The Klein-Gordon string A tool I ve never heard of before Sergej Faletič

Transcription

1 The Klein-Gordon string A tool I ve never heard of before Sergej Faletič,, Jadranska 9, Ljubljana, sergej.faletic@fmf.uni-lj.si

2 Klein-Gordon string SEEMPE 5 Sergej Faletič springs/ elastic bands The Klein-Gordon string The Klein-Gordon string is best described as a long series of strongly coupled oscillators, but it can also be thought of as a string braced in transversal springs. A discrete version is easier to describe. beads springs spring coefficient Wave equation: the Klein-Gordon equation Fdx y K( x) y = dm x t y

3 Klein-Gordon string SEEMPE 5 Sergej Faletič The Klein-Gordon string Fdx y K( x) y = dm x t y We can derive the dispersion relation from which we can write the expression for phase velocity. c = c ω ω We notice a term that we can identify as the natural frequency of one of the coupled oscillators.

4 Klein-Gordon string SEEMPE 5 Sergej Faletič Dispersion Concept of phase and group velocities.

5 Klein-Gordon string SEEMPE 5 Sergej Faletič Dispersion Concept of phase and group velocities. The phase velocity is associated with an infinite wave of known frequency. The group velocity is associated with an envelope of this wave. Due to this, the actual waveform changes with time. envelope group velocity phase velocity,,,,,8,8,8,8,4,4,4,4 -, , , , ,8 -,8 -,8 -,8 -, -, -, -, This is simplified, but it communicates the main idea.

6 Klein-Gordon string SEEMPE 5 Sergej Faletič The correct approach: We decompose the initial pulse into its Fourier components, time evolve each component, taking into account its phase velocity, and then recompose the waveform at a later time., t (s) Measured, t (s) Calculated from FT,9,9,8,8,7,7,6,6,5,5,4,4,3 x (cm),3 x (cm) The pictures compare the measured waveform with the one derived by Fourier analysis.

7 Klein-Gordon string SEEMPE 5 Sergej Faletič How do we measure phase velocity,,9,8,7,6,5,4,3 t (s) We can measure the wave velocity by following the crests and troughs. We come to the conclusion that there are two velocities. One is the velocity of an individual crest or trough 8 (phase)(red and blue). x (cm) position (c cm) min, reflected, with fit max, with fit min, with fit min max, with fit t (s) The other is the velocity of the disturbance as a whole (group) (green).

8 Klein-Gordon string SEEMPE 5 Sergej Faletič A tentative explanation of the mechanics of how the trough appears after we only created a crest.,5,5,5,5,5,5,5,5,5,5-4 -,5-4 -,5-4 -,5-4 -,5-4 -, ,5 -,5 -,5 -,5 -,5 The first rises. The second rises less due to additional springs. The third rises even less. The second overshoots equilibrium Due to the additional force from springs. The third overshoots even more.

9 Klein-Gordon string SEEMPE 5 Sergej Faletič Ellyptical polarization A Klein-Gordon string can be used to show how a linear polarization can turn into an ellyptical one. Due to anisotropy of the coupled oscillators they have different natural frequencies for perpendiculat directions. c v = c ω, v ω c h = c ω, h ω The initial pulse can be decomposed into two perpendicular directions, waves in each direction have different velocities.

10 Klein-Gordon string SEEMPE 5 Sergej Faletič What happens, if I make a pulse at 45? (Unfortunately in a PDF version the animated gifs are static) We analysed the mootion by tracing the motion of each bead. (next slide) We also decomposed the initial pulse into Fourier components, time-evolved the components and recomposed them at subsequent times (next slide)

11 Klein-Gordon string SEEMPE 5 Sergej Faletič 4 3 z (a.u.) y, with offset (a.u.) -3 The measured tracks of each bead.,3,, -,,,,3,4,5,6,7,8,9,,,,3,4,5,6,7,8,9 -, Tracks predicted by Fourier analysis based on the shape of the initial pulse. The direction of propagation was z. The graphs are shown in the x,y plane. There is an offset addead to each bead so that its z position shows as an offset on the y axis.

12 Klein-Gordon string SEEMPE 5 Sergej Faletič Quantum Wavefunctions We can use the Klein-Gordon string to reproduce shapes of quantum wavefunctions with a mechanical wave. Schroedinger equation h m x ψ + V ( x) ψ = ih ψ t Klein-Gordon (wave) equation Fdx y K( x) y = dm x t y The two equationshave the same spatial part, but different temporal parts. So we can reproduce the waveshapes, but the frequencies will be different.

13 Klein-Gordon string SEEMPE 5 Sergej Faletič ω We expect to see similar shapes as we would expect for wavefunctions. ω ω Dispersion k( ω) = c ω relation < ω k i ω > ω Imaginary k. Real k. Exponential waveform. Always smaller than without dispersion. Sinusoidal waveform, greater wavelength. k k < k

14 Klein-Gordon string SEEMPE 5 Sergej Faletič Low frequency, exponential waveform Higher frequency, still exponential, but longer tail. Some energy passes to the other side. High frequency, Sinusoidal waveform, Greater wavelength

15 Klein-Gordon string SEEMPE 5 Sergej Faletič Here is the confirmation of a tunnelling waveform. The points are mesured displacements and the line is the mathematical result for the given parameters. Tunnelling Here is an opportunity to discuss what the potential means for classical waves. It represents an area where the wave must transfer some energy to an external storage (springs). They return the energy and we get reflection. How much energy the springs take from the wave depends on how much they are deformed. So if the wave vanished, there would be no deformation and no energy transfer to the springs. Hence no reason for the wave to change shape. So we have to have some deformation of the springs (some aplitude of the wave) inside the potential, even if it is very strong.

16 Klein-Gordon string SEEMPE 5 Sergej Faletič With appropriate placement of the springs and appropriate spring coefficients that represent potential, we can achieve familiar waveforms. Finite potential well Quantum harmonic oscillator

17 Klein-Gordon string SEEMPE 5 Sergej Faletič Conclusions I just wanted to show a versatile device. Mission accomplished?

18 Klein-Gordon string SEEMPE 5 Sergej Faletič References Gravel P, Gauthier C, Classical applications of the Klein-Gordon equation Am. J. Phys. 79p 447 Mouchet A, 8, Interaction with a field: a simple integrable model with backreaction Eur. J. Phys.9p 33