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Time Dependent Wave Equation 1D One Dimensional Optical Resonator Modes
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Inelastic Collision Conversion to Internal Energy Brownian Motion Animation 2D Using Hard Sphere Scattering Large Disc Kinetics 2D Condensation of a Gas on a 2D Solid Maxwell's Demon Energy Seperator 2D Minimize Final Energy 2D and 3D Gas Diffusion 2D Gas Expansion 2D Linear Gas Expansion 2D Constant Pressure Gas Expansion 2D Circular Gas Explosion 2D Circular MagnetoCaloric Effect 2D Convection Loops in a Numerical Gas Chapter 3: Gas Energy Equipartition Energy Equipartition of a Two Dimensional Gas with Two Different Masses Energy Equipartition of a Three Dimensional Gas with Two Different Masses Gas Energy Equipartition 3D
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Photoelectric Effect Alpha Particle Emission from a Large Nucleus 2D Black Body Emission from Charged Harmonic Oscillators Quantum Animation of Energy Band Gap 1D Animation of Quantum Wave Packets in a pn Junction 1D Quantum Mirror Focusing by Iterating the Schrodinger Equation Quantum Lens Focusing by Iterating the Schrodinger Equation Quantum Two Mirror Resonator by Iterating the Schrodinger Equation Entangled Quantum Object Propagation in a Parabolic Potential Classic and Quantum Object Propagation in a Parabolic Potential Accurate Wave Packet Propagation in a Parabolic Potential Comparison of Quantum to Classical Physics 1D-Time Dependent Accurate Wave Packet Propagation in a Swaged Potential Series Solution of Wave Packet Propagation in a Infinite Square Potential Propagation of a Wave Packet in a Parabolic Potential Propagation of a 2D Wave Packet in a 2D Parabolic Potential Propagation of an Asymmetric Shaped Wave Packet in a 2D Parabolic Potential Two Slit Diffraction by Iterating the Schrodinger Equation Two Potential Slit Diffraction by Iterating the Schrodinger Equation Propagation of an Electron Wave Packet in an Infinite Square Well (Series Solution) Propagation of a Free Wave Packet (Series Solution) Series and Algebraic Solutions for Propagation of a Wave Packet in a Parabolic Potential Comparison of Quantum to Classical Physics 1D-Time Independent Mexican Hat Stationary Quantum States of a Finite Square Potential Well Various Wave Packet Motion in a Finite Square Potential Well Various Wave Packet Motion in a Swaged Potential Well Comparison of Quantum to Classical Physics 1D-Time Independent Cosine/Parabola Electromagnetic Modes of a 2D Cavity Black Body Radiation Experiment and Theory Emission of the Inner Walls of a Black Body Cavity Multiple Slit Diffraction Statistics of Indistinguishable Dice or Particles Quantum Field Collapse Visualization Matrix Operations and Eigenmodes
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Tumbling Block on Inclined Plane New Roller on Inclined Plane Sliding Block on Inclined Plane Classic Objects Motion 1D Generation of Curve for Fastest Time between Two Points Curve for Fastest Time between Two Points Bead Sliding On a Stiff Wire Sound Wave in Two Solid Media Mechanical Properties of a 2D Numerically Modeled Lattice Vibrating Reed Numerical Model 2D Laminar Flow of a Fluid in a 2D Bearing Animation of a Fountain 2D Gravity Leveling of a 2D Liquid Longitudinal and Shear Wave Sound Speed in a 2D Lattice Mechanical Properties of a 2D Numerically Modeled Lattice Vibration Frequencies of a 2D Numerically Modeled Lattice Damped Harmonic Oscillator Motion and Modes of a Particle in a Mexican Hat Potential Forced Oscillator Roller Bearing Cam and Roller Follower Simple Hybrid Gear Train Ratchet Animation Child's Swing Height Increase Mechanics Roller Chain Animation Elasticity of 3D Gas Driving a Piston Free to Forced Harmonic Oscillator Transition Physics of Mounting a Tire on a Rim Physics of a Pendulum Clock Escapement Physics of a Gas Pressure Gauge
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E=mc^2 and Mexican Jumping Bean Radar Pulse Return Time Interval Experiment to Determine Gamma Current Loop with a Rotating Test Particle Moving Fabry-Perot Interferometer Space Time Invariant Intervals of a Moving Clock Relativity Transformations by Requiring Simulataneity Accelerated Space Trip Clock Times and Rates Minguzzi Invariant Inertial Frame Time Accelerated Space Trip with Light Pulse Signaling
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Semiconductor Electron Energies and Band Gap 1D Semiconductor Electron Energies and Band Gap 2D Simple Fermi Surface p-nJunctionDiode Depletion Width Temperature Dependence of 2D Fermi-Dirac Particles
Chapter 4
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Assembly of 3D Parts Using Javascript Web Graphics Library 3D Compared to Javascript 3D Graphics 3D Graphics Tutorial Beginnings Proof of Pythagoras a^2+b^2=c^2
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Multiple Slider Animation Template for Plotting with HTML Enhanced Sliders or Ranges Collapsible Dropdown Menu Example Arrow Functions Fast Fourier Transform (FFT) Demonstration
Animation of a Progress Bar
Template for Scientific Plots
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Time Dependent Wave Equation 1D One Dimensional Optical Resonator Modes
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Probability of Birthday Match Random Creation of Probability Function Profiles Two Dies Probability Distribution Three Dies Probability Distribution
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Light Rays in a Transparent Isosceles Prism Light Ray in a Series of Wedges of Various Index of Refraction Snells Law of Refraction Refraction Due to Dipoles in Two Media Refraction and Reflection Two Media Animation of Concave Mirror Reflection Animation of Convex Lens Refraction Prism Refraction-Wave Picture Thick Lens Ray Trace Monochromator (Czerny-Turner Type) Wavelets from Concave Mirror> Thick Lens Focusing
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Sagnac Effect Fiber Optical Gyro (FOG) Ring Laser Gyro (RLG)
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Paticle Motion in a Venturi: Wall Reflections Paticle Motion in a Venturi Potential Elliptical Ion Trap Using Boundary Potential Distribution Prediction of Hard Disc Collision Time and Position Rigid Rotor Collisions
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Acoustic Wave with Built-in Pressure Acoustic Waves in 2D Numerically Modeled Solid Wave 1D on a Linear Array Waves on Discrete Model of Vibrating String Linear Acoustic Wave in a 3D Gas Spherical Acoustic Wave in a 3D Gas Laterally Perturbed Two Dimensional Gas with Lennard Jones Repelling Forces Longitudinally Perturbed Two Dimensional Gas with Lennard Jones Repelling Forces
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Bicep and Forearm Animation Blood Pressure Measurement Multiple Dose Residual Medicine Vs Time The Eye and its Resolution Breathing Animation Scoliosis Animation Muscle Animation
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Scattering from a Crystalline Target 2D Scattering from a Crystalline Target 3D
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
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Chapter 12
Energy Conduction in a 2D Numerically Modeled Solid Lattice Unbalanced Gas Dynamics:Energy Flow in a Gas
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12

