F11QT - Quantum Theory

Course leader(s):

Aims

We introduce the main mathematical set up of discrete quantum mechanics as well as basic quantum mechanical concepts, such as quantum states, observables and time evolution of quantum systems. We also present the notions of spin operators, spin states and composite systems and basic concepts on quantum computing. We then discuss the time dependent and time independent Schrödinger’s equation for space continuous systems and solve prototypical quantum mechanical systems.

Syllabus

1. Fundamental concepts (1.1 Introduce the basic notions in formulating quantum mechanics: linear algebra, vector spaces, eigenvalues & eigenstates. 6 lectures)

2. Basic quantum mechanics (2.1 quantum states and observables, time evolution of quantum systems 6 lectures)

3. Spin operators & Composite systems (3.1 Introduce the notion of spin operators and spin states. Derive the tensor product of matrices and vectors and construct composite quantum mechanical systems. Introduce Schmidt’s decomposition 6 lectures)

4. Quantum computing (4.1 introduce basic quantum circuits and quantum algorithms. 6 lectures)

5. The Schrödinger equation (5.1 Study the time dependent and the time independent Schrödinger equation for continuous quantum mechanical systems. Study fundamental quantum mechanical systems 6 lectures)

6. Extra material (6.1 Schrödinger equation in 3 dimensions and separation of variables 5 lectures.)

Learning outcomes

By the end of the course, students should be able to do the following:

Explain the mathematical formalism of quantum mechanics. Linear algebra, vector spaces

Describe the notions of quantum states, observables, and time evolution in quantum quantum systems

Explain the notions of spin operators and spin states

Define the notions of tensor products and composite systems and Shcmidt’s decomposition

Describe basic notions in quantum computing such as quantum circuits and quantum algorithms

Analyse the time dependent and time independent Schrödinger equation for space continuous systems

Illustrate prototypical, fundamental quantum mechanical systems, such as the harmonic oscillator and the infinite square well.

Analyse the Schrödinger equation in 3 dimensions: special examples (special topic)

1. Explain the mathematical formalism of quantum mechanics. Linear algebra, vector spaces 2. Describe the notions of quantum states, observables, and time evolution in quantum quantum systems 3. Explain the notions of spin operators and spin states 4. Define the notions of tensor products and composite systems and Shcmidt’s decomposition 5. Describe basic notions in quantum computing such as quantum circuits and quantum algorithms 6. Analyse the time dependent and time independent Schrödinger equation for space continuous systems 7. Illustrate prototypical, fundamental quantum mechanical systems, such as the harmonic oscillator and the infinite square well. 8. Analyse the Schrödinger equation in 3 dimensions: special examples (special topic)