Quantum Theory

Micro-world Macro-worldLecture 14

Light propagates like a wave,& interacts as a particle

E=hf

hp =

Louis deBroglie

If light behaves as particles, maybe other “particles” (such as electrons) behave as waves h

p = Photons:

hp =

particles:

hp =

Wave-particle duality is

a universal phenomenon

“deBroglie

wavelength”

Ordinary-sized objects have tiny wavelengths

0.2kg30m/s

hp =

hmv

= 6.6x10-34Js

0.2kg x 30 m/s =

6.6x10-34Js6.0 kg m/s = 1.1x10-34m =

Incredibly small

the wavelength of an electronis not so small

9x10-31 kg 6x106 m/s

hp =

hmv

= 6.6x10-34Js

9x10-31kg x 6x106 m/s =

6.6x10-34Js5.4x10-24 kg m/s = 1.2x10-10m =

-

About the size of an atom

Send low-momentum electrons thru narrow slits

See a diffraction pattern characteristic of

wavelength =h/p as predicted by de Broglie

or through a small hole

“Diffraction”rings

Matter waves(electrons through a crystal)

“Diffraction”rings

Electron waves through a narrow slit acquire some py

py

py

y

x

y

Waves thru a narrower slit

wider

py

py

y

When the slit becomes narrower, thespread in vertical momentum increases

x

y

y/2

x

y

/2

hyp

hyp

yp

h

yph

y

y

sin2

2sin

Heisenberg Uncertainty Principle

y py > h

Uncertaintyin location

Uncertainty in

momentum in

that direction

If you make one of these smaller, the other has to become bigger

Heisenberg tries to measure the location of an atom

For better precision, usea shorter wavelength

But then the momentumchange is higher

x px > h

Localize a baseball

0.2kg x px > h px > hx

Suppose x= 1x10-10m

px > 6.6x10-34Js

1x10-10m

vx >

= 6.6x10-24kgm/s

A very tiny uncertainty

About the size of a single atom

px m

6.6x10-44Js

0.2kg = = 3.3x10-23 m/s

Localize an electron

x px > h px > hx

Suppose x= 1x10-10m

px > 6.6x10-34Js

1x10-10m

vx >

= 6.6x10-24kgm/s

Huge, about 2% of c

About the size of a single atom

px me

6.6x10-24Js

9x10-31kg = = 7x106 m/s

-me=9x10-31kg

uncertainty is inherentin the quantum world

The Bohr-Rutherford Atom

Physics 100

Chapt 23

Nils Bohr

Ernest Rutherford

1895 J.J. Thomson discovered electron

+-

- -

-

+ +

+ +

cathode

anode

“cathode rays”

Vacuum flask

Cathode rays have negative chargeand very small mass

+-

- -

-

+ +

+ +

cathode

anode N

S

m=0.0005MHydrogen

Plum pudding?

-

-

--

-

--

-

-

10-10m

+

+

+

+

+

+

+

+

+

+

++

+

+

+

+

Positively chargedporridge

Negatively chargedraisins (plums)

Planetary-like?

10-10m

+

-

-

-

-

-

Positively chargeddense central nucleus

Negatively chargedorbiting electrons

Rutherford Experiment

Vacuumflask

-rays

What’s in the box?

Is all the mass spreadthroughout as in a box

of marshmallows”?

or is all the massconcentrated in a dense

“ball-bearing”?

Figure it out without opening (or shaking)

Shoot bullets randomly through the box. If it is filled with marshmallows, all the bullets will go straight through without (much) deflection

Figure it out without opening (or shaking)

If it contains a ball-bearing most the bullets will go straight through without deflection---but not all

Occasionally, a bullet will collide nearly head-on to the ball-bearing and be deflected by a large angle

Rutherford used -ray “bullets” to distinguish between the plum-pudding & planetary models

-

-

--

-

--

-

-

+

+

+

+

+

+

+

+

+

+

++

+

+

+

+

no way for -rays to scatter at wide angles

Plum-pudding:

+

-

-

-

-

-

Occasionally, an-rays will be pointed head-on to a nucleus & will scatter at a wide angle

distinguishing between the plum-pudding & planetary models

Rutherford saw ~1/10,000a-rays scatter at wide angles

+

-

-

-

-

-

from this he inferred a nuclearsize of about 10-14m

Rutherford atom

+10-14m

10-10m

Not to scale!!!If it were to scale,the nucleus wouldbe too small to seeEven though it has more than 99.9% of the atom’s mass

Relative scalesAloha stadium

Golf ball

AtomNucleus

99.97% of the mass

x10-4

Classical theory had trouble with Rutherford’s atom

Orbiting electrons are accelerating

Accelerating electrons should radiate light

According to Maxwell’stheory, a Rutherford

atom would only survivefor only about 10-12 secs

Fraunhofer discovered that some wavelengths aremissing from the sun’s black-body spectrum

sola

Other peculiar discoveries:

Solar light spectrum:

Other discoveries…Low pressure gasses, when heated,

do not radiate black-body-like spectra;instead they radiate only a few specific colors

bright colors from hydrogen match the missing colors in sunlight

Hydrogen spectrum

Solar spectrum

Bohr’s idea

“Allowed”orbits

Hydrogen energy levels

+1

3

4

Hydrogen energy levels

+

1

2

What makes Bohr’s allowed energy levels allowed?

Recall what happens when we force waves into confined spaces:

Confined waves

Only waves with wavelengths that just fit in survive(all others cancel themselves out)

Electrons in atoms are confined matter waves ala deBroglie

This wave, as it goes around, will interferewith itself destructively and cancel itself out

However, if the circumference is exactly an integer number of wavelengths, successive

turns will interfere constructively

Bohr’s allowed energy states correspond to thosewith orbits that are integer numbers of wavelengths

Bohr orbits

Bohr orbits

Quantum Mechanics

Erwin Schrodinger

Schrodinger’s equation

ExV

dx

d

m )(

2 2

222h

Matter waves are “probability” waves

Probability to detect theelectron at some place

is 2 at that spot

Electrons are most likely

to be detected hereor here

Electrons will never be detected here

Electron “clouds” in Hydrogen

Nobel prizes for Quantum Theory

1918 Max Planck E=hf1921 Albert Einstein photons1922 Niels Bohr atomic orbits1929 Louis de Broglie matter waves1932 Werner Heisenberg uncertainty principle…