QUANTUM MECHANICS

MISCELLANEOUS

BIOPHYSICS         NEUROSCIENCE

            Physics of neurons

• A brief introduction to neural systems


• Neuron membrane as a capacitor


• Neuron membrane: ion channels and gate variables


• Neuron membrane currents and reversal potentials (Nernst Equation)



            Single neuron models

• Geometrical methods of analysis of [1D] dynamical systems


• The Fitzhugh-Nagumo Model for Spiking Neurons: Phase Plane Analysis


• Hodgkin-Huxley model for membrane currents and action potentials


• Leaky integrate and fire model


• Izhikevich model for action potentials and spike trains


• Izhikevich quadratic model for spiking neurons


• Bursting Neuron model using three coupled first order ODEs



            Neural network models

• Izhikevich models for networks of spiking neurons

GETTING STARTED WITH MATLAB

• Matlab Basics


• XY Plots


• Advanced XY plotting techniques


• Creating a simple GUI with input boxes: SHM and the Sine Function


• Creating and Saving Sounds: BEATS

APP DESIGNER: GUI SIMULATIONS

• OSCILLATIONS: Mass Spring System (free, damped, forced motion and resonance)


• OSCILLATIONS: Sinusoidal functions


• QUANTUM MECHANICS: Square / Sloping Potential Well (eigenvalues, eigenfunctions, expectation values)

DATA ANALYSIS and MATHEMATICAL ROUTINES

• Curve Fitting: Least squares fit to a straight line for linear, power and exponential relationships


• Curve Fitting: Weighted least squares fit for uncertainties in the y-data


• Solving Quadratic Equations: GUI and Function call


• Stationary Points of a function: Function to find the points of maximima and minima of a curve


• VECTOR ANALYSIS: coordinates, vector algebra, dot product, cross product, [3D] rotation of axes


• DIFFERENTIAL CALCULUS: 1st and 2nd derivatives; gradient; divergence; curl; Laplacian


• INTEGRATION: Numerical computation of one-dimensional integrals


• INTEGRATION: Estimate the value of the integral of the function y = f(x) between x = a and y = b


• INTEGRATION: Numerical computation of two-dimensional integrals - double or surface integrals


• Fourth-order Runge-Kutta Method: Initial value problems


• Runge-Kutta Method for solutions to the differential equations governing oscillating and chaotic systems


• ODE: Solving second order differential equations with the ode45 solver (mass/spring system and van der Pol oscillator)


• SINC FUNCTION: Characteristics of the unnormalized sinc function


• BESSEL FUNCTION OF THE FIRST KIND: Fraunhofer Diffraction from a circular aperture


• SOLIDS OF REVOLUTION: [2D] and [3D] plots and an animation


• Finite Difference Time Development Method: [1D] Scalar Wave Equation


• PERCOLATION THEORY: A Matlab simulation of site percolation


• Visualization of the phase of complex functions using COLOR


• A Visual Approach to Understanding Complex Functions

  John Sims Biomedical Engineering Department, Federal University of ABC, Sao Bernardo Campus Brasil



• FOURIER TRANSFORMS made easy (calculating a Fourier Transform has never been so easy)


• A VISUAL APPROACH: complex numbers and the Fourier Transform>

  John Sims Biomedical Engineering Department, Federal University of ABC, Sao Bernardo Campus Brasil



• Signal Analysis: ALIASING (Sergio Furuie, School of Engineering, University of Sao Paulo, Brazil)


• Solving ODEs with the Laplace Transform made simple


• 1st order nonlinear ODEs: slope (direction) field, nullcline, isocline, solution (integral) curve

MECHANICS / MECHANICAL SYSTEMS

• Horizontal Motion with Resistance: drag and viscous drag force


• Analytical and Numerical Analysis of Vertical Motion with Resistance: drag and viscous drag force, falling, drag coefficient


• Animation of a vibrating spring: time variation of KE and PE


• Animation of the Slingshot Effect


• Physics of bungee jumps and bridge swings


• Animation of the free fall vertical motion of an object: trajectory, displacement, velocity and acceleration time graphs



• Satellite Motion GUI used for inputs. Finite difference method used to calculate a detailed description of the satellite motion


