Physics 2D Lecture Slides Lecture 17: Feb 8th 2005 Vivek Sharma - PDF document

Physics 2D Lecture Slides Lecture 17: Feb 8th 2005 Vivek Sharma UCSD Physics A PhD Thesis Fit For a Prince • Matter Wave ! – “Pilot wave” of λ = h/p = h / ( γ mv) – frequency f = E/h • Consequence: – If matter has wave like properties then there would be interference (destructive & constructive) • Use analogy of standing waves on a plucked string to explain the quantization condition of Bohr orbits 1

De Broglie’s Explanation of Bohr’s Quantization Standing waves in H atom: Constructive interference when � � n = 2 r h h � = s ince = p ...... ( NR ) m v n = 3 nh � = � 2 r m v � = n � mvr Angular momentum Quantization condit io ! n This is too intense ! Must verify such “loony tunes” with experiment Just What is Waving in Matter Waves ? For waves in an ocean, it’s the water that “waves” For sound waves, it’s the molecules in medium For light it’s the E & B vectors that oscillate • What’s “waving” for matter waves ? – It’s the PROBABLILITY OF FINDING THE PARTICLE that waves ! – Particle can be represented by a wave packet • At a certain location (x) • At a certain time (t) • Made by superposition of many sinusoidal waves of different amplitudes, wavelengths λ and frequency f • It’s a “pulse” of probability in spacetime 2

What Wave Does Not Describe a Particle � 2 = � � + � = = � y A cos ( kx t ) k , w 2 f y � x ,t • What wave form can be associated with particle’s pilot wave? = � � + � y A cos ( kx t ) • A traveling sinusoidal wave? • Since de Broglie “pilot wave” represents particle, it must travel with same speed as particle ……(like me and my shadow) = � Phase velocity (v ) of sinusoid a l wave: v f Single sinusoidal wave of infinite p p In Matter: extent does not represent particle Conflicts with h h � = ( ) = a localized in space Relativity  � p mv � 2 E m c Unphysical = (b) f = Need “wave packets” localized h h � 2 2 E mc c Spatially (x) and Temporally (t) � = � = = = > v f c ! p � p m v v Wave Group or Wave Pulse • Wave Group/packet: Imagine Wave pulse moving along – Superposition of many a string: its localized in time and sinusoidal waves with different wavelengths and frequencies space (unlike a pure harmonic wave) – Localized in space, time – Size designated by • Δ x or Δ t – Wave groups travel with the speed v g = v 0 of particle • Constructing Wave Packets – Add waves of diff λ , – For each wave, pick Wave packet represents particle prob • Amplitude • Phase – Constructive interference over the space-time of particle – Destructive interference elsewhere ! localized 3

How To Make Wave Packets : Just Beat it ! • Superposition of two sound waves of slightly different frequencies f 1 and f 2 , f 1 ≅ f 2 • Pattern of beats is a series of wave packets • Beat frequency f beat = f 2 – f 1 = Δ f • Δ f = range of frequencies that are superimposed to form the wave packet Addition of 2 Waves with slightly + Resulting wave's "displacement " y = y y : different wavelengths and 1 2 [ ] = � + � slightly different frequencies y A cos( k x w t ) cos( k x w t ) 1 1 2 2 A+B A-B Trignometry : cosA+cos B =2cos( )cos( ) 2 2 � � � + + � � k k w w � � k k w w � � = � � y 2 A cos( 2 1 x 2 1 t ) cos( 2 1 x 2 1 t ) � � � � � � � 2 2 � � 2 2 � � � � � � � � � since k k k , w w w , k � k , w � w 2 1 ave 2 1 ave � � � � � k w � � = � � � ' � y 2 A cos( x t ) cos( kx w t ) y = A cos( kx wt ) , A' oscillates in x,t � � � � � � 2 2 � � � � � � k w t = � A ' 2 A cos( x ) = modulated amplit ud e � � � 2 2 � w = Phase Vel V ave p k ave � w = � Group Vel V g k dw V : Vel of envelope= Wave Group Or packet g dk 4

Non-repeating wave packet can be created thru superposition Of many waves of similar (but different) frequencies and wavelengths Wave Packet : Localization •Finite # of diff. Monochromatic waves always produce INFINTE sequence of repeating wave groups  can’t describe (localized) particle •To make localized wave packet, add “ infinite” # of waves with Well chosen Ampl A, Wave# k, ang. Freq. w � � � = � ( , ) x t A ( ) k e i k ( x wt ) dk �� = A ( ) k Amplitude Fn x � diff waves of diff k have different amplitudes A(k) v g t w = w(k), depends on type of wave, media dw = Group Velocity V g dk k = k 0 localized 5

