Quantum Physics 2013-14

Prof Val GibsonPrerequisites

A working knowledge of the classical mechanics of large objects, including rotational mechanics and angular momentum. An appreciation of the mathematical description of waves in one, two and three dimensions. A basic knowledge of classical electromagnetism and optics. Basic probability theory, both discrete and continuous. Complex numbers and functions of complex variables. Fourier transforms and Fourier series. Elementary notions of vector and operator methods: mappings, matrix algebra, inner products, eigenvectors and eigenvalues.

Learning Outcomes and assessment

To appreaciate how classical physics breaks down at microscopic scales. To understand the conceptual foundations of quantum mechanics, and the notion of the state of a system. To gain a working knowledge of Schrodinger's equation, and to understand how to solve the equation for a variety of one-dimensional potentials. To appreciate the difference between time-dependent and time-independent quantum mechanics. To understand the meaning of non-commuting observables, and the origin and meaning of Heisenberg's uncertainty principle. To gain a fluency in operator algebra, and a knowledge of how it underpins quantum mechanics at a fundamental level. To appreciate the meaning of Dirac's bra-ket notation, to develop the mathematical skills associated with the algebra of linear operators, to understand the conceptual meaning of operator methods, and their use in solving problems. To understand the quantisation of angular momentum, the representation of angular momentum, and to see how it can be used to describe the detailed behaviour of the hydrogen atom. To appreaciate the unique role of spin. To start to gain an insight into the description of multi-particle systems, and to gain an appreciation of the rich physics that inevitably follows from the basic principles of quantum mechanics. To understand the difference between Ferions and Bosons. To appreciate how quantum mechnics necessarily leads to concepts such as entanglment.

Synopsis

**The Quantum Revolution: **Photoelectric effect. de Broglie Hypothesis. Bohr's atom and atomic structure. Electron diffraction, Davisson and Germer. Compton scattering. Blackbody radiation, u-v catastrophe. The central role of Planck’s constant.

**Wave-Particle Duality and the Uncertainty Principle: **Young’s double-slit experiment. Free particle in one dimension: wavefunctions and wave-packets. The Heisenberg Uncertainty Principle. Time evolution of wave-packets: dispersion and propagation.

**The Schrödinger Equation: **Time-independent and time-dependent Schrödinger Equation.

**Wave Mechanics of Unbound Particles: **Particle flux. One dimensional potentials and boundary conditions. The potential step: reflection and transmission. The potential barrier: tunnelling. Radioactivity : a, b and g-decay.

**Wave Mechanics of Bound Particles: **The infinite square well potential and bound states. Normalization, parity and orthogonality. The Correspondence principle. The finite square well potential. The harmonic oscillator: vibrational specific heat. Wave mechanics in 3D: particle in a box, degeneracy and 3D harmonic oscillator.

**Operator Methods: **Operators, observables, linear hermitian operators and operator algebra. Dirac notation: eigenstates and eigenvalues. Orthogonality, degeneracy and completeness of eigenstates. Compatible and incompatible observables: commuting operators and simultaneous eigenstates, non-commuting operators, generalized uncertainty relations, minimum-uncertainty states. Ladder operators: the harmonic oscillator, equipartition.

**Time-Dependent Quantum Mechanics: **Time-dependence: expectation values, Ehrenfest’s theorem, stationary states, the time-evolution operator, time-energy uncertainty relation, conserved quantities.

**Quantum Mechanics in Three Dimensions: **General formulation. Orbital angular momentum: eigenfunctions and parity. The 3D harmonic oscillator. The rigid rotator: rotational specific heats of gases. Central potentials: conservation of angular momentum, quantum numbers, separation of variables. The hydrogen atom. Non-central potentials and hybridization.

**Spin: **The Stern-Gerlach experiment. Spin angular momentum, spin operators, spin eigenstates. Combining spin and orbital angular momentum, combining spins. Matrix methods. Pauli spin matrix and spinors.

**Identical Particles: **Identical particle symmetry: multiparticle states, fermions and bosons, exchange operator, exclusion principle, symmetry and interacting particles, counting states. Entanglement. Two-electron system: helium ground and excited states.