Physics 2D Lecture Slides Feb 12 Vivek Sharma UCSD Physics Wave - - PowerPoint PPT Presentation
Physics 2D Lecture Slides Feb 12 Vivek Sharma UCSD Physics Wave Packet : Localization To make localized wave packet, add infinite # of waves with Well chosen Amplitudes A, Wave# k & angular Freq. w = i k ( x
Vivek Sharma UCSD Physics
vgt x Wave Packet : Localization To make localized wave packet, add “ infinite” # of waves with Well chosen Amplitudes A, Wave# k & angular Freq. w localized
( )
( ) 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
∞ − −∞ =
= = = ⇒
∫
2 cos( ) cos( ) 2 2 k w y A x t kx wt ∆ ∆ = − −
Amplitude Modulation
w Node at y = 0 = 2A cos ( ), Examine x or t behavior 2 2 in space x: Need to combine many to make small pulse k x= , for smal . l x k & Vice Ve r I ca k k x k t x x and π π ∆ ∆ − ⇒ ⇒ ∆ ∆ ∆ ∆ → ⇒ ∆ → ∞ ∆ = ∆ n time t : Need to combine many to make small pulse = , for small & Vice Ve rca . t w t t t ω π ω ω π ∆ ∆ → ⇒ ∆ → ∞ ∆ ∆ ∆ = ⇒ ∆
We added two Sinusoidal waves What can we learn from this simple model
Wave Packets & Uncertainty Principle in space x: since usual 2 h k = , p = approximate relation ly one writes In time t : since =2 , . .
k x w f E hf t
π π π λ ω π λ ∆ ∆ = ∆ ∆ ⇒ ⇒ ⇒ = =
usually approximate re
lation s
⇒ ∆
What do these inequalities mean physically?
Know the Error of Thy Ways: Measurement Error: ∆
energy
– 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 ± 500) kg = m ± ∆m
– 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
Where in the World is Carmen San Diego?
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
Baby Pictures of Our Universe Revealed Yesterday
temperature & polarization in cosmic microwave background
years old
than ever, propeled by a mysterious (unknown) DARK ENERGY
indication of the “dynamite” behind the “big bang” Microwave Anisotropy Probe (MAP)
Back to Heisenberg’s Uncertainty Principle
– If the measurement of the position of a particle is made with a precision ∆x and a SIMULTANEOUS measurement of its momentum px in the X direction , then the product of the two uncertainties (measurement errors) can never be smaller than ≅h/4π irrespective of how precise the measurement tools
– If the measurement of the energy E of a particle is made with a precision ∆E and it took time ∆t to make that measurement, then the product of the two uncertainties (measurement errors) can never be smaller than ≅h/4π irrespective of how precise the measurement tools
These rules arise from the way we constructed the Wave packets describing Matter “pilot” waves
Perhaps these rules Are bogus, can we verify this with some physical picture ??
The Act of Observation (Compton Scattering)
Act of observation disturbs the observed system
Compton Scattering: Shining light to observe electron
Light (photon) scattering off an electron I watch the photon as it enters my eye hgg g The act of Observation DISTURBS the object being watched, here the electron moves away from where it was originally λ=h/p= hc/E = c/f
Eye
Photons that go thru are restricted to this region of lens
Observed Diffraction pattern
Diffraction By a Circular Aperture (Lens)
See Resnick, Halliday Walker 6th Ed (on S.Reserve), Ch 37, pages 898-900
Diffracted image of a point source of light thru a lens ( circular aperture of size d ) First minimum of diffraction pattern is located by
See previous picture for definitions of ϑ, λ, d
Resolving Power of Light Thru a Lens
Resolving power x 2sin λ θ ∆
diameter d, ability to resolve them depends on λ & d because of the Inherent diffraction in image formation
Not resolved resolved barely resolved
∆X d ϑ Depends on d
somewhere on screen, image (of electron) is fuzzy
resolvable points is :
Putting it all together: act of Observing an electron Eye
Photons that go thru are restricted to this region of lens
Observed Diffraction pattern
sin sin electron momentum uncertainty is ~2h p sin
x
h h P θ θ λ λ θ λ − ≤ ≤ ∆ ≅
2sin x λ θ ∆ =
2 sin . 2sin . 2 / p h p x h x θ λ λ θ ∆ ∆ = ⇒ ⇒ ∆ ∆ ≥
Pseudo-Philosophical Aftermath of Uncertainty Principle
– Newton’s laws told you all you needed to know about trajectory of a particle
– Know every thing ! X, v, p , F, a – Can predict exact trajectory of particle if you had perfect device
– Of small momenta, forces, energies – Cant predict anything exactly
– There is so much chance that the particle landed here or there – Cant be sure !....cognizant of the errors of thy observations
Philosophers went nuts !...what has happened to nature Philosophers just talk, don’t do real life experiments!
Can Electrons Exist Within the Nucleus?
– ∆p ≥ h / (4π. ∆x )……..now do the numbers
16 8 14 7 2 2 2 2
6.58 10 . 3.0 10 / 1.0 10 2.0 10 so electron momentum can be Looks large, lets go relativistic in calculation (cant hurt) E ( ) ( ) 20 MeV/c s 2 , sub t
x x e
eV s m s m c eV c pc p m c x p
− −
≤ ≤ × × = × ≥ × + ∆ ∆ = ≥
2 2
itute #s E 400( ) E>> 13.6 eV, even larger than typical energy in radioactivit 20 , Kinetic energy KE = E - 19.2 MeV y larger than Bohr's Ionization energy for Hydrogen atom
e
MeV Much E MeV m c ⇒ > ⇒ ≥ ⇒ ≥