Physics Lecture 8 2/18/ Andrew Brandt Wednesday February 18, 2009 Dr. Andrew Brandt 1.Phase Velocity and Group Velocity 2.Particle in a box 3.Uncertainty Principle

Wave Equation 2/18/ Andrew Brandt2 General Wave Equation: where is the angular frequency (a particle moving in a circle  times /second sweeps over 2  radians and wave number is the number of radians in a wave train 1 m long, since 2  radians is one wavelength (or how many wavelengths fit in 2  )

The wave equation: represents an infinite series of waves with same amplitude and thus clearly does not represent a moving particle, which should be localized and represented by a wave packet or group A wave group is the superposition of waves with different wavelengths or frequencies If different wavelengths do not proceed together and there is dispersion Example of wave group is beats: 2 waves with similar amplitude and slightly different frequency: example 440 and 442 Hz. We hear fluctuation sound of 441 Hz with two loudness peaks (beats) per second (tune violin) 2/18/ Andrew Brandt3 Phase Velocity vs. Group Velocity

Beats Example Consider two waves, y 1 and y 2 with slightly different frequency What happens if we add them? To evaluate this we need to use some trig identities: and The first term is the sum of the two waves and basically has twice the amplitude and same frequency (precisely it would have the average frequency), while the second term is the modulation term of the orginal wave, which gives the wah wah sound (successive wave groups) [board] 2/18/ Andrew Brandt4

More about Wave Velocities Definition of phase velocity: Definition of group velocity: For light waves : For de Broglie waves where v is velocity of particle 2/18/ Andrew Brandt5 Note that since both  and v have velocity dependence, there are two terms that must be added together and simplified (as in Ex. 1.5) Wave group travels with velocity of the particle!

Diffraction: Davisson and Germer Davisson and Germer in U.S. and Thomson in UK confirmed de Broglie hypothesis in 1927 by demonstrating diffraction (wave phenomena) of electron beams off crystal Classically electrons scattered in all directions with minor dependence on intensity and energy Initial results for scattering off block of nickel favored classical interpretation! Then accident allowed air in to setup which oxidized the surface of the nickel block, so they baked (heated) it to get rid of impurities Suddenly results differed: distinct maxima and minima with position dependent on incident electron energy 2/18/ Andrew Brandt6

How can the results be explained and why did the change occur? Heating causes nickel to form a single large crystal instead of many small ones Electron “waves” diffracted off single crystal Bragg equation for diffractive maxima: d=0.091 from x-ray diffraction measurement angle of incidence and scattering relative to Bragg planes (families of parallel planes in crystal—see Sec. 2.6) for n=1 maxima Obtain …and the point is? From with obtain Plugging in gives ! Hello Stockholm! Why non-relativistic? 54 eV small compared to MeV 2/18/ Andrew Brandt7 Davisson and Germer (cont.)

Particle in a Box Particle moves back and forth bouncing off infinitely hard walls (we’ve all been there) with non-relativistic velocity Why infinitely hard walls? So particle loses no energy in collision From wave POV particle in a box is like a standing wave in a string stretched between walls: transverse displacement of string or  for particle must be zero at walls since wave stops there General formula for permitted wavelengths: Limits on wavelength imply limits on momentum through and consequently on KE 2/18/ Andrew Brandt8

Particle in a Box Still General expression for non-rel Kinetic Energy: With no potential energy in this model, and applying constraint on wavelength gives: Each permitted E is an energy level, and n is the quantum number General Conclusions: 1) Trapped particle cannot have arbitrary energy like a free particle—only specific energies allowed depending on mass and size of box 2) Zero energy not allowed! v=0 implies infinite wavelength, which means particle is not trapped 3) h is very small so quantization only noticeable when m and L are also very small 2/18/ Andrew Brandt9

Example g marble in 10 cm box, find energy levels For n=1 and Looks suspiciously like a stationary marble!! At reasonable speeds n=10 30 Quantum effects not noticeable for classical phenomena 2/18/ Andrew Brandt10

Uncertainty Principle a)For a narrow wave group the position is accurately measured, but wavelength and thus momentum cannot be precisely determined b)Conversely for extended wave group it is easy to measure wavelength, but position uncertainty is large Werner Heisenberg 1927, it is impossible to know exact momentum and position of an object at the same time: This is “derived” in book using wave approach Note this is not an apparatus error, but an unknowability of quantities Can’t know perfectly where a particle is and where it’s going: future is not determined, just have probabilities! Sounds like philosophy! (Why is this not a problem for NASA?) 2/18/ Andrew Brandt11

Gaussian Distribution 2/18/ Andrew Brandt12 Integrating over 2 standard deviations gives 95.4%, generally need 4-5  for discovery (fluctuation probability less than 1/100000) Ex.: top quark discovery