Physics 2D Lecture Slides Feb 11 Vivek Sharma UCSD Physics Just - - PowerPoint PPT Presentation

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

water that “waves”

• For sound waves, it’s the

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

Imagine Wave pulse moving along a string: its localized in time and space (unlike a pure harmonic wave)

What Wave Does Not Describe a Particle

• What wave form can be associated with particle’s pilot wave?
• A traveling sinusoidal wave?
• Since de Broglie “pilot wave” represents particle, it must travel with same speed

as particle ……(like me and my shadow)

cos ( ) y A kx t ω = − + Φ cos ( ) y A kx t ω = − + Φ x,t y

2 , 2 k w f π π λ = =

p 2 2 p 2 p

In Matter: h ( ) = Phase velocity

• f sinusoid

E (b) f = a l wave: (v ) v h ! v E mc c f c p h a p mv v m m h f v c λ γ γ γ λ λ γ = = = = = = > = ⇒

Conflicts with Relativity Unphysical Single sinusoidal wave of infinite extent does not represent particle localized in space Need “wave packets” localized Spatially (x) and Temporally (t)

Wave Group or Wave Pulse

• Wave Group/packet:

– Superposition of many sinusoidal waves with different wavelengths and frequencies – Localized in space, time – Size designated by

• ∆x or ∆t

– Wave groups travel with the speed vg = v0 of particle

• Constructing Wave Packets

– Add waves of diff λ, – For each wave, pick

• Amplitude
• Phase

– Constructive interference over the space-time of particle – Destructive interference elsewhere !

Wave packet represents particle prob localized Imagine Wave pulse moving along a string: its localized in time and space (unlike a pure harmonic wave)

Making Wave Packets: Simple Model with 2 waves

1 1 1 1 2 1 2 ave 2

f f f -f Wave with : f = Ex: Phenomenon of "Be Add two waves of slightly different , f Start with two waves y ( ), y , ating" in S Amplitude A 2

• un

2 d: ACos k x w t A λ +     ⇒ ∝        = − = 

2 2

2 ( ) : , 2 Cos k x w t k w f π π λ − = =

[ ]

2 1 1 2 1 1 2 2 2 1 2 1 2 1 2 2 1

Resulting wave's "displacement " y = y : cos( ) cos( ) A+B A-B Trignometry : cosA+cos B =2cos( )cos( ) 2 2 2 cos( ) 2 2 since , k cos( ) 2 2

ave

k y y A k k w w x k k w x w t k x w t k k w w y A x t t + +   −  + = − + − − −     ∴ = −        ≅    ≅  ≅

' 1

y = A cos( ) ' 2 cos( ) = modulated amplit cos( ) A' oscillates in x,t ud 2 cos( ) , e 2 2 , 2 , 2

ave

ks wt k w y A x kx w w w k k w t A A x w k w t t − − ∆ ∆   = − ∆ ∆         ∴ = − ≡         ≅ ∆ ∆

• g

Phase Vel V Group Vel V : Vel of envelope=

ave p ave g

w k w k dw V dk = ∆ = ∆

wave Group Or packet

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 localized vgt x

( )

( ) Amplitude Fn diff waves of diff k have different amplitudes A(k) w = w(k), depends on type of wave, media ( , ) Group Velocity ( )

i k g x k wt k

e dk A x t dw V k dk k A ψ

∞ − −∞ =

= = = ⇒

∫

Group, Velocity, Phase Velocity and Dispersion

p

In a Wave Packet: ( ) Group Velocity Since V ( )

g k k p p g p k k k k

w w k dw V dk wk def w k dV dw V V k dk dk V

= = =

= = = ⇒ = = = + ∴

p p p

Material in which V varies with are said to be Dispersive Individual harmonic waves making a wave pulse travel at different V thus changing shape of pulse an usu d b ally V ( ecome spread out )

p

V k orλ λ =

g g

In non-dispersive media, V In dispersive media V ,depends on

p p p

V dV V dk = ≠ 1ns laser pulse disperse By x30 after travelling 1km in optical fiber

Matter Wave Packets

2 g 2

Energy E = hf = mc Consider An Electron: mass = m velocity = v, momentum = p ; 2 = 2 mc h 2 2 k h Wavelength = ; = Group Velocity / / : p 2 V dw dw dv dk dk dv dw d dv f k mv h dv π ω π γ π π γ λ λ γ ⇒ = = = = =

2 1/ g 2 2 1/ 2 2 3/ 2 2 3/ 2 2

/ V mc 2 mv 2 m h & v v v [1- Group velocity of electron Wave packet "pilot wave" ( ) ] h 2 v [1-( ) ] [1-( ) ] h[1-( ) ] / c c c c dk d dv dv dw dw dv v dk dk m h dv v π π π π     = =           =      =  = = ⇒

2 p

But velocity of individual waves is same as el making up the wave packet ect V ron's physical v (not physical e ) ! i y loc t w c c k v = = >

Wave Packets & Uncertainty Principle

• Distance ∆X between adjacent minima = (X2)node - (X1)node
• Define X1=0 then phase diff from X1 X2 =π

2 cos( ) cos( ) 2 2 k w y A x t kx wt ∆ ∆     = − −        

Amplitude Modulation

w Node at y = 0 = 2A cos ( ) 2 2 . Need to combine more to make small packet also implies . / 2 . Need to combine more to make small packet a k k t x k x x p x h and w t t π π ω ∆ ∆ − ⇒ ∆ ∆ = ⇒ ∆ ⇒ ∆ ∆ = ∆ ∆ = ⇒ ∆ lso . / 2 E t h ⇒ ∆ ∆ =

What does This mean?