Wave Properties of ParticlesLouis deBroglie:

If light is both a wave and a particle,

why not electrons?

In 1924 Louis de In 1924 Louis de In 1924 Louis de In 1924 Louis de BroglieBroglieBroglieBroglie suggested in his doctoral suggested in his doctoral suggested in his doctoral suggested in his doctoraldissertation that there is a wave connected with the movingdissertation that there is a wave connected with the movingdissertation that there is a wave connected with the movingdissertation that there is a wave connected with the movingelectron and that the electron wave had a wavelength ofelectron and that the electron wave had a wavelength ofelectron and that the electron wave had a wavelength ofelectron and that the electron wave had a wavelength of

.λλλλ =hp

For a photon:

pEc

hfc

h= = = λλλλ →→→→ λλλλ =

hp

E hf= →→→→ fEh

=

Wave properties for an electron (or any particle):

called the DeBroglie wavelengthλλλλ γγγγ= =hp

hmv

and .fEh

=

If electrons are waves, orbits are only stable

if they result inconstructive interference.

Since they can travel eitherdirection, they must

produce standing waves.

2 r n nh

m v

m vr nh 2e

e

ππππ λλλλ

ππππ

= =

= /

Clint Davisson and LesterGermer had been investigating the scatteringof electrons from crystalsurfaces at Bell TelephoneLaboratories since 1917 andgetting perplexing results.

At a meeting in England in1926, Davisson hadconversations with otherscientists which led him torealize that the preidctedwave properties of electronscould be responsible.

He also realized that theirexperiments could be usedto test DeBroglie’shypothesis.

Davisson-Germer ExperimentThey showed that interference occurs when electrons scatter from a crystal surface.

The nickel crystal acts like a two-dimensional diffraction grating. The surface atomsact as an array of sources since scattering occurs only where a nickel atom sits.

By varying the electron energy, they change the electron wavelength.

λλλλ

λλλλ

λλλλ

= =

=

=

hp

hmv

h

2mE

150VV

K

0

Ao

When the spacing of atoms on the surface is D, constructive interference will occur at angles given by

n Dsinλλλλ φφφφ=

D

Low Energy Electron Diffraction (LEED)from a Hydrogenated

Diamond C(100) Surface

Wave PacketsWave PacketsWave PacketsWave Packets

Linear Wave Equation : ∂∂∂∂∂∂∂∂

∂∂∂∂∂∂∂∂

2

2 2

2

2

1yx v

yt

====

For waves on a string or water waves, y is the displacement of themedium from the equilibrium position.

For electromagnetic waves, y is E or B.

Question: For an electron, what is oscillating?

Answer: Probability!

One solution to the wave equation is a harmonic wave:

( )y y cos2

x2T

t y cos kx t0 0=

−

= −ππππλλλλ

ππππωωωω

where k is the wave number and is the angular frequency.ωωωωThis solution is periodic and infinite in extent.

The wave speed or phase velocity is

vT

fkp = = =

λλλλλλλλ

ωωωω

Most often waves occur as pulses, short in time and limited inextent, they contain a mixture of harmonic waves at differentwavelengths and frequencies. This group of waves makes up awave packet.

A wave packet can beresolved into harmonic waves

using Fourier analysis.

vkg =

∆ω∆ω∆ω∆ω∆∆∆∆

v fk

vddk

p

g

= =

=

λλλλωωωω

ωωωω

Wave FunctionsWave FunctionsWave FunctionsWave Functions

Linear Wave Equation : ∂∂∂∂∂∂∂∂

∂∂∂∂∂∂∂∂

2

2 2

2

2

1yx v

yt

====

The function y(x,t) that satisfies this equation is the wave function.

For electromagnetic waves, y is , the electric field.

The energy density at any position and time is . U 12= ε0

2

Since the energy is actually in packets of energy hf, the energy densityis proportional to particle density.

So, the probability of finding a photon in a particular place isproportional to the energy density which is proportional to the electricfield (wave function) squared.

Wave FunctionsWave FunctionsWave FunctionsWave Functions

The wave function squared tells us the probability of observing aparticle (photon or electron) at a place and time.

In general,

P x dx dx2( ) = ΨΨΨΨ

where

ΨΨΨΨ ΨΨΨΨΨΨΨΨ2 = ∗

and is the complex conjugate of .ΨΨΨΨ∗ ΨΨΨΨ

To get the complex conjugate of a function, you replace i with -i everywhere it appears.

