1Mon. Mar. 26, 2007 Phy107 Lect 261From Last Time…Matter particles are waveswith wavelength λ=h/p.Matter waves can behave as particles by making asuperposition (wave packet)Leads to uncertainty principle:Spread in position and spread in momentum ‘trade off’Reminder: essay topic and paragraph due Wed. March 28Mon. Mar. 26, 2007 Phy107 Lect 262Making a particle out of waves440 Hz +439 Hz440 Hz +439 Hz +438 Hz440 Hz +439 Hz +438 Hz +437 Hz +436 HzMon. Mar. 26, 2007 Phy107 Lect 263Spatial extentof localized sound wave• Δx = spatial spread of ‘wave packet’• Spatial extent decreases as the spread inincluded wavelengths increases.-8-4048-15 -10 -5 0 5 10 15JΔxMon. Mar. 26, 2007 Phy107 Lect 264Same occurs for a matter wave• Construct a localized particle by adding togetherwaves with slightly different wavelengths.• Since de Broglie says λ = h /p, each of thesecomponents has slightly different momentum.– We say that there is some ‘uncertainty’ in the momentum• And still don’t know exact location of the particle!– Wave still is spread over Δx (‘uncertainty’ in position)– Can reduce Δx, but at the cost of increasing the spread inwavelength (giving a spread in momentum).Mon. Mar. 26, 2007 Phy107 Lect 265Interpreting• For sound, we would just say that the sound pulse iscentered at some position, but has a spread.• Can’t do that for a quantum-mechanical particle.• Many measurements indicate that the electron isindeed a point particle.• Interpretation is that the magnitude of electron ‘wave-pulse’ at some point in space determines theprobability of finding the electron at that point.-8-4048-15 -10 -5 0 5 10 15JMon. Mar. 26, 2007 Phy107 Lect 266Heisenberg Uncertainty Principle• Using– Δx = position uncertainty– Δp = momentum uncertainty• Heisenberg showed that the product ( Δx ) • ( Δp ) is always greater than ( h / 4π )Often write this aswhere is pronounced ‘h-bar’Planck’sconstant ! "x( )"p( )~ h /2 ! h =h2"2Mon. Mar. 26, 2007 Phy107 Lect 267Example: ‘Particle in a box’Particle confined to a fixed region of spacee.g. ball in a tube- ball moves only along length L• Classically, ball bounces back and forth in tube.– No friction, so ball continues to bounce back and forth,retaining its initial speed.– This is a ‘classical state’ of the ball. A different classical state wouldbe ball bouncing back and forth with different speed.– Could label each state with a speed,momentum=(mass)x(speed), or kinetic energy.– Any momentum, energy is possible.Can increase momentum in arbitrarily small increments.LMon. Mar. 26, 2007 Phy107 Lect 268How fast is it moving?• Box is stationary, so average speed is zero.• But remember the classical version• Particle bounces back and forth.– On average, velocity is zero.– But not instantaneously– Sometimes velocity is to left, sometimes to rightLMon. Mar. 26, 2007 Phy107 Lect 269Quantum momentum• Quantum version is similar. Both contributions• Ground state is a standing wave, made equally of– Wave traveling right ( positive momentum +h/λ )– Wave traveling left ( negative momentum - h/λ )! "= 2LOne half-wavelength! p =h"=h2LmomentumLMon. Mar. 26, 2007 Phy107 Lect 2610Where is the particle?• Can say that the particle is inside the box,(since the probability is zero outside the box),but that’s about it.• The wavefunction extends throughout the box,so particle can be found anywhere inside.• Can’t say exactly where the particle is,but I can tell you how likely you are to find at aparticular location.Mon. Mar. 26, 2007 Phy107 Lect 2611Uncertainty and WavefunctionsPosition uncertainty = LMomentum uncertainty from! "h# to +h# = 2h#=hL(Since λ=2L)Reducing the box size reduces position uncertainty,but the momentum uncertainty goes up!L! "= 2LOne half-wavelengthThe product is constant:(position uncertainty)x(momentum uncertainty) ~ hMon. Mar. 26, 2007 Phy107 Lect 2612Other quantum mechanical states?• Wave nature of electron gives constraints onquantum mechanical states in same way asclassical mechanical waves are constrained• Basically same as resonance condition inclassical objects.3Mon. Mar. 26, 2007 Phy107 Lect 2613Resonance• Most physical objects will vibrate at some set ofnatural frequencies– Ringing bell– Wine glass– Musical instrument• The electrons in an atom analogous tosound waves in a musical instrument.• In instrument, only certain pitches produced,corresponding to particular vibration wavelengths.• Since the electrons in the box are waves, onlycertain wavelengths are allowed.Mon. Mar. 26, 2007 Phy107 Lect 2614Resonance on a string• Think about in a normal wind instrument, orvibrations of a string.• Wind instrument with particular fingering plays aparticular pitch, particular wavelength.• Guitar string vibrates at frequency, wavelengthdetermined by string length.λ=L/2f=v/λMon. Mar. 26, 2007 Phy107 Lect 2615Resonances of a stringFundamental,wavelength 2L/1=2L,frequency f1st harmonic,wavelength 2L/2=L,frequency 2f2nd harmonic,wavelength 2L/3,frequency 3fλ/2λ/2λ/2n=1n=2n=3n=4frequency. . .Vibrational modes equallyspaced in frequencyMon. Mar. 26, 2007 Phy107 Lect 2616String resonancesA string has a fundamental frequency of 440 Hz.If I pluck it so that it vibrates at the firstharmonic (half the wavelength) what is thefrequency?A. 440 HzB. 220 HzC. 880 HzWavelength has decreased byfactor of 2. Since f=v/λ,frequency has gone up byfactor of two.Mon. Mar. 26, 2007 Phy107 Lect 2617Not always equally spaced• Vibrational modes are accurately shown by means ofholographic interferometry, which displays a contour map ofthe vibration. Several modes of vibration of a wine glass areshown in the photo below. Points of maximum motion, whichoccur around the rim, appear as "bull's eyes." The vibrationalamplitude changes by half a wavelength of light (316 nm) inmoving from one bright fringe to the next one. Thefundamental mode, designated as the (2,0) mode, has foursuch regions, with the glass moving in alternate directions as itvibrates.n=2n=3n=4n=5n=6n=7frequencyVibrational modesunequally spacedMon. Mar. 26, 2007 Phy107 Lect 2618Quantum Particle in a Box• In Quantum Mechanics, ball represented by wave– Wave reflects back and forth from the walls.– Reflections cancel unless wavelength meets thestanding wave condition:integer number of half-wavelengths fit in the tube.! "= LTwo half-wavelengths! p =h"=hL= 2 pomomentum!
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