Group, Velocity, Phase Velocity and Dispersion = In a Wave Packet: w w k ( ) dw = Group Velocity V g dk = k k 0 = � = Since V wk ( def ) w k V p p dV dw � = = + p V V k g p k k dk dk = 0 = k k 0 = k or � usu ally V V ( ) 1ns laser pulse disperse p p � Material in which V varies with are said to be Dispersive By x30 after travelling p Individual harmonic waves making a wave pulse travel at 1km in optical fiber different V thus changing shape of pulse an d b ecome spread out p = In non-dispersive media, V V g p dV � p In dispersive media V V ,depends on g p dk Group Velocity of Wave Packets: V g Consider An Electron: mass = m velocity = v, momentum = p � 2 � 2 � � = � 2 Energy E = hf = mc ; = 2 f mc h � � � h 2 2 � � = Wavelength = ; k = k mv x � p h dw dw dv / = = Group Velocity : V v g t g dk dk dv / � � 2 � � � mc 2 � � = � � � � � dw d 2 mv dk d 2 2 m h = = = & m v � � � � v v v v dv dv dv dv � [1- ( ) ] 2 1/ 2 � h [1-( ) ] 2 3/ 2 � h [1-( ) ] 2 1/ 2 � h[1-( ) 2 3/ 2 ] � � � � c c c c dw dw dv / = = = � V v Group velocity of electron Wave packet "pilot wave" g dk dk / dv is same as el ect ron's physical v e loc t i y w c 2 = = > But velocity of individual waves making up the wave packet V c ! (not physical ) p k v 6

Wave Packets & Uncertainty Principles We added two Sinusoidal waves � � � � � k w � = � � y 2 A cos( x t ) cos( kx wt ) � � � � � 2 2 � � � Amplitude Modulation x 2 x 1 • Distance Δ X between adjacent minima = (X 2 ) node - (X 1 ) node • Define X 1 =0 then phase diff from X 1  X 2 = π ( similarly for t 1  t 2 ) � � w k � Node at y = 0 = 2A cos ( t x ), Examine x or t behavior What can 2 2 we � � � = � � � in x: k . x Need to combine many waves of diff. to make small k x pulse learn � from � � � � � � � x= , for small x 0 k & Vi ce Verca � k this simple � � = � � � � a d n In t : w . t Need to combine many waves of diff to make small t pulse model � � � � � � � � � ? t = , for small t 0 & Vice V e r ca � � Signal Transmission and Bandwidth Theory • Short duration pulses are used to transmit digital info – Over phone line as brief tone pulses – Over satellite link as brief radio pulses – Over optical fiber as brief laser light pulses • Ragardless of type of wave or medium, any wave pulse must obey the fundamental relation » ΔωΔ t ≅ π • Range of frequencies that can be transmitted are called bandwidth of the medium • Shortest possible pulse that can be transmitted thru a medium is Δ t min ≅ π / Δω • Higher bandwidths transmits shorter pulses & allows high data rate 7

Wave Packets & Uncertainty Principles of Subatomic Physics � 2 h � � = � � in space x: k . x since k = , p = � � � � = � p . x h / 2 � � � � p . x / 2 usual ly one writes approximate relation � � = � � � � = In time t : w . t since =2 f E , hf � � � = E . t h / 2 � � � � E . t / 2 usually one write s approximate re lation What do these inequalities mean physically? Know the Error of Thy Ways: Measurement Error  Δ • Measurements are made by observing something : length, time, momentum, energy • All measurements have some (limited) precision`…no matter the instrument used • Examples: How long is a desk ? L = (5 ± 0.1) m = L ± Δ L (depends on ruler used) – How long was this lecture ? T = (50 ± 1)minutes = T ± Δ T (depends on the accuracy of – your watch) How much does Prof. Sharma weigh ? M = (1000 ± 700) kg = m ± Δ m – • Is this a correct measure of my weight ? – Correct (because of large error reported) but imprecise – My correct weight is covered by the (large) error in observation Length Measure Voltage (or time) Measure 8

Measurement Error : x ± Δ x • r • Measurement errors are unavoidable since the measurement procedure is an experimental one • True value of an measurable quantity is an abstract concept • In a set of repeated measurements with random errors, the distribution of measurements resembles a Gaussian distribution characterized by the parameter σ or Δ characterizing the width of the distribution Measurement error smaller Measurement error large Interpreting Measurements with random Error : Δ True value 9