Physics 102: Lecture 23, Slide 1

De Broglie Waves, Uncertainty, and Atoms

Physics 102: Lecture 23

Physics 102: Lecture 23, Slide 2

Three Early Indications of Problems with Classical Physics

• Blackbody radiation• Photoelectric effect• Wave-particle duality

Lecture 22:Quantum Mechanics

• Compton scattering• DeBroglie• Heisenberg Uncertainty Principle

Today

Physics 102: Lecture 23, Slide 3

Experiment: Outgoing photon has longer wavelength λ′

Recoil electron carries some momentum and KE

Incoming photon has momentum p, and wavelength λ

This experiment really shows photon momentum!

Electron at rest

Compton Scattering

Pincoming photon + 0 = Poutgoing photon + Pelectron

λhchfE ==

λhp =

Photon energy Photon momentum

⇒ E = pc

Physics 102: Lecture 23, Slide 4

Compton Scattering

• Incident photon loses momentum, since it transfers momentum to the electron

• Lower momentum means longer wavelength• This is proof that a photon has momentum

λhp =

Physics 102: Lecture 23, Slide 5

Is Light a Wave or a Particle?• Wave

– Electric and Magnetic fields act like waves– Superposition, Interference, and Diffraction

• Particle– Photons– Collision with electrons in photo-electric effect– Compton scattering from electrons

BOTH Particle AND Wave

Physics 102: Lecture 23, Slide 6

ACT: Photon CollisionsPhotons with equal energy and momentum hit both

sides of the plate. The photon from the left sticks to the plate, the photon from the right bounces off the plate. What is the direction of the net impulse on the plate?

1) Left 2) Right 3) Zero

Physics 102: Lecture 23, Slide 7

Incident photons

Radiometer

Preflight 23.1Photon A strikes a black surface and is absorbed. Photon B strikes a shiny surface and is reflected back. Which photon imparts more momentum to the surface?

Photon A Photon B

Black side (absorbs)

Shiny side (reflects)

Physics 102: Lecture 23, Slide 8

Photons bouncing off shiny side and sticking to black side. Shiny side gets more momentum so it should rotate with the black side leading

Ideal Radiometer

Physics 102: Lecture 23, Slide 9

Our RadiometerBlack side is hotter: gas molecules bounce off it with more momentum than on shiny side-this is a bigger effect than the photon momentum

Physics 102: Lecture 23, Slide 10

Electrons are Particles and Waves!• Depending on the experiment electron can behave

like– wave (interference) – particle (localized mass and charge)

• Recall Young’s double slit experiment: – If we measure which slit the electron went through,

then there is no interference pattern!!

Physics 102: Lecture 23, Slide 11

λhp =

So far only photons have wavelength, but De Broglie postulated that it holds for any object with momentum- an electron, a nucleus, an atom, a baseball,…...

Explains why we can see interference and diffraction for material particles like electrons!!

De Broglie Waves

ph

=λ

Physics 102: Lecture 23, Slide 12

Which baseball has the longest De Broglie wavelength?

(1) A fastball (100 mph)

(2) A knuckleball (60 mph)

(3) Neither - only curveballs have a wavelength

Preflight 23.3

Physics 102: Lecture 23, Slide 13

ACT: De Broglie Wavelength

A stone is dropped from the top of a building.

1. It decreases

2. It stays the same

3. It increases

What happens to the de Broglie wavelength of the stone as it falls?

Physics 102: Lecture 23, Slide 14

Some Numerology

• 1 eV = energy gained by a charge +e when accelerated through a potential difference of 1 Volt– e = 1.6 x 10-19 C so 1 eV = 1.6 x 10-19 J

• h = 6.626 x 10-34 J·sec• c = 3 x 108 m/s

– hc = 1.988 x 10-25 J·m = 1240 eV·nm• mass of electron m = 9.1 x 10-34 kg

– mc2 = 8.2 x 10-13 J = 511,000 eV = 511 keV

Standard units (m, kg, s) are not convenient for talking about photons & electrons

Physics 102: Lecture 23, Slide 15

• Photon with 1 eV energy:

Comparison:Wavelength of Photon vs. Electron

λhcE =

Ehc

=⇒ λ nm 1240eV1

nmeV1240==

You have a photon and an electron, both with 1 eV of energy. Find the de Broglie wavelength of each.

• Electron with 1 eV kinetic energy:

KE =

12

mv 2 and p = mv, so KE = p2

2mK.E.)(2mp =Solve for

KE)(2mh

=λKE)(2 2mc

hc=

eV) 1)(eV 000,511(2nmeV1240

= nm23.1=

Big difference!

Eq

uat

ion

s a

re d

iffe

ren

t -

be

ca

refu

l!

Physics 102: Lecture 23, Slide 16

X-ray diffraction e– diffraction

From College Physics, Vol. Two

Identical pattern emerges if de Broglie wavelength of e– equals the X-ray wavelength!

X-ray vs. electron diffraction

Demo

Physics 102: Lecture 23, Slide 17

Preflights 23.4, 23.5

Photon A has twice as much momentum as Photon B. Compare their energies.

• EA = EB

• EA = 2 EB

• EA = 4 EB

Electron A has twice as much momentum as Electron B. Compare their energies.

• EA = EB

• EA = 2 EB

• EA = 4 EB

Physics 102: Lecture 23, Slide 18

ACT: De Broglie

Compare the wavelength of a bowling ball with the wavelength of a golf ball, if each has 10 Joules of kinetic energy.

(1) λbowling > λgolf

(2) λbowling = λgolf

(3) λbowling < λgolf

Physics 102: Lecture 23, Slide 19

Rough idea: if we know momentum very precisely, we lose knowledge of location, and vice versa.

Heisenberg Uncertainty Principle

π2hypy ≥ΔΔ

Recall: Quantum Mechanics tells us nothing is certain, everything is probability

Uncertainty in position (along y)

Uncertainty in momentum (along y)

Physics 102: Lecture 23, Slide 20

Number of electrons arriving at screen

screen

w

x

y

θΔpy = p sinθ

θ

= Δy

θλθ

sinsin pypy =ΔΔ pλ= h=

w = λ/sinθ

electron beam

Electron diffractionElectron beam traveling through slit will diffract

Recall single-slit diffraction 1st minimum:sinθ = λ/w

Using de Broglie λ

Single slit diffraction pattern

Physics 102: Lecture 23, Slide 21

Electron entered slit with momentum along x direction and no momentum in the y direction. When it is diffracted it acquires a pywhich can be as big as h/w.

The “Uncertainty in py” is Δpy ≈ h/w.An electron passed through the slit somewhere along the y direction. The “Uncertainty in y” is Δy ≈ w.

electron beam

screen

Number of electrons arriving at screen

w

x

y py ⋅w = h

py

∴ Δpy ⋅ Δy ≈h

Physics 102: Lecture 23, Slide 22

electron beam

screen

Number of electrons arriving at screen

w

x

y

py

∴ Δpy ⋅ Δy ≈h

If we make the slit narrower (decrease w =Δy) the diffraction peak gets broader (Δpy increases).

“If we know location very precisely, we lose knowledge of momentum, and vice versa.”

Physics 102: Lecture 23, Slide 23

to be precise... ΔpyΔy ≥ h

2π

Of course if we try to locate the position of the particle along the x axis to Δx we will not know its x component of momentum better than Δpx, where

ΔpxΔx ≥ h

2πand the same for z.

Preflight 23.7According to the H.U.P., if we know the x-position of a particle, we can not know its:

(1) y-position (2) x-momentum

(3) y-momentum (4) Energy