Everything is a wave

1. Everything is a wave “Okay…what you have to realize is…to first order…everything is a simple harmonic oscillator. Once you’ve got that, it’s all downhill from there.” - Some Random Physicist

2. Why quantization of angular momentum? an integer number of wavelengths fits into the circular orbit where l is the de Broglie wavelength In order to understand quantum mechanics, you must understand waves!

3. Travelling Waves wave A wave is a time-varying disturbance that also propagates in space. y = A sin k x y = A sin k (x-v t) = A sin ( k x – ω t) Lets play with some waves: https://phet.colorado.edu/sims/html/wave-on-a-string/latest/wave-on-a-string_en.html

4. "Freshman" waves go on forever... (but they are nonetheless instructive) A wave that propagates forever in one dimension is described by: in shorthand: angular frequency wave number

5. Screen shot of the wave Screen shot of a wave at a fixed time is a sine wave: y = A sin (k x- w 0) = A sin kx For most of our purposes the reverse is also true – a wave is a traveling sine wave: y = A sin (k (x- v t)) = A sin (kx – w t) So understanding the sine function well is crucial to understanding waves

6. Interference waves can interfere (add or cancel) But one can play more complicated games: http://www.falstad.com/fourier/

7. Interefering waves, generally… “Beats” occur when you add two waves of slightly different frequency. They will interfere constructively in some areas and destructively in others. Can be interpreted as a sinusoidal envelope: Modulating a high frequency wave within the envelope:

8. adding varying amounts of an infinite number of waves spatially localized wave group sinusoidal expression for harmonics amplitude of wave with wavenumber k=2p/l FOURIER THEOREM: any wave packet can be expressed as a superposition of an infinite number of harmonic waves Adding several waves of different wavelengths together will produce an interference pattern which begins to localize the wave. To form a pulse that is zero everywhere outside of a finite spatial range Dx requires adding together an infinite number of waves with continuously varying wavelengths and amplitudes.

9. WAVE PACKETS The word “particle” in the phrase “wave-particle duality” suggests that this wave is somewhat localized. How do we describe this mathematically? …or this …or this

10. The Uncertainty Principle Remember our sine wave that went on “forever”? We knew its momentum very precisely, because the momentum is a function of the frequency, and the frequency was very well defined. But what is the frequency of our localized wave packet? We had to add a bunch of waves of different frequencies to produce it. Consequence: The more localized the wave packet, the less precisely defined the momentum.

11. “Beats” occur when you add two waves of slightly different frequency. They will interfere constructively in some areas and destructively in others. Can be interpreted as a sinusoidal envelope: Modulating a high frequency wave within the envelope: the group velocity the phase velocity if For dispersive waves group and phase velocities are different See also simulation: Fun with wavepackets

12. Dispersion

13. Waves with boundaries Standing waves (harmonics) Ends (or edges) must stay fixed. That’s what we call a boundary condition. This is an example of a Bessel function.

14. Reflection How does this wave behave at a boundary? at a free (soft) boundary, the restoring force is zero and the reflected wave has the same polarity (no phase change) as the incident wave at a fixed (hard) boundary, the displacement remains zero and the reflected wave changes its polarity (undergoes a 180o phase change) Lets play with some waves: https://phet.colorado.edu/sims/html/wave-on-a-string/latest/wave-on-a-string_en.html

15. When a wave encounters a boundary which is neither rigid (hard) nor free (soft) but instead somewhere in between, part of the wave is reflected from the boundary and part of the wave is transmitted across the boundary In this animation, the density of the thick string is four times that of the thin string …

16. Bessel Functions: are simply the solution to Bessel’s equations: Occurs in problems with cylindrical symmetry involving electric fields, vibrations, heat conduction, optical diffraction. Spherical Bessel functions arise in problems with spherical symmetry. de Broglie’s concept of an atom… Legendre’s equation: comes up in solving the hydrogen atom It has solutions of:

17. Next time: the realization that matter has wavelike properties

18. Double Slit experiment for waves

19. Double Slit experiment for particles 2 Slits, 1 electron 2 Slits, arbitrary energy electrons 1 Right Slit, many electrons 2 Slits, tuned energy electrons 1 Left Slit, many electrons Uncertainty Principle -> fuzziness

20. What does it mean? • A large number of electrons going through a double slit will produce an interference pattern, like a wave. • However, each electron makes a single impact on a phosphorescent screen-like a particle. • Electrons have indivisible (as far as we know) mass and electric charge, so if you suddenly closed one of the slits, you couldn’t chop the electron in half-because it clearly is a particle. • A large number of electrons fired at two simultaneously open slits, however, will eventually, once you have enough statistics, form an intereference pattern. Their cumulative impact is wavelike. • This leads us to believe that the behavior of electrons is governed by probabilistic laws. --The wavefunction describes the probability that an electron will be found in a particular location. (see animation https://www.youtube.com/watch?v=Xmq_FJd1oUQ)