Physics 3313 - Lecture 8

3313 Andrew Brandt 1

Physics 3313 - Lecture 8

2/18/2009

Wednesday February 18, 2009Dr. Andrew Brandt

1. Phase Velocity and Group Velocity2. Particle in a box3. Uncertainty Principle


Wave Equation

2/18/2009

cos( )y A t kx 2

2k

• 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)

2/18/2009

cos( )y A t kx

Phase Velocity vs. Group Velocity

( )pv g pv v


Beats Example• Consider two waves, y1 and y2 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]

2/18/2009

1 cos( )y A t kx 2 cos( )y A t k k x

1 1cos cos 2cos ( ) cos ( )2 2

1 21 12 cos (2 ) (2 ) cos ( )2 2

y y y A t k k x t kx

cos( ) cos( )

2 cos cos( / 2 / 2)y A t kx t kx


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

2/18/2009

2 / 2pv k

/gd d dkvdk dv dv

2

2 2 mcvh

32 2 2

2

(1 / )

d mvdv

h v c

2 2 mvkh

gdv

k dk

g pv v c

32 2 2

2

(1 / )

dk mdv

h v c

Note that since both and v have velocity dependence, there are twoterms that must be added together andsimplified (as in Ex. 1.5)

/gd dkv vdv dv

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

2/18/2009


• 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 0.511 MeV

2/18/2009

Davisson and Germer (cont.)

2 sinn d

65

0.165nm 2KEvm

2mv mKEhmv

0.166nm


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

2/18/2009

2n

Ln

hp


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 box2) 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

2/18/2009

2 22

2

12 2 2

p hKE mvm m

22 2 2

2 / 82 (2 / )n

hE n h mLm L n


Example 3.5• 10 g marble in 10 cm box, find energy levels

• For n=1 and• Looks suspiciously like a stationary marble!!• At reasonable speeds n=1030

• Quantum effects not noticeable for classical phenomena

2/18/2009

2 34 264 2

2 1 2

(6.63 10 ) 5.5 108 10 (10 )n J s n J

kg m

2 2 2/ 8nE n h mL

64min 5.5 10E J 313.3 10 /v m s


Uncertainty Principlea) 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?)2/18/2009

2x p

2E t


Gaussian Distribution

2/18/2009

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

Documents

Physics 3313 - Lecture 8