Selected chapters of quantum mechanics for modern engineering

The course is for physics majors.

You will learn how to gain insight in contemporary fields in condensed matter physics.

The main topics include:

1. Bose-condensates and selected topics from superconductivity.
2. Introduction to quantum information theory.
3. Quantum computing algorithms.
4. Quantum measurements and entanglement.
5. Quantum teleportation.
6. Aharonov-Bohm effect and its modern use.
7. Eigenstate thermalization hypothesis.

The course instructors are active researchers in a theoretical solid state physics. Armed with the tools mastered while attending the course, the students will have solid understanding of the principles of quantum mechanics, gain insight in contemporary fields in condensed matter physics.

Week 1:

History of quantum mechanics. Wave packet. Schrödinger equation. Properties of wave function, normalization. Averages and operators. Superposition of states, measurement, commutators. Schrödinger equation

Week 2:

Infinite well. Delta-barrier, matching wave functions. Dirac’s bra-ket notation. Operators in Dirac’s notation, Hermitian conjugation. Harmonic oscillator via ladder operators.

Week 3:

Qubit and Bloch sphere. Quantum superposition of N qubits. Overview of quantum computing algorithms and their advantage as compared to classical algorithms. Quantum teleportation. Quantum decoherence of many-qubit system and quantum error correction. Physical realization of quantum computers: difficulties and advances.

Week 4:

Superconductivity: Discovery. Main Properties, Meissner's and Josephson's effect. Types of supercoductors, Vortices. Applications of superconductivity. Summary and theoretical explanation.

Week 5:

Modeling a realistic quantum system: resonant microwave cavity coupled to qubits array via the gauge-invariant quantum phases. Mapping on the infinitely coordinated Ising spin-chain of spins-½. Holstein-Primakoff representation. 1st-order quantum phase transition into dipolar ordered state. Metastable states of the spin-chain: bound states of light.

Week 6:

Emergence of instantonic ‘pairing boson’ and high-Tc superconductivity. Negative energy of the ‘antiferromagnetic’ instantons. Zero-mode of instantonic ‘crystal’ along Matsubara time axis as the ’pairing boson’ in the Eliasberg-like equations. Why AF instantons behave as a ‘hidden order’. Spin excitations of the instantonic ‘crystal’ — hourglass modes?.

Week 7: Final Exam