Christopher Crawford 2016-08-24 PHY 520 Introduction Christopher Crawford 2016-08-24

Course mechanics Introductions / class list Canvas / course webpage http://www.pa.uky.edu/~crawford/phy520_fa16 Syllabus

What is physics? Study of … 4 pillars of physics: Matter and interactions Symmetry and conservation principles 4 pillars of physics: Classical mechanics – Electrodynamics Statistical mechanics – Quantum mechanics Classical vs. modern physics What is the difference and why is it called classical?

18th century optimism:

But two clouds on the horizon…

But two clouds on the horizon…

… (wavy clouds)

Modern Revolution October 1927 Fifth Solvay International Conference on Electrons and Photons

The Extensions of Modern Physics

Key principles of QM Quantization (Planck, Einstein, deBroglie) Waves are quantized as packets of energy (particles) Particles have quantized energy from modes of their wave functions. Correspondence (Bohr) Agreement with classical mechanics for large quantum numbers Duality / Complementarity / Uncertainty (Heisenberg) Complementary variables cannot be simultaneously measured Symmetry / Exclusion (Pauli) Identical particles cannot be distinguished-> symmetric wavefunction

State and Equation of Motion Classical mechanics The initial state of a particle is it’s position x0 and velocity v0 The equation of motion is Newton’s 2nd law: F=m d2x/dt2 Integrate this ODE with initial conditions to determine trajectory x(t) In principle, exact position, velocity known at all times Conservation: it is usually easier to work energy & momentum Quantum mechanics The initial state of a particle is it’s initial wave function 0 (x) Equation of motion: TDSE: –ħ2/2m d2/dx2 + V(x)  = iħ d/dt Solve this PDE boundary value problem to evolve (x,t) in time A measurement of position yields a random value according to the probability distribution |(x,t)|2dx Measurement “collapses the wavefunction” so that a subsequent measurement is certain to yield the same value

General course outline Ch. 1+: Historical underpinnings -> TDSE & wave function Blackbody radiation, de Broglie mater waves, Bohr model Quantization and dispersion: propagation of wave functions (TDSE) Complementarity and the uncertainty principle Ch. 2: Solutions of the time-independent Schrödinger Eq. Infinite square well, harmonic oscillator, free particle, delta function Finite square well: bound/unbound states; transmission/ reflection Ch. 3: Formalism of Quantum Mechanics Mathematical review; Dirac bra-ket formalism Hilbert space, operators(observables), eigenfunctions Postulates of Quantum Mechanics Ch. 4: 3-d systems Angular momentum, hydrogen atom

Mathematics needed for 520 Probability distributions weighted average (expectation) Fourier decomposition Wave particle duality General linear spaces Vectors, functional, inner product, operators Eigenvectors Sturm-Louisville theory, Hermitian operators Symmetries Transformations, Unitary operators