Download Waves

“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
mevr  n n  1, 2, 3...
  h / 2
an integer number of wavelengths
fits into the circular orbit
n  2r
where
h

p
 is the de Broglie wavelength
In order to understand quantum mechanics, you must
understand waves!
An oscillation is a time-varying disturbance.
oscillation
restoring force : F  kx
wave
A wave is a time-varying disturbance that also propagates in space.
y  A sin t becomes
y  A sin kx  t 
(but they are nonetheless instructive)
A wave that propagates forever in
one dimension is described by:
 2x

y  A cos 
 2ft 
 

v p  f
in shorthand:
y  A cos kx  t 
  2f , k  2 / 
angular
frequency
wave
number
waves can interfere (add or cancel)
Interefering waves, generally…
y  y1  y2  A cosk1 x  1t   A cosk2 x  2t 

y  2 A cos
1
k2  k1 x  (2  1 t cos 1 k1  k2 x  1  2 t
2
2
“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:
 
 k
2 A cos
x
t
2
2


1
1



Modulating a high frequency wave within the envelope: cos  k1  k 2 x  1  2 t 
2
2

2  1  / 2  1  
the group
the phase  p 

2  1  / 2 
1
g 


k 2  k1  / 2 k1
velocity
velocity
k2  k1  / 2 k
if
1  2
the group
velocity
g 
2  1  / 2  
k2  k1  / 2 k
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.
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:
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
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.
spatially
localized
wave group
adding varying amounts of
an infinite number of waves
1
f ( x) 
2



To form a pulse that is
zero everywhere outside
of a finite spatial range
x requires adding
together an infinite
number of waves with
continuously varying
wavelengths and
amplitudes.
sinusoidal expression
for harmonics
a(k )eikx dk
amplitude of wave with
wavenumber k=2/
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 …