QUANTUM BEHAVIOUR

Chapter 7

Equations

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Warning!

“What I am going to tell you is what we teach our physics students in the third or fourth year of graduate school - and you think I am going to explain it to you so you can understand it? No, your not going to be able to understand it….you see my physics students don’t understand it either, that’s because I don’t understand it. Nobody does.”

Richard P. Feynman - QED

Feynman Lectures

http://vega.org.uk/video/subseries/8

Q.E.D.If its so difficult why do we use it?

I would again like to impress you with the vast range of phenomena that the theory of quantum electrodynamics describes; It’s easier to say it backwards: the theory describes all the phenomena of the physical world except the gravitational effect,..,and radioactive phenomena, which involves nuclei shifting in their energy levels.

So if we leave out gravity and radioactivity (more properly, nuclear physics), what have we got left? Gasoline burning in automobiles, foams and bubbles, the hardness of salt or copper, the stiffness of steel. In fact, biologists are trying to interpret as much as they can about life in terms of chemistry, and as I already explained, the theory behind chemistry is quantum electrodynamics.’

“Richard P. Feynman - QED”

Q.E.D

The most accurate scientific theory ever developed

For example it predicts the value of a particular constant to bePredicted value 1.00115965221±4Measured value 1.00115965265±20

This is the same as measuring the distance from London to New York with an error equal to the thickness of human hair

Photography with photons

A frustratingly unpredictable process – except on average!

Old expectations

Smooth arrival of energy

moreexposure

This does not happen! This happens

Photons arriverandomly in spaceand time

Only averagingover a largenumber of arrivalsis predictable

New reality

Photons arriving

the CCD

making a photo usinga 3 bit greyscale CCD

Evidence for the graininess of light

Light-emitting diodes

LEDs are engineered to drop eachelectron by a fixed p.d. and to emita photon of a definite colour. A rangeof LEDs, of different colours, illustratethe relationship between energy andfrequency for photons.

Striking p.d. fixesenergy

Constant slope, E/f. The number of joules per hertz is uniform for allelectromagnetic radiation.h, the gradient, is 6.634 10–34 J Hz–1. More often written ash = 6.634 10–34 J s

fblue light

+

f/Hz

E/J

fred light fgreen light

E = e Vblue

E = e Vgreen

E = e Vred

True as long as the LEDdoes not warm up.

Increase the p.d. until the LED justglows. This is the striking p.d.

V

V

energy transferredto each electron= e V

energytransferred toeach photon= e V

E = qV

+

+

Spectral lines and energy levels

Particular colours of light are associated with certainenergies.

The same pattern extends beyond thevisible, to all parts of theelectromagnetic spectrum.

That there are sharp spectral linesmeans some rungs of an energyladder exist – a clue about thestructure of atoms.

Constant slope, E/f. The number of joules perhertz is uniform for all radiation.

h, the gradient, is 6.634 10–34 J Hz–1

More often written as h = 6.634 10–34 J s

E/Jatom fixesenergy

Frequency determines colour. Frequency = speed/wavelength.

fred light fgreen light fblue light

+

+

+

f/Hz

E/J

Eblue light

Egreen light

Ered light

Photoelectric effect

The ejection of electrons from metals by photons was important in establishing the photon description.

Constant slope, E/f. The number of joules perhertz is uniform for all electromagnetic radiation.

h, the gradient, is 6.634 10–34 J Hz–1

More often written as h = 6.634 10–34 J s

too low a frequency toprovide the energy toeject electron

Stopping p.d.measures electronenergy E = qV

= hf0 energy neededto eject one electron fromthe metal

f/Hz

E/J

fblue light

E = e Vblue

E = e Vgreen

originalmetal

metal that givesup electrons moreeasily

f/Hz

E/J

The energy from a single photonis transferred to a single electron.

energy to just climbpotential hill = e V

energy transferred by eachphoton = e V +

A V

potential differencejust stops electrons

fgreen lightfred light

+

+

f0

How a path is explored

S

D

‘waypoints’ define pathsfor the photons to explore

arrow moves atthe speed of thephoton

an arrow spins atthe frequency ofthe photon as thepath is explored

‘path’ is one ofthe many routesthat the photonmust explore tocalculate thefraction of theemitted photonsfound at thedetector