• Satellites Orbits: Solving the equation of motion using ode45



• Secrets of Computer Graphics and Animations

THERMAL PHYSICS

• Thermal Conduction: Temperature gradient and energy flux calculations through a composite rod


• Blackbody Radiation: Sun and Earth, Hot Objects, Efficiency Incandescent Globe, Star Temperatures


• Second Law of Thermodynamics - arrow of time: animation for the motion of gas molecules released from a corner of a room


• Newton's Law of Cooling: CSI Murder - time of death / Coffee cooling problem / First Order Differential Equations


• Live Editor:   Maxwell Distributuion of Speeds

A NUMERICAL APPROACH TO THE PHYSICS OF THE ENVIRONMENT AND CLIMATE

• The vertical profile of the atmosphere


• Chaos in the atmosphere: Damped Drived Pendulum


• Solar radiation modelling: Blackbody radiation

A COMPUTATIONAL APPROACH TO ELECTROMAGNETIC THEORY

• 1. VECTOR ANALYSIS: Coordinate Systems, Vector Algebra


• 2. VECTOR DIFFERENTIAL CALCULUS: derivatives, div, curl, Laplacian


• 3. VECTOR INTEGRAL CALCULUS: Integration: line, surface, volume; Green's & Stokes' Theorems


• 4. VECTOR ANALYSIS: Problems with Matlab solutions


• 5. EM FIELDS: Visualisation of electric fields and potentials due to different charge distributions


• 6. [2D] Surface Integration: Electrostatic potential along symmetry axis of a uniformly charged square plate


• 7. GENERATING EM WAVES BY AN ELECTRIC DIPOLE: Graphical and animated analysis


• Electric Field and Electric Potential due to various charge distributions


• Electric Field and Electric Potential:     Line Integral of the electric field


• Electric Field and Electric Potential:     Method of Images


• Electric Field and Electric Potential:     Gauss's Law and Electric Flux


• Electrostatics:    Diververgence     /     Curl     /     Radial electric Fields


• Numerical solution of [2D] Poisson's and Laplace's equations: introduction and examples


• Numerical solution of [2D] Laplace's equation: Two concentric conductive squares


• Numerical solution of [2D] Poisson's and Laplace's equations: Examples using Dirchlet and Neumann boundary conditions


• Molecular Simulations of Cathodic Arc Plasmas (Cathode Spots)


• Animation: Electromagnetic Induction - induced current as magnet moves through a coil


• Animation: DC Motor - Armature current, torque and rotation


• Physics of DC motors - How a motor works in lifting a load


• Motion of charged particles through magnetic and electric fields: Numerical and graphical analysis


• Electromagnetic Induction:    Faraday's Law     /     Mutual Inductance     /     Self Inductance


• Static Magnetic Fields:    Divergence     /     Curl     /     Biot-Savart Law


• ANIMATIONS: GIF and AVI FILES: POLARIZED LIGHT: Linear, Circular, Elliptical

  John Sims     Biomedical Engineering Department, Universidade Federal do ABC, Brazil



• Color plot of the visible spectrum for the wavelength range 380 nm to 780 nm


• Engineering Electrodynamics: House of Maxwell's Electrodynamics (Vladimir Volman): link to an online textbook


• Advanced electromagnetism problem sets using Matlab (Vladimir Volman)

DYNAMICS OF OSCILLATING AND CHAOTIC SYSTEMS / SIGNAL ANALYSIS

• Lorenz Attractor: chaos dynamics


• Logistic Difference Equation


• Creating a simple GUI with input boxes: SHM and the Sine Function


• Physics of bungee jumps and bridge swings


• Forced vibration of a string: RESONANCE - animations of a vibrating string due a vibrator


• COUPLED OSCILLATORS:> Part 1   Single Oscillator (Fourier Transform of Motion)


• COUPLED OSCILLATORS: Part 2   Motion of Two Coupled Oscillators (Fourier Transform of Motion)


• COUPLED OSCILLATORS: Part 3   N oscillators - [1D] monatomic lattice


• COUPLED OSCILLATORS: Part 4   N oscillators - [1D] diatomic lattice


• Modelling a Mass / Spring System: free oscillations, damping, force oscillations (impulsive and sinusoidal)


• Chaos in the atmosphere: Damped Drived Pendulum


• Runge-Kutta Method for solutions to the differential equations governing oscillating and chaotic systems