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Quantum Modes of Particle in a 1D Mexican Hat Potential

The goal of this program is to explore the boundary between classical physics and quantum physics. This boundary was first hypothesized by Niels Bohr in 1920 and is called the Bohr Correspondence Principle . To do this I will show the classical motion of the particle at the eigenmode energy as well as the usually periodic variation of the particle wave function in this eigenmode. The modes are the eigenvalues and eigenvectors of the second order Schrodinger equation (SE)

`-(ℏ^2)/(2m)(del^2Psi)/(delx^2)+V(x)Psi=EPsi`

, where ℏ is the reduced Planck constant, m is the mass and E is the energy of the particular mode whose wave function is the eigenvector. The reader might want to refer to Finding Eigenvectors from Eigenvalues.

For this program I have chosen to set the Planck constant (divided by 2*pi), ℏ, equal to 1 Joule second and the mass, m, equal to 1 kilogram. Although the SE can also express the time dependence of the wave function, only the steady state eigenmodes will be computed here. The second derivative operator can be expressed in tridiagonal matrix form as

The entire matrix expression of the time independent SE is then In order to solve this equation efficiently using the Jacobi Method, I have derived Jacobi Method Modifications. The potential V(x) here is a standard harmonic oscillator parabola with a guassian "bump" in the middle so it looks like an upside down Mexican hat. This potential is believed to be the "symmetry breaking" feature that mediates the Higgs boson in the Standard Model of quantum field theory. Of course the potential in quantum field theory is 3 dimensional but three dimensions are hard to depict so I have chosen 1 dimension, x, here. As labeled, the black trace is the potential that the particle is in, the red trace is the force it feels at any given location, the orange trace is the signed velocity of the particle, and the green trace is the time spent per unit distance moved by the particle, 1/|Velocity|. The latter would equal the value of the magnitude squared of the wave function for very large wave function quantum numbers. The green trace value should be proportional to the envelope of the square of the quantum mechanical wave function, psi, for the quantum mode which corresponds to the initial potential energy of the particle. But in quantum field theory, the particle will sometimes "tunnel" through the center hill of the potential, so a simple relation to the classical value if 1/|vx| cannot be expected. The eigenvalues of the particular setup are depicted as a purple trace with small purple circles at each eigenvalue. By placing the mouse on one of these circles, and clicking the eigenvector (spatial mode) of that eigenvalue will be drawn in a color corresponding to the index number of the eigenvalue. If this circle is clicked again, the eigenvector corresponding to that eigenvalue will be removed from the list of drawn eigenvectors, reducing the clutter so that new eigenvectors can be viewed. The eigenvalues can be computed quite efficiently using the Jacobi method which successively "zeros out" the off diagonal elements by simple two dimensional rotations. These rotations change only two rows and two columns of the matrix so, instead of full matrix multiplications, we only need to compute the changes to these two rows and two columns.

