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Logistic Equation Feigenbaum-Diagram QM Harmonic Oscillator

MODELLING OF PHYSICAL PHENOMENA Using LabView Invited Talk, Bucharest, October 1999 Prof. Dr. R. Lincke Inst. für Experimentelle und Angewandte Physik der Universität Kiel. Lissajous-Figures Mother-Daughter-Problem Simple Harmonic Oscillator Fourier-Analysis. Logistic Equation

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Logistic Equation Feigenbaum-Diagram QM Harmonic Oscillator

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  1. MODELLING OF PHYSICAL PHENOMENA Using LabView Invited Talk, Bucharest, October 1999Prof. Dr. R. LinckeInst. für Experimentelle und Angewandte Physik der Universität Kiel • Lissajous-Figures • Mother-Daughter-Problem • Simple Harmonic Oscillator • Fourier-Analysis • Logistic Equation • Feigenbaum-Diagram • QM Harmonic Oscillator • QM Radial Wavefunction of H-Atom

  2. LISSAJOUS-FIGURESProgramming x/y-Graphs One of the simplest exercises is plotting functions. Lissajous-Figures are a suitable subject. The trigonometric functions can be calculated with the formula box or with the graphical operators. Different numerical inputs can be studied here.

  3. Mother-Daughter-Problem of RadioactivitySolving 2 Coupled Differential Equations This standard problem of introductory physics is very suited to teach the method of iteration and 2-channel display in LabView.

  4. Simple Harmonic Oscillatordry and viscous damping The simple harmonic Oscillator is a very good topic for learning numerical integration. Comparing different types of friction is another good reason for this application. This standard problem of introductory physics is very suited to teach the method of iteration and 2-channel display in LabView. As far as LabView is concerned, here one can study the differences between a graph and a chart.

  5. Introduction to ChaosIterating the Logistic Equation Solving the Logistic Equation is a very suitable exercise to learn the concept of iteration: Xn+1 = P · Xn · ( 1 - Xn ) The program is exceedingly simple: P = 2,90 P = 3,5

  6. Introduction to ChaosFeigenbaum Diagram Xn+1 = P · Xn · ( 1 - Xn ) Here we vary P; after 40 iterations to eliminate transients 60 values are plotted. Rather than using 2 nested loops for n and P, only one loop is beeing used, and P is incremented after 100 steps. This is much faster.

  7. The Harmonic OscillatorIntegrating the Schrödinger-Equation Using the EULER-Method E=1,0005eV E=5,06eV E=4 eV: this is no stationary state E=3,007eV The spring constant is set so that the zero-point energy is 1eV. When running the program one has to adjust the energy until the wave function shows the required symmetry. This gives the discrete energy levels 1,0005, 3.007 and 5,06 eV (errors due to the EULER-method!).

  8. ns-Wave Functions of the H-Atomby Integrating the Schrödinger-Equation Using the EULER-Method Here the Schrödinger-equation is integrated by the Euler-method using shift registers and the formula node of LabView. When running the program, one has to adjust the energy until the wave function vanishes for r  . This yields the energy eigenvalues. n = 1 E = -13,607 eV Step 10-10 m n = 2 E = -3,467 eV Step 2·10-10 m n = 3 E = -1,511 eV Step 2·10-10 m

  9. FOURIER AnalysisA Standard Application For LabView This standard task can be solved very easily in LabView since all required procedures are available as .VIs: Signal Generator, Amplitude and Phase Spectrum, and Array to Bar Graph. All relevant parameters can be varied over a wide range. This program will play an important role in connection with many experiments.

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