•     RIGID PENDULUM:   Free damped and forced motions, resonance, chaotic motion


•     DUFFING TWO-WELL OSCILLATOR:   damped and forced motions, resonance, chaotic motion


•     POINCARE SECTIONS   Duffing two-well oscillator


•     PHASE PLANE ANALYSIS:   Linear Systems


•     PHASE PLANE ANALYSIS:   Model of the retina: C-cell/H-cell negative feedback interaction


•     PHASE PLANE ANALYSIS:   Nonlinear Systems


•     The Fitzhugh-Nagumo Model for Spiking Neurons: Phase Plane Analysis


• ODE: Solving second order differential equations with the ode45 solver (mass/spring system and van der Pol oscillator)


• Signal Analysis: ALIASING (Sergio Furuie, School of Engineering, University of Sao Paulo, Brazil)


• Physics of Neurones: [1D] Nonlinear Dynamical Systems


• Solving ODEs with the Laplace Transform made simple

WAVE MOTION

• ANIMATIONS: GIF and AVI FILES: [1D] Travelling Harmonic Transverse Wave

  John Sims     Biomedical Engineering Department, Universidade Federal do ABC, Brazil



• WAVE MOTION ANIMATED GIFS: Travelling, Transverse, Longitudinal Waves


• BEATS: Superposition of two sinusoidal waves of slightly different frequencies: Creating and saving sounds


• DOPPLER EFFECT: [2D] simulation for a moving source with sound effects and creation of animated gifs


• Travelling Wave Animations: Phase and Group Velocities


• Travelling waves along a string: Solving the wave equation for the transmission of energy along a string


• Frequency spectrum of an excited string (wave equation & Fourier Tranform)


• Refraction: Animations of a plane wave incident upon an interface separating two different medium


• HELMHOLTZ EQUATION: Eigenvalue Problem: animation of standing waves on a rod


• Transverse Normal Modes of Vibration of a Rod: Eigenvalues and Eigenfunctions


• HELMHOLTZ EQUATION: Eigenvalue Problem: animation of standing waves on a rectangular membrane


• Normal modes for air columns of varying cross-sectional area - musical instruments


• Animated solutions of the [1D] Scalar Wave Equation using the Finite Difference Time Development Method


• DIFFRACTION PHYICS: Single, Double, Particle with colour display corresponding with wavelength color


• Visualization of the phase of complex functions using COLOR


• FOURIER TRASFORMS made easy (calculating a Fourier transform has never been so easy)

CIRCUITS

• DC CIRCUIT ANALYSIS: Voltage divider, Max power transfer, Wheatstone Bridge


• Transient responses of RC circuits


• ac Circuits: A complete analysis of ac circuits without the trogonometry and algebra.


• RC FILTERS: A complete analysis of low, high and band pass filters without all the messy algebra.


• SERIES RESONANCE RLC series resonance circuit.

  An in-depth analysis of ac resonance circuits without all the algebra



• PARALLEL RESONANCE RLC parallel voltage divider resonance circuit.

  An in-depth analysis of ac resonance circuits without all the algebra



• Finite Difference Method: Resistors, Capacitors and Inductors


• Finite Difference Method: RC circuits


• Finite Difference Method: RL circuits


• Finite Difference Method: Series RCL circuits


• Finite Difference Method: Parallel RCL circuits

ATOMIC PHYSICS, NUCLEAR PHYSICS, QUANTUM PHYSICS, SPECIAL RELATIVITY

• Wavefunction Statistical Interpretation: Wave particle duality


• Wavefunction Statistical Interpretation: Probability


• Wavefunction: Observables, expectation values, uncertainties, Heisenbergy Uncertainty Principle


• Time Dependent Schrodinger Equation: Finite Difference Time Development Method


• Schrodinger Equation: Particle in a box model


• Schrodinger Equation: Infinite square well


• Schrodinger Equation: Harmonic Oscillator - Matrix method (eigenvalues & eigenvectors


• Schrodinger Equation: Harmonic Oscillator - Hermite polynomials


• Schrodinger Equation: Anharmonic oscillator, Morse potential, HCl molecule


• Free particle: Wavepacket spreading (TDSE / FDTD)


• Free particle: Gaussian wavepacket propagation (TDSE / FDTD)


