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PHYS 420-SPRING 2006 Dennis Papadopoulos LECTURE 13 MATTER WAVES WAVEPACKETS Fig. 4-22, p. 132 Fig. 4-21, p. 131 Fig. 4-15, p. 127 C n2 (n 2 22 ) Balmer Series Fig. 4-20, p. 129 Fig. 4-23, p. 133 Fig. 4-24, p. 134 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? Phase cancellation me vr n n 1, 2, 3... h / 2 an integer number of wavelengths fits into the circular orbit n 2r where Photons p=hn/c= h/ h p is the de Broglie wavelength In order to understand quantum mechanics, you must understand 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.” An oscillation is a time-varying disturbance. oscillation restoringforce: 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: 2x y A cos 2ft v p f in shorthand: y A coskx t 2f , k 2 / angular frequency wave number waves can interfere (add or cancel) Standing Waves Interefering waves, generally… y y1 y2 A cosk1 x 1t A cosk 2 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 k1 k2 x 1 1 2 t 2 2 / 2 1 p 2 1 k2 k1 / 2 k1 1 Modulating a high frequency wave within the envelope: cos the group velocity / 2 g 2 1 k2 k1 / 2 k the phase velocity if 1 2 the group velocity g 2 1 / 2 k2 k1 / 2 k 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. 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 )eikxdk 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. E 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. p k 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 …