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The Wave Nature of Matter
• de Broglie matter waves• The Davisson-Germer experiment • Matter wave packets • Heisenberg uncertainty principle • Particle wave duality
de Broglie’s hypothesis (1923)
The electron of the Bohr atom forms a standing wave around the nucleus.
The photon has a dual character:wave and corpuscle
How about the electron ?
(After all, Nature is full of symmetries!)
Electron has a dual character too!
The associated de Brogliewave length and frequency:
hEph /,/ =ν=λλ=π nr2hnrmvL ==⇒
Physics 215Winter 2002
Prof. Ioan KosztinLecture #9Introduction to Modern Physics
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Davisson-Germer Experiment (1927)direct experimental proof that electron has wavelength ph /=λ
Scattered electrons energy E = 54 eV
Scat
tere
d in
tens
ity
Polar plot
Scattering of electrons from a crystalline Ni target leads to electron diffraction.
Confirmation of de Broglie’s hypothesis
1. Experiment: Condition for diffraction maximum
Reflection diffraction grating
λ=φ= ndAB sin
Elec
tron
dif
frac
tion X
-ray diffraction
d=2.15Å, Φ=50°, n=1 ⇒ λ=1.65Å
2. Theory:
meVhmvh 2// ==λ
V=54V ⇒ λ=1.67Å
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Matter wave packets
The wave packed attached to a particle of mass m moving with velocity v0 can be represented as a superposition of many plane wave with λ distributed about λ0=h/mv0.
)](exp[)(),( tkxikAkdxt kω−∫=ψ
)( kk ω≡ω=ωDispersion relation:
Phase velocity: 00 / kvp ω=Group velocity:
0|)/( kkg dkdv =ω=
A wave packet is a superposition of many monochromatic plane waves:
Space-time extent of a wave packet
)](exp[),(),( tkxixtdxtdkA ω∆−∆−ψ∫=ωSince A(ω,k) is non zero only in the vicinity of ω0 and k0, the temporal (spatial) extent ∆t (∆x) of the wave packet is subject to the conditions
11 ≥∆⋅∆≥∆⋅ω∆ xktExp:
A(k)
k0
1
k0+∆kk0-∆k kxk
ikxdkx kkkk
∆∆=
∫=ψ ∆+∆−
)sin(
|)exp(||)(| 00
-7.5 -5 -2.5 0 2.5 5 7.5
-0.2
0
0.2
0.4
0.6
0.8
1
2∆x
1~0)( π=∆∆⇒=∆ψ xkx
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The Heisenberg uncertainty principle (1927)
2/h≥∆⋅∆ xpxPosition and conjugate momentum can not be measured simultaneously with any degree of accuracy (i.e., the concept of trajectory looses its meaning in quantum mechanics).
2/h≥∆⋅∆ tEEnergy-time uncertainty:
∆t = duration of measurement; ∆E = precision of measured energy
⇒ Excited energy levels with finite lifetime are broadened!
The Heisenberg uncertainty principle
hpxxhp eeee ~sin/~,sin)/(~ ∆∆⇒θλ∆θλ∆
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Wave-particle dualityElectrons behave like either particles or waves, depending on the experimental circumstances. However, it is impossible to measure both the particle and wave properties simultaneously.
Electron double slit experiment:
Dphph
D
x
x
2/sin/
2/sin
=θ≈θ=λ
λ=θ
Condition for diffraction minimum:
Wave-particle dualityOnly upper slit opened
Only lower slit opened Both slits opened
Ψδ+Ψ+Ψ=Ψ
Ψ+Ψ=Ψ2
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|||||| interference term
Electrons are detected as particles at localized spots, at certain t, BUT their distribution function |Ψ|2 is determined by superposition of waves.
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Through which slit the electron passes ?
At the same time (simultaneously) it is impossible to see the interference (wave property) and to determine which slit the electron goes through (particle property), without violating the uncertainty principle.
• For detecting the electron as a particle:
∆y<<D• Diffraction pattern will be observed if
∆py~pxθ~h/2DThus,
∆y ∆py<<h/2Which contradicts Heisenberg uncertainty principle
Dy <<∆