The Uncertainty Principle

For all wave packets

∆∆∆∆ ∆∆∆∆

∆∆∆∆∆∆∆∆

∆∆∆∆ ∆∆∆∆

x k 1

xp

1

x p

≈

≈

≈

h

h

∆ω∆∆ω∆∆ω∆∆ω∆

∆∆∆∆∆∆∆∆

∆∆∆∆ ∆∆∆∆

t 1

Et 1

E t

≈

≈

≈

h

h

If we define the uncertainties as equal to the standard deviation:

∆∆∆∆ ∆∆∆∆∆∆∆∆ ∆∆∆∆

x p 2

E t 2

≥≥

h

h

/

/

Equal sign holds for Gaussian wave packets only.

Determining the position and momentum of an electron by scattering a photon off of it

This is a Compton scattering and momentum must be conserved.So the x-component of the momentum of the photon must be equal to

minus the change in the x-component of the electron’s momentum.

We can observe a photon afterscattering if it reaches our lens.

θθθθ ≤≤≤≤ θθθθ′

So the uncertainty in themomentum of the electron will be

∆∆∆∆ θθθθ λλλλθθθθp psin

hsinx ≥ =

The uncertainty in the position ofthe electron is determined by thediffraction limit for resolution.

∆∆∆∆λλλλ

θθθθx=

2sin

∆∆∆∆ ∆∆∆∆λλλλ

θθθθ λλλλ θθθθx p2sin

hsin

h2x ≥

=

A particle confined in a box of length LThe particle can travel back and forth but can’t leave the box.

∆∆∆∆x L=So the uncertainty principle tells us that

∆∆∆∆ ∆∆∆∆px

=L

≥h h

We’ll take the standard deviation of the momentum as our measure of its uncertainty.

( ) ( ) ( ) ( ) ( ) ( )( ) ( )∆∆∆∆

∆∆∆∆

p p p p 2p p p p 2p p p

p p 2 p p p p 2 p p p p p

2

avg

2 2 2

avg

2

avg avg

2

avg

2 2

avg

2 2 2 2 2 2 2

= − = − + = + − +

= − + = − + = − =

So the square of the average momentum is

( )p pL

2 22

= ≥

∆∆∆∆

h

And the average kinetic energy must be

This is called the zero-point energy.Ep2m 2mL

K = ≥2

h 2

2

Similarly, for an electron confined to an orbit around an atom

( )p pr

2 22

= ≈

∆∆∆∆

h

And the average kinetic energy must be

Ep2m 2mr

K = ≈2

h 2

2

The total energy for the hydrogen atom would be

E E Ep2m

ker 2mr

kerK P

2 2

= + = − ≈ −2

h 2

2

The minimum value of the energy is found by setting dE/dr equal to zero.

dEdr mr

ker

0

rke m

a

k e m2

13 6eV

2

min 2 0

2 4

= − + =

= =

= − = −

h

h

h

2

3 2

2

min 2E .

Spectral Widths

The decay of any excited state to a lower energy state can be describedby a lifetime, JJJJ. The uncertainty principle tells us that this lifetime

leads to an uncertainty in the energy of the excited state, ))))E, which isnamed the natural line width of the state, ''''0.

∆∆∆∆ ∆∆∆∆E t≈ � ∆∆∆∆ ∆∆∆∆E

t≈

�ΓΓΓΓ ττττ0 ≈

�

So if this excited state with energy E decays to a lower state of energyE0 and emits a photon in the process, there will also be an uncertainty

in the wavelength of the emitted light.

E E hfhc

E hc

0

2

−−−− λλλλ

∆∆∆∆∆λ∆λ∆λ∆λλλλλ ττττ

= =

≈ ≈�

∆∆∆∆−−−−

∆λ∆λ∆λ∆λλλλλ

∆λ∆λ∆λ∆λλλλλππππ ττττ

EE E

2 c

02

≈

≈

A Z0 particle, the most massive fundamental particle known, has arest energy of 91.2 GeV. Measurements of its rest energy show a

natural line width of 2.5 GeV. What is the lifetime of the Z0 particle?

The lifetime of the lowest electronic excited state in polyene is 1.9 ps. This excitation is performed using light at a wavelength of 610 nm.

What is the energy spacing between the ground and excited states?What is the uncertainty in the energy of the excited state?

What spread of wavelengths would you expect for light emittedduring the decay of the exited state?