able to;
- Explain which difficulties arise for quantum systems of more than two particles
- Apply the rules for coupling angular momenta to practical problems
- Understand the fundamental connection between symmetries and conservation laws and how the consequences of this connection differ from classical mechanics
- State the main difference between identical particles in classical and quantum mechanics and its consequences for the structure of the solution to the Schrödinger equation
- Explain the orbital model of atomic structure
- Use perturbation theory
- Distinguish degenerate and non-degenerate time-independent perturbation theory and know when and how to apply it
- Derive Fermi’s Golden Rule
- Explain the difference between spontaneous, stimulated emission and absorption
- Use and explain differences between the WKB approximation, partial wave analysis and Born rule to solve 1d and 3d scattering problems
- Take effective notes and use supplementary resources and discussions with peers to fill in possible gaps in the notes
- Translate and develop problem statements efficiently into problem solutions
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In this course, we ask the question: How can we apply the fundamental principles of quantum mechanics to systems beyond the hydrogen atom and to systems that interact with electromagnetic radiation? We start by recapping the structure of the solution of the hydrogen atom and define the ingredients that are needed to describe systems that consist of more than one electron and proton: coupling of angular momenta, symmetries, and particle-particle interactions. The latter are neglected in this course and will be introduced elsewhere. The former two and their intimate relationship with each other will be discussed in detail. This will then allow us to gain an approximate understanding of the periodic table of elements and even of the electronic structure of some molecules.
We will then use the concept of perturbation theory (time-independent and time-dependent) to understand effects like the fine structure of the hydrogen atom and how matter interacts with electromagnetic radiation, deriving Fermi’s famous Golden Rule. Finally, we will look at scattering problems and approximate ways for solving them with many important applications in diverse areas such as X-rays and particle physics.
- Introduction and motivation for lecture
- Symmetries and conservation laws
- Symmetries and conservation laws in quantum mechanics
- Conservation of momentum
- Conservation of energy
- Conservation of angular momentum
- Parity and time-reversal
- Many-particle quantum mechanics
- Central potential problems: Use of symmetries
- Addition of angular momenta
- Identical non-interacting particles
- Time-independent perturbation theory
- Reminder: non-degenerate case
- The Helium atom
- Degenerate case
- Linear Stark effect
- Time-dependent perturbation theory
- General formulation
- Constant perturbation and Fermi’s Golden Rule
- Periodic perturbation and Fermi’s Golden Rule
- Interaction of atoms with electromagnetic fields
- Scattering with 1D potentials
- Scattering with a potential barrier
- Scattering with a delta potential
- The WKB approximation
- Scattering with 3D potentials
- General scattering theory
- The Born approximation
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Assessment type: Written exam and homework assignments
Assessment description:
Exam: The course will be assessed by a written exam. The exam consists of two parts: Short problems and Advanced problems. The short problems will cover every chapter of the lecture and are graded with between 2 and 4 points. There are meant to test your conceptional knowledge. Advanced problems give between 16 and 24 points and required more extensive calculations and/or derivations. The exam is not open book. Instead, you are allowed to bring one A4 page with everything that you think might be useful for the exam (no restrictions). Standard mathematical formulas and expressions will be provided by us.
Homework assignments: There are weekly graded homework assignments. When you first received the assignment, you will be asked to provide a solution that demonstrates your ability for problem translation. This means that instead of providing a fully worked out solution, you should hand in a “solution recipe”, demonstrating that you have comprehended the problem, know which facts/ingredients from the lecture you need to apply to solve it, which possible pathways to the full solution could be taken, and how these need to be executed technically. We will grade your solution recipe until the next tutorial session. During the tutorial session, you will receive a model solution recipe and will have time to ask questions and work on a full solution. This solution will be graded for accuracy. The final homework grade is composed of your solution recipe grade (30%) and the full solution grade (70%).
The final grade for the course will be calculated via G = H+ E(10-H)/10 where G is the course grade, H is the grade for the homework (maximum 2 points) and E is the grade for the exam (maximum 10 points).
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