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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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12

21

|||||| 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 <<∆