Quantum Mechanics has been at the root of the enormous advances and major discoveries in Physics over the last century, having strong influence on the development of modern technology, like lasers, transistors and superconductors. It gives a description of nature that is very different from the one of Classical Mechanics, in particular being non-deterministic and "non-local", and seeming to give a crucial role to the presence of "observers".

The course begins with a brief historical and conceptual introduction, explaining the reasons for the failure of classical ideas, and the corresponding changes needed to describe the physical world. Then it presents the Schrodinger picture in Quantum mechanics. Furthermore, the course will develop the mathematical formalism of Quantum mechanics The formalism will be applied to a wide class of one-dimensional problems, including potential barriers and potential wells. The problem of measurement in Quantum mechanics will be discussed as well. In addition, the alternative approach, known as "secondary quantisation" will be presented. Its main advantage is that it has a purely algebraic nature, based on the commutation relations between the raising and lowering (ladder) operators. The course will outline at the end the Heisenberg (interaction) picture of the unitary evolution in Quantum mechanics.

Through the whole module, a clear analogy with the Hamiltonian formalism in Classical mechanics will be made. The module can serve as a solid base for studying further subjects in Applied Mathematics and Mathematical Physics, like (Classical and Quantum) Statistical Mechanics, Quantum Field Theory, Cosmology, Quantum Information and Quantum Computing, as well as Operator Algebras and Functional Analysis.

Aims

The module aims to provide students with a conceptual understanding of Quantum Mechanics, highlighting the break with "classical physics" concepts. Students will learn a fundamental theory describing the physical world through combining their mathematical skills learned in earlier stages. Through lectures, problems classes and seminars, students will develop graduate skills, like ability to assimilate, process and engage with new material quickly and efficiently. Finally, they will develop problem-solving skills and learn how to apply learned techniques to unseen problems.

Learning Outcomes

On successful completion of this module, students should be able to

a) Demonstrate systematic understanding of the basic principles of quantum mechanics and be able to apply them in simple physical situations.

b) Solve simple eigenfunction problems for Hermitian operators.

c) Calculate the quantum mechanical wave function, probability distribution in a given state.

d) Calculate quantum mechanical expectation values of observables. Predict how the state of an undisturbed quantum system evolves with time.

e) Demonstrate the capability accurately to analyse and solve problems using a reasonable level of skill in in the following areas: potential wells and barriers in one dimension and the treatment of eigenvalue problems in quantum mechanics.

f) Apply key aspects of quantum mechanics in well-defined contexts, showing judgement in the selection and application of tools and techniques.

g) Gain knowledge and understanding of basic results about theory of linear operators in a Hilbert space.

Syllabus

1. Early quantum physics: quanta and particles. Bohr model of the hydrogen atom.

2. De Broglie waves.

3. The Schrodinger equation and the wave function.

4. The time-independent Schrodinger equation: potential barriers and potential wells.

5. Formalism of quantum mechanics: states and observables. Postulates of Quantum mechanics. Dirac bra- and ket- vectors. Hilbert space (non-rigorous treatment).

6. Expectation values and commutation relations.

7. Quantum harmonic oscillators. Raising and lowering operators.

8. Time evolution in Quantum Mechanics. Symmetries and conserved quantities.

9. Schrodinger versus Heisenberg (interaction) picture.

- Module Supervisor: Murat Akman