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Heisenberg Uncertainty Principle Heisenberg U.P. ω
Planck: E
=
hf = h
= ω
2π
h
h
2
π
de Broglie: p =
=
= k
λ 2π λ
Heisenberg: ΔEΔt ≥ ! 2
ΔpΔx ≥ ! 2
Heisenberg U.P. h
Planck’s constant (reduced): 
≡
2π
2
π
Wave vector: k≡
λ
Angular frequency: ω ≡ 2π f
Heisenberg U.P. examples 1. Confine electron to atom: 2. Confining electron to slit translates to uncertainty in perpendicular velocity: 3. Intro to wave packets (real quantum mechanics!) hOp://en.wikipedia.org/wiki/File:Doubleslit.svg Detect electron path with photon?! ScaOer electron off a photon y Electron interference paOern Width b π d sin θ min π
Δycentral :
=
λe
2
Spacing d
Δycentral = 2Lθ min
Electron wave λe L
≈
d
h
λe =
mve,x
⎡ π d sin θ ⎤ ⎡ ⎡ π bsin θ ⎤ π bsin θ ⎤
I (θ )  cos ⎢
⎢sin ⎢
⎥
⎥
⎥
λ
λ
λ
⎦
e
e
⎣
⎦⎣ ⎣
e
⎦
2
2
hOp://en.wikipedia.org/wiki/File:Doubleslit.svg Detect electron path with photon?! ScaOer electron off a photon To resolve slit, photon scaOered y λ
<<
d
γ
Width b Electron recoil (in y) Δpe = pγ
Spacing d h
mΔvy =
Electron wave Electron travels in y: ß L à Compare! Δycentral
λe L
=
d
λγ
L
λe L
Δy = Δvyt = Δvy
=
ve,x
λγ
What happens to fringes? One frequency Frequency Spectrum Time domain -­‐ components Time domain -­‐ sum hOp://phet.colorado.edu/en/simulaXon/fourier Many different frequencies Frequency Spectrum Time domain -­‐ components Time domain -­‐ sum One wavelength (one k-­‐value) k (momentum) domain Spectrum Space domain -­‐ components Space domain -­‐ sum delocalized Many wavelengths (many k-­‐values) k (momentum) domain Spectrum Space domain -­‐ components Space domain -­‐ sum localized CheaXng … (but just a bit) k (momentum) domain Spectrum Space domain -­‐ components Space domain -­‐ sum Localized? Fourier bandwidth theorem •  Disturbance localized in Xme/space means require many f-­‐components/k-­‐components. •  Disturbance spread out (delocalized) in Xme/
space means require few f-­‐components/k-­‐
components. ΔωΔt  2π
ΔkΔx  2π