‘source’ iswhere thephotons comefrom

‘detector’ is wherewe look to find outthe fraction of theemitted photonsarriving

One arrow by itself means nothing - youneed to sum arrows from all possible paths

An arrow is theoutput from thisprocess

The spinningarrow freezeswhen it arrivesat the detectorto give an arrow

I am describing to you how nature works you won’t understand why nature works that way. But you see nobody understands that

“Richard P. Feynman QED”

Calculating probabilities from arrows

D

Each path exploreddelivers one arrow

0.72 = 0.49

Square the amplitude togive a numberproportional to theprobability that a photonis detected

Add these nose to tail togive the amplitude

DS

DS

Exploring three paths to calculate an amplitude

DS

Intensity

a b c d e f g h i j k l m n o p q r s t u v w x y z

31

30

29

28

27

26

25

way point

arows lining up

arows curling up

31

30

29

28

27

26

25

Reflection - explorations over a surface

length = 0.4chance = 0.42

= 0.16

S D

S D

S D

Place the source,detector and mirror.

Fix the frequency ofthe photon anddefine a set of pathsfor each photon toexplore by flaggingwaypoints. Allchosen paths go viathe mirror.

Explore each path bymoving a phasoralong that path. Startwith a fresh phasoreach time and recordthe final arrow.Record these arrowsin order.

Place all thesearrows nose to tail inorder. The sum ofthese arrows is theamplitude. Squarethe amplitude to findthe chance that aphoton ends up atthis detector.

Exploring more pathsgives more arrows,which increases theprecision of thecalculation.

Use a restricted setof paths (with onlyone waypoint each)to keep thingssimple.

Reflection occurs -quantum mechanicssays mirrors shouldwork. Most of thefinal amplitudecomes from pathswith waypoints onthe middle of themirror.

The pattern is clear.Almost all theamplitude comesfrom the centre of themirror, only a littlefrom the ends.The intensity, equalto the number ofphotons per second,does not changemuch if the ends ofthe mirror are cut off.

Thesephasors allcome from anarrow sliceat the middleof the mirror

from end of mirror

from middle of mirror

from end of mirror

Making a grating from a mirror

We aim to make the ends count

concentrate on thispiece and see howto get the arrows toline up

Now remove the middle one

Take out the middle oneby eliminating that path

best detector to have a chance offinding red photons

best detector to have a chance offinding green photons

best detector to have a chance offinding blue photons

S

This is a reflection grating - useful for analysing spectra

S D

Source Detector

A = z2+x2

B = z2+(y–x)2

separation of source and detector (y)

perpendicular distancedetector to mirror (z)

x mirror

time = A+B

speedtime

position

set updetectorwhere wewould liketo get afocus

starting with a plane mirror

start bending the mirror to get the arrows to line up

not much chanceof gettingphotons here

S

D

keep bending untilthe arrows line up

?

up a little heredown a little here

up a little more here

S

D

Photon trip time for refraction

separation of source and detector, y

height of sourceabove surface, h

water

x depth ofdetector belowsurface, d

y–x

A =h2+ x2

B = d2+ (y–x)2

time A = speed in air

h2+x2

time B = speed in water

d2+(y–x)2

position of impacton surface

trip time = time A+ time B

Refraction – explorations through a surface

Choose a photon frequencyand define a characteristicset of paths going via thesurface.

S

S

Explore each path by movinga phasor along the path.Start with a fresh phasor eachtime and record the finalarrow.Record these arrows in order.

S

Refraction occurs – quantummechanics says that there isa large chance that the photonbe found at the detector.Most of the final amplitudecomes from paths just to theright of the straight line path;paths close to the path of leasttime.

Place the source, detectorand surface.

Light appears to travel moreslowly below the surface, sowe reduce the speed of theexploring phasor. Thefrequency is unchanged.

The trip time is calculated intwo parts: above and belowthe surface. The phasor spinsat the same frequency. Thetime taken determines theangle through which it hasturned.

Obtain and square theamplitude to find the chancethat a photