Physics 2D Lecture Slides Lecture 17: Feb 10 th Vivek Sharma UCSD - PDF document

Physics 2D Lecture Slides Lecture 17: Feb 10 th Vivek Sharma UCSD Physics

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 What Wave Does Not Describe a Particle π 2 = − ω + Φ = = π y A cos ( kx t ) k , w 2 f y λ x ,t • What wave form can be associated with particle’s pilot wave? = − ω + Φ 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) = λ 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 λ = ( ) = a Relativity � γ localized in space p mv γ 2 E m c = Unphysical (b) f = Need “wave packets” localized h h γ 2 2 E mc c ⇒ = λ = = = > Spatially (x) and Temporally (t) v f c ! γ p p m v v

Wave Group or Wave Pulse • Wave Group/packet: Imagine Wave pulse moving along – Superposition of many sinusoidal a string: its localized in time and waves with different wavelengths and frequencies space (unlike a pure harmonic wave) – Localized in space, time – Size designated by • ∆ x or ∆ t – Wave groups travel with the speed v g = v 0 of particle • Constructing Wave Packets – Add waves of diff λ , Wave packet represents particle prob – For each wave, pick • Amplitude • Phase – Constructive interference over the space-time of particle – Destructive interference elsewhere ! localized 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 = ∆ f • ∆ 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 ∴ = − − 2 1 2 1 2 1 2 1 ⎜ ⎟ ⎜ ⎟ y 2 A ⎢ cos( x t ) cos( x 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 = ave Phase Vel V p k ave ∆ w = ∆ Group Vel V g k dw V : Vel of envelope= Wave Group Or packet g dk 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 ∞ ∫ ψ = − i k ( x wt ) ( , ) x t A ( ) k e 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 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 λ usu ally V V ( ) p p 1ns laser pulse disperse λ 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 π 2 γ ω π = γ 2 2 Energy E = hf = mc ; = 2 f mc h π π γ h 2 2 λ ⇒ = Wavelength = ; k = k mv x λ p h dw dw dv / = = Group Velocity : V g v g t dk dk dv / π ⎡ ⎤ ⎡ ⎤ 2 2 mc ⎢ ⎥ = ⎢ ⎥ π π π dw d 2 mv dk d 2 2 m h = = = ⎢ ⎥ & ⎢ m v ⎥ v v v v dv dv dv dv ⎢ ⎥ ⎢ ⎥ 2 1/ 2 2 3/ 2 2 1/ 2 2 3/ 2 [1- ( ) ] h [1-( ) ] h [1-( ) ] h[1-( ) ] ⎣ ⎦ ⎣ ⎦ 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 2 w c = = > But velocity of individual waves making up the wave packet V c ! (not physical ) p k v 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 ∆ 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 2 2 What can we ⇒ ∆ ∆ = π ⇒ ∆ in x: k . x Need to combine many waves of diff. to make small k x pulse learn π ∆ ∆ → ⇒ ∆ → ∞ from x= , for small x 0 k & Vi ce Verca ∆ k this ∆ ∆ = π ⇒ ω ∆ simple a d n In t : w . t Need to combine many waves of diff to make small t pulse model π ∆ ∆ → ⇒ ∆ ω → ∞ t = , for small t 0 & Vice V e r ca ? ∆ ω

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 ∆ω∆ 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 ∆ t min ≅ π / ∆ω • Higher bandwidths transmits shorter pulses & allows high data rate Wave Packets & Uncertainty Principles of Subatomic Physics π 2 h ∆ ∆ = π ⇒ in space x: k . x since k = , p = λ λ ∆ ∆ = ⇒ p x . h / 2 ∆ ∆ ≥ � p x . / 2 usual ly one writes approximate relation ∆ ∆ = π ⇒ ω π = 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 � ∆ • 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 ± ∆ L (depends on ruler used) – How long was this lecture ? T = (50 ± 1)minutes = T ± ∆ T (depends on the accuracy of – your watch) How much does Prof. Sharma weigh ? M = (1000 ± 700) kg = m ± ∆ 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 Measurement Error : x ± ∆ 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 ∆ characterizing the width of the distribution Measurement error smaller Measurement error large

Interpreting Measurements with random Error : ∆ True value Where in the World is Carmen San Diego? • Carmen San Diego hidden inside a big box of length L • Suppose you can’t see thru the (blue) box, what is you best estimate of her location inside box (she could be anywhere inside the box) x X=0 X=L Your best unbiased measure would be x = L/2 ± L/2 There is no perfect measurement, there are always measurement error