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7.2 Wave Nature of MatterDe Broglie WavelengthDiffraction of electronsUncertainty PrincipleWave FunctionTunneling

Wave properties of matterMaterial particles behave as waves with a wavelength given by the De

Broglie wavelength (Planck’s constant/momentum)

The particles are diffracted by passing through an aperture in a similar manner as light waves.

The wave properties of particles mean that when you confine it in a very small space its momentum (and kinetic energy) must increase. (uncertainty principle) This is responsible for the size of the atom.

Wave properties are only dominant in very small particles.

hp

λ =

De Broglie WavelengthMomentum of a photon - inverse to wavelength.

Epc

= Einstein’s specialrelativity theory

hp =λ

since hcE =λ

Wavelength of a particle- inverse to momentum.

hp

λ =

Lois De Boglie

De Broglie proposed that this wavelengthapplied to material particles as well as for photons. (1924)

Wave properties

a

λ

θ

Diffraction of waves Increases as the ratio λ/a

de Broglie wavelength of a baseball

mvh

ph==λ

A baseball with a mass of 0.15 kg is pitched at 45 m/sWhat is its De Broglie wavelength?

m= 0.15 kgv= 45 m/s

m10x8.9)s/m45)(kg15.0(

10x6.6 35s.j34

−−

==

Diffraction effects of a baseball are negligible

de Broglie wavelength of an electron

meV2h

ph==λ

eVm2

pm2vmmv

21KE

2222 ====

V = 1000 V

+-

e-

Find the de Broglie wavelength of a 1000 eVelectron.

m=9.1x10-31 kg

m10x0.4)C/J1000)(C10x6.1)(kg10x1.9(2

Js10x6.6 11

1931

34−

−−

−

==λ

atomic dimensions

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Proof of the wave nature of electrons by Electron Diffraction

• Davisson – Germer Experiment• Thompson Experiment

Showed the wave nature of light by diffraction of electrons bycrystals.

Diffraction of light from crystals

Crystals act as a three-dimensional diffraction gratingLight with wavelength close to the inter-atomic spacing(x-rays) is diffracted.

Diffraction of electrons from crystals

Ni crystal

An electron beam is scattered from a crystal

The scattered beam shows a diffraction patternexpected for the crystal spacing.

Davisson-Germer Experiment (1927)

Comparison between electron diffraction and x-ray diffraction

George Thomson (1927)

X-ray pattern electron pattern

Either x-raysor electrons

The electron wavelengthwas adjusted to the samevalue as the x-ray by varyingthe voltage.The diffraction pattern forx-rays and electrons arevery similar.

Aluminumpowder

Diffractionpattern

Material particles have wave properties.

Electron microscopy

hmv

λ =

Shorter wavelength can be obtained by increasing v, the speed of the electron.

Electron microscopy

electron microscopepictures of ribosomes

model of ribosome

Electron microscopy can be used to image structures of molecules.

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Diffraction of particlesProbabilistic Interpretation of the

wave amplitude.

particle withwavelengthλ

Intensity ∝ Amplitude2

of the diffracted wave.

But each particle hitsa certain point on thescreen

The amplitude2 is interpreted as the probability of the particle hitting the screen at a certain positionThis is true for electrons as well as photons.

WavefunctionIn quantum mechanics the result of an

experiment is given in terms of a wavefunction Ψ. The square of the wavefunction Ψ2 is the probability of the particle being at a certain position.

The wavefunction can be calculated using using the Schrödinger Equation. For instance for electrons in an atom.

Wave property of particles

λ

∆x ∆vx

x direction

When ∆x decreases, ∆px increases.

Decreasing the slit reduces ∆xBut increases the width of the diffraction, ∆vx

Uncertainty Principle

λ

The uncertainty in position is ∆x

∆x

The uncertainty in the x component of momentum is x xp m v∆ = ∆

∆vx

x directionThe particle is diffracted

hx sinmv

∆ θ = λ =

xvsinv∆

θ = θ =v

xv hxv mv∆

∆ =

most often written as an inequalityThe position and velocitycannot be know withunlimited certainty.

xx p h∆ ∆ =

Particle passing through a slit --

Therefore

π≥∆∆

4hpx x

The size of an atom

What accounts for the size of the hydrogen atom?

+-

Classical electrostatics predicts that the potentialenergy of the hydrogen atom should go to – infinity

The finite size of the atom is a quantum mechanical effect.

electron proton2

ok ePEr

= −

PE-> - infinityas r -> zero

As the size of the atom r decreases – The uncertainty in the momentum increases until the uncertainty in momentum limits the momentum. Then the kinetic energy must increase due to the uncertainty principle.

x xhp p2r

∆ = ≈2

2 xx

p1KE mv2 2m

= =2

2h

8r m≈

+-

r =∆xxh hpx r

∆ ≈ =∆

px cannot be smaller than ∆px

E= KE+PE goes through a minimumas a function of r

Use linear momentum as a rough estimate.

r

E

KE

PE

E

hpx x ≈∆∆

KE increases as 1/r2

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Tunneling across a barrier

Wall

a macroscopic objectimpinging on a barrierthe object cannot penetratewithin the barrier.

e-

a wave particle impinging on abarrier can penetrate within thebarrier for distance. and go through the barrier if it is thin enough.

potential energybarrier

distanceThe probability of tunneling decreases exponentially with thewidth of the barrier.

wave functiondecays

Electron Tunneling Microscope

r22o

2 eobabilityPr_Electron α−Ψ≈Ψ∝

α ~10 nm-1

1E-27

1E-22

1E-17

1E-12

1E-07

0.01

0 0.5 1 1.5 2 2.5 3

r (nm)

elec

tron

pro

abab

ility

The tunneling probability falls off exponentially with r and is a sensitive function of distance

Tunneling in Photosynthesis

Bacterial Reaction Center

Electron transfer is due to tunneling. Transfer times decay exponentially as a function of distance.

hf

Electron transfer time vs distance

10-12 s

10-12 s

10-10 s10-1 s

e-

2.0 nm