Physics 2D Lecture Slides Feb 11 Vivek Sharma UCSD Physics
Just What is Waving in Matter Waves ? • For waves in an ocean, it’s the Imagine Wave pulse moving along water that “waves” a string: its localized in time and • For sound waves, it’s the space (unlike a pure harmonic wave) molecules in medium • For light it’s the E & B vectors • What’s waving for matter waves ? – It’s the PROBABLILITY OF FINDING THE PARTICLE that waves ! – Particle can be represented by a wave packet in • Space • Time • Made by superposition of many sinusoidal waves of different λ • It’s a “pulse” of probability
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 localized in space Relativity � γ p mv γ 2 E m c Unphysical = (b) f = Need “wave packets” localized h h γ 2 2 E mc c Spatially (x) and Temporally (t) ⇒ = λ = = = > v f c ! p γ 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 space (unlike a pure harmonic wave) and frequencies – 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
Making Wave Packets: Simple Model with 2 waves Ex: Phenomenon of "Be ating" in S oun d: λ Add two waves of slightly different , f + f f f -f ⇒ ∝ Wave with : f = 1 2 , Amplitude A 1 2 ave 2 2 Start with two waves π 2 = − = − = = π y ACos k x ( w t ), y A Cos k x ( w t ) : k , w 2 f 1 1 1 2 2 2 λ
+ Resulting wave's "displacement " y = y y : 1 2 [ ] = − + − 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( ks 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 wave ∆ w = ∆ Group Group Vel V g k Or packet dw V : Vel of envelope= g dk
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 ( ) 1ns laser pulse disperse p p 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
Matter Wave Packets 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 λ p h dw dw dv / = = Group Velocity : V g 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 Principle ∆ ∆ k w = − − y 2 A cos( x t ) cos( kx wt ) 2 2 Amplitude Modulation • Distance ∆ X between adjacent minima = (X 2 ) node - (X 1 ) node • Define X 1 =0 then phase diff from X 1 � X 2 = π ∆ ∆ w k − Node at y = 0 = 2A cos ( t x ) 2 2 ⇒ ∆ ∆ = π ⇒ ∆ What does k . x Need to combine more to make small k x packet ⇒ ∆ ∆ = also implies p . x h / 2 This mean? and ∆ ∆ = π ⇒ ω ∆ w t . Need to combine more to make small t packet ⇒ ∆ ∆ = a lso E . t h / 2
Recommend
More recommend