• Free particle: Motion of an electron in a uniform electric field (TDSE / FDTD)


• Wave Particle Duality


• Solving the time independent Schrodinger Equation for bound states using the finite difference method


• Solving the time independent Schrodinger Equation for bound states using a matirx method for finding the eigenvalues and eigenvectors of the energy operator


• Time evolution of the wavefunction: stationary and compound states


• Hydrogen-like atoms and ions: solutions to the [3D] Schrodinger Equation


• Hydrogen atom - Transition Rates, Liftimes, Selection Rules, Animations of Compund States (Corrections March 2018: Duncan Carlsmith, University of Wisconsin-Madison)



• Electron Beams in Field-free space: Travelling and Standing waves


• Scattering and Beam Penetration: STEP POTENTIAL beam energy less than step height



• Theory of Alpha Decay: Solving the Schrodinger Equation - tunnelling - half-life as a function of the energy of the alpha particle (Badr Eddine El Fatehy Chfira University of Murcia (UMU, Spain) )


• Solving the [1D] Time Dependent Schrodinger Equation with the Finite Difference Time Development method


• Time Dependent Schrodinger Equation: Animation of the Motion of a Wave Packet


• Time Dependent Quantum-Mechanical Scattering in Two Dimensions - Animated motion of wave packets


• Two Particles in a Box / Symmetric and Antisymmetric states / Bosons and Ferimons


• Nuclear Physics Calculations Made Simple : Binding energies, Q-values, Coulomb barrier, lifetime of Sun


• Alpha Particle Scattering: Solving the equation of motion using ode45


• Radioactive Decay Chains: Rate equations solved using ode45


• Special Relativity: Notes with animations created in Matlab


• Special Relativity: Minkowski spacetime diagrams

NUMERICAL ANALYSIS OF OPTICAL AND ELECTROMAGNETIC PHENOMENA

            Electromagnetic waves

•     Propagation of electromagnetic waves in free space

            [1D] Finite Difference Time Development method (EM waves in non-magnetic media)

•     Propagation of electromagnetic waves and the finite difference time development method


•     Gaussian EM pulse propagating in non-lossy media


•     Sinusoidal and modulated EM waves propagating in non-lossy media


•     Propagating EM waves striking a medium with a different dielectric constant


•     Propagating EM waves in lossy dielectric media


•     Propagating EM waves and interference effects with thin films

            Polarization

•     Vectorial nature of light


•     Mathematical foundations


•     Linear polarized light


•     Circular and elliptical polarized light


•     Optical elements and the manipulation of the state of polarization


•     Matlab exercises

            Coherence and Interference

•     Fabry-Perot Interferometer


•     Michelson Interferometer: Fringe Patterns

            Scalar diffraction theory

•     Scalar Diffraction theory: Diffraction Integrals


•     Poynting Vector and Irradiance


•     NUMERICAL INTEGRATION METHODS:: Rayleigh-Somerfeld Diffraction Integral of the First Kind


•     BESSEL FUNCTION OF THE FIRST KIND: Fraunhofer Diffraction from a circular aperture


•     Diffraction from RECTANGULAR APERTURES


•     Diffraction from rectangular apertures: DOUBLE SLIT


•     Diffraction from RECTANGULAR APERTURES: CROSS SHAPED APERTURE


•     Diffraction from TRIANGULAR APERTURES


•     Diffraction from CIRCULAR APERTURES: Rayleigh-Sommerfield Integral - near and far field calculations / energy enclosed within circles


•     Diffraction from CIRCULAR APERTURES: POINT SOURCE illumination of aperture


•     Diffraction from ANNULAR APERTURES: Rayleigh-Sommerfeld Integral


•     Diffraction from FRESNEL ZONE PLATES: Rayleigh-Sommerfeld Integral


•     GUI SIMULATION: Diffraction by a circular aperture


•     Convergent beams: IRRADIANCE IN THE FOCAL REGION: Rayleigh-Sommerfeld Integral


•     GUI SIMULATION: Focussed Beam - Single source, Two Sources and Resolution


•     DIFFRACTION PHYICS: Single, Double, Particle with colour display corresponding with wavelength color

        Beam Propagation



• Modelling the Behaviour of Gaussian Beams (Paraxial Regime)