Interference
waves can interfere (add or cancel)
But one can play more complicated
games:
http://www.falstad.com/fourier/
“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:
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
D
x requires adding
together an infinite
number of waves with
continuously varying
wavelengths and
amplitudes.
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
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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.
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http://demonstrations.wolfram.com/EvolutionOfAGaussianWavePacket/
Delve into the fascinating world of interference patterns created by waves and discover the mathematical description of waves with localized characteristics in the wave-particle duality concept. Uncover the implications of the Uncertainty Principle on wave packets and Gaussian wavepackets. Explore the Fourier Theorem and how any wave packet can be expressed as a superposition of harmonic waves, leading to a better understanding of wave behavior in different scenarios.
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Presentation Transcript
Interference waves can interfere (add or cancel) But one can play more complicated games: http://www.falstad.com/fourier/
Interefering waves, generally ( ) ( ) = + = + cos cos y y y A k x t A k x t 1 2 1 1 2 2 1 1 ( ) ) ( ) ( ) t = + + 2 cos ( cos y A k k x t k k x 2 1 2 1 1 2 1 2 2 2 Beats occur when you add two waves of slightly different frequency. They will interfere constructively in some areas and destructively in others. k 2 cos A x t Can be interpreted as a sinusoidal envelope: 2 2 1 1 ( ) ( ) + + cos k k x t Modulating a high frequency wave within the envelope: 1 2 1 2 2 2
FOURIER THEOREM: any wave packet can be expressed as a superposition of an infinite number of harmonic waves 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. Adding several waves of different wavelengths together will produce an interference pattern which begins to localize the wave. adding varying amounts of an infinite number of waves spatially localized wave group sinusoidal expression for harmonics 1 + = ikx ( ) ( ) f x a k e dk 2 amplitude of wave with wavenumber k=2 /
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
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.
Uncertainty Relation for Gaussian Wavepackets ???? /? http://demonstrations.wolfram.com/EvolutionOfAGaussianWavePacket/
