LECTURE 13 MATTER WAVES WAVEPACKETS PHYS 420-SPRING 2006 Dennis Papadopoulos.
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Transcript of LECTURE 13 MATTER WAVES WAVEPACKETS PHYS 420-SPRING 2006 Dennis Papadopoulos.
We said earlier that Bohr was mostly right…so where did he go
wrong?
• Failed to account for why some spectral lines are stronger than others. (To determine transition probabilities, you need QUANTUM MECHANICS!) Auugh!
• Treats an electron like a miniature planet…but is an electron a particle…or a wave?
2/
...3,2,1
h
nnvrme
In order to understand quantum mechanics, you must understand waves!
an integer number of wavelengths fits into the circular orbit
rn 2
where
p
h
is the de Broglie wavelength
Photonsp=h/c=
h/
Phase cancellation
“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.”
(but they are nonetheless instructive)
fv
ftx
Ay
p
2
2cos
/2,2
cos
kf
tkxAy
A wave that propagates forever in one dimension is described by:
in shorthand:
angular frequency
wave number
“Beats” occur when you add two waves of slightly different
frequency. They will interfere constructively
in some areas and destructively in others.
txkktxkkAy
txkAtxkAyyy
21211212
221121
2
1cos(
2
1cos2
coscos
Interefering waves, generally…
Can be interpreted as a sinusoidal envelope:
txk
A22
cos2
Modulating a high frequency wave within the envelope:
txkk 2121 2
1
2
1cos
kkkg
2/
2/
12
12the group velocity
1
1
1
12
12
2/
2/
kkkpthe phase
velocity
if 21
kkkg
2/
2/
12
12the group velocity
Beats
http://www.school-for-champions.com/science/soundbeat.htm
http://library.thinkquest.org/19537/java/Beats.html
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
…or this
FOURIER THEOREM: any wave packet can be expressed as a superposition of an infinite
number of harmonic waves
FOURIER THEOREM: any wave packet can be expressed as a superposition of an infinite
number of harmonic waves
dkekaxf ikx)(
2
1)(
spatially localized wave group
amplitude of wave with wavenumber k=2/
adding varying amounts of an infinite number of
wavessinusoidal 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 x 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.
Consequence: The more localized the wave packet, the less precisely defined the momentum.
kp
E
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 …