From the Schrodinger Equation solutions we obtain three important types of information about how a particle moves in the potential V(x).
1. The energies, Ei, which are the eigenvalues of the stationary state modes.
2. The square of the amplitudes of the eigenvectors, vi, which is the probability density for the position, x, of the particle.
3. The spatial frequencies, k(x) Vs position x of the eigenvectors to which the particle momentum is proportional.
All three of these quantities are computed and displayed in this program except k(x) is not computed for eigenvalue mode numbers less than about 12. The learner should note that the probability density is usually lower where the spatial frequency k(x) is higher as shown in the following Figure:

Figure 1: Eigenvector amplitudes and spatial frequencies k(x) of Eigenmode 39 where the total range of x was 100. Note that the k(x) is larger where psi(x) amplitude is smaller. Also note that the eigenvalues (energies) occur in pairs (doublets) of equal energy up to and including mode 35. This doublet behavior is called degeneracy. Degeneracy is where a mode has different spatial distribution but the same energy bevause of the symmetry of the potential energy. At mode 35 the mode energy is just below the center peak of the potential energy. All modes below that have equal probability of appearing mainly in the left or right potential well


Figure 2: Eigenvector plot by finite element method (FEM) of Eigenmode 35 where all parameters except the mode number are the same as in Figure 1. Note that the periodic wavefore is much smoother than that of Figure 1. That is to be expected for the FEM since it uses many more spatial increments than the finite difference method (FDM). An important similarity between the two methods is that the transition from doublet energy modes to singlet modes (to non-degeneracy) occurs when the mode energy just exceeds the peak center potential.


Figure 3: Similar to Figure 1 but with the classical velocity (orange), classical 1/|velocity| (green), and a gaussian wave packet (red) plotted. The quantum velocity is also plotted in orange with a dashed line. Note that the classic velocity and quantum velocity agree quite well. However, one would expect that the envelope of the magnitude squared of the eigenvector (blue) should agree reasonably with the classic value of 1/|velocity| At best one might say that there is qualitative agreement in that the x value of the maximum of the envelope coincides with the maximum of the 1/|Velocity| curve. One must conclude that the quantum effect is to greatly "soften" the infinite result for 1/|v| of the classical turn around at each end of the classic trajectory. Either that or one might argue that I have not chosen a large enough quantum number for the Copenhagen Interpretation (probability)to be valid. This softening is to be expected from the Heisenberg Uncertainty Principle since that principle would preclude determinining the exact point of turnaround. The gaussian wave packet moves in the x direction at the exact speed of the particles. If this wave packet were immersed in this potential there would be significant distortion near the ends of its trajectory. But this distortion is a subject of the time dependent Schrodinger equation and is reserved for a later animation.

Interpretation of Time Independent Schrodinger Equation Solution

Note that the solution is in the form of a standing wave. Since the wave is standing and has regularly spaced nodes, there can be only one way of interpreting it: The interference pattern of two oppositely traveing waves desceibed by


where A and k are constantly varying functions of x. Obviously this sum addes up to a cosine function in x. Classically this would correspond to oppositely moving particles of equivalent speed. That is what I show when the learner selects the "Start Particle Motion" button. We expect that that maximum classical speed should be equal to the maximum of the quantum product hbar*k where k is 2*pi/lambda and lambda is twice the space between zero crossings of the Schrodinger wave function, I provide a comparison of these two quantities and the difference is usually less than 10% of the average. Also, the average kinetic energy of the mode is its eigenvalue and we should expect that this eigenvalue should be equal to <|hbar*k|^2/(2*mass) where the <> symbols indicate average. The differennce between these two quantities is uusually less than 20% of the average of the quantities, This concludes my interpretation of the time independent wave function. Note that this interpretation is much more detailed than the Copenhagen Interpretration which usually only asserts that the magnitude squared of the wave function is the probability of that the particle will be in a given location. The learner has the option of downloading any of these graphics by right-clicking and selecting "Copy".