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PHY 520 Introduction

Explore the foundations of quantum mechanics, including key principles, such as quantization and duality, and the equations of motion in comparison to classical mechanics. Covering topics such as the historical underpinnings, solutions of the Schrödinger equation, formalism, and 3-dimensional systems.

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PHY 520 Introduction

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

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

  3. What is physics? • Study of … • 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?

  4. 18th century optimism:

  5. But two clouds on the horizon…

  6. But two clouds on the horizon…

  7. … (wavy clouds)

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

  9. The Extensions of Modern Physics

  10. 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

  11. 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 valueaccording to the probability distribution |(x,t)|2dx • Measurement “collapses the wavefunction” so that a subsequent measurement is certain to yield the same value

  12. 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

  13. 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

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