“ 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.” - a crazy friend and colleague of Professor Hoffman

In order to understand quantum mechanics, you must understand waves! an integer number of wavelengths fits into the circular orbit where l is the de Broglie wavelength

An oscillation is a time-varying disturbance. oscillation

wave A wave is a time-varying disturbance that also propagates in space.

(but they are nonetheless instructive) A wave that propagates forever in one dimension is described by: in shorthand: angular frequency wave number

waves can interfere (add or cancel)

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

the group velocity Listen to the beats!

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.

Legendre’s equation: comes up in solving the hydrogen atom It has solutions of: de Broglie’s concept of an atom… 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.

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

FOURIER THEOREM: any wave packet can be expressed as a superposition of an infinite number of harmonic waves spatially localized wave group amplitude of wave with wavenumber k=2 p / l adding varying amounts of an infinite number of waves sinusoidal expression for harmonics 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.

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.

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)

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 …