Quantum MechanicsThe other great theory of modern physics

Deals with very small objects

Electrons, atoms, molecules

Grew out of problems that seemed simple

Black-body radiation

Photoelectric Effect

Atomic Spectra

Produces some very strange results…

Quantum Hypothesis

𝜆𝑛=2𝑛𝐿𝐸𝑛=h𝑓=h

𝑐𝜆𝑛

Planck’s trick: Each mode has a minimum energy depending on frequency

Can only contain an integer multiple of fundamental energy

Modes with very short wavelength would need more than theirshare of thermal energy

Amount of radiation drops off very sharply at short wavelength

Photoelectric Effect: EinsteinObservations:

1) Number of electrons depends on intensity

2) Energy of electrons DOES NOT depend on intensity

3) Cut-off frequency: minimum frequency to get any emission

4) Above cut-off, energy increases linearly with frequency

Higher intensity More quanta

Only one photon to eject

𝐾𝐸=h𝑓 −𝜙 Einstein in 1921Nobel Prize portraitCited for PE Effect

Bohr Model1913: Neils Bohr comes up with “solar system” model

1) Electrons orbit nucleus in certain “allowed states”

2) Electrons radiate only when moving between allowed states

3) Frequency of emitted/absorbed light determined by Planck rule

Works great for hydrogen, but no reason for ad hoc assumptions

Matter WavesLouis de Broglie: Particles are Waves

Electrons occupy standing wave orbits

Orbit allowed only if integral number of electron wavelengths

Wavelength determined by momentumh

p

Same rule as for light…

Big Molecules

430 ATOMS

Light as a ClockLight: Electromagnetic wave

Extremely regular oscillation

No moving parts

Use atoms as a reference:

Performance: Lose 1s in 100,000,000 years

Defining TimeHow do you define a second?

Initial formal definition:

“the fraction 1/86,400 of the mean solar day”

Update (1960):

“the fraction 1/31,556,925.9747 of the tropical year for 1900 January 0 at 12 hours ephemeris time.”

More specific, recognizes changing length of year

Precision limited by astronomical observations

Difficult to measure locally

Quality FactorWant a good standard reference fortimekeeping

How to characterize clocks?

Common method: “Q factor”

Regular oscillation at some frequencySome small range about average resonance frequency

Q = ratio of central frequency to spread in frequency

How to quantify performance?

Quality Factorfrequency

Qspread

Two ways to get high Q:

1) Decrease frequency spread

improve measurementimprove stability

2) Increase average frequency

“Best” oscillator has high frequency, narrow range in frequency

(Practical limit: Must be able to convert frequency to useful signal)

Light as a ClockLight: Electromagnetic wave

Extremely regular oscillation

No moving parts

Use atoms as a reference:

Performance: Lose 1s in 100,000,000 years

AmmoniaN

HH

H

First standard based on quantum mechanics:

N

HH

H

NH3 molecule: tetrahedral shapeTwo possible arrangements

Leads to pairs of states with slightenergy separation

(23,870 )E hf h MHz

First used as time reference at US National Bureau of Standards in 1949

AmmoniaN

HH

H

(23,870 )E hf h MHz

NH3Oscillator

Operation:

1) Reference oscillator generates signal

2) See if NH3 absorbs

3) Adjust frequency as needed

4) Reference oscillator drives clock (divide frequency electronically)

Ammonia ClockN

HH

H

(23,870 )E hf h MHz

NH3Oscillator

Advantages:

1) Cheap, readily available molecule

2) Convenient frequency for electronics

Disadvantages

1) Doppler effect limits measurement

2) Relatively low frequency

Q ~ 100,000-1,000,000

CesiumDefinition of second since 1967:

the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom.

(Updated to specify at sea level, temperature of absolute zero)

+

-

+

-

“Hyperfine Level” Lowest energy state splitin two by intrinsic magnetic moments ofnucleus and electron

Cesium ClockEarly Cs clocks use atomic beam, magnets:

Csoven

N

S

Microwave Cavity

N

S

Oscillator

Q ~ 107-108

Basic Scheme: I. I. Rabi

Cesium ClockEarly Cs clocks use atomic beam, magnets:

Csoven

N

S

Microwave Cavity

N

S

Advantages:

1) Atoms move perpendicular to light reduces Doppler shift

2) Lower frequency than NH3, but better intrinsic uncertainty

Limitations1) Size of cavity limits measurement time, resolution

2) Still not that high a frequency

Separated Fields

oven

RF

NIST-7: lose 1s in 3,000,000 years

Improved method by Norman Ramsey:

Break cavity in two

Free flight in between

Allows longer measurement

Limitations of Beam Clocks

oven

RF

What determined best performance of NIST-7?

1) Doppler shifts

2) Cavity shifts

3) Time of flight

Atoms moving at >100m/s

Hard to make identical

Only ~100 ms between

Fountain Clock

RF

Zacharias (1953) proposed solution to cavity and time-of-flight problems

Launch atoms vertically

Only one cavity, interact twice

Long time-of flight above cavity

Problem: Hot atoms High velocitiesspray all over the place

Very few make it back through cavity

Laser-Cooled Fountain Clock

Performance: Lose 1s in ~100,000,000 years

Use lasers to slow motion of atoms

Reduce velocity to ~cm/stemperature to 10-6 K

(Lots of cool physics, different class)

Use single microwave cavity

Around 1s interaction time

Primary standards in France, US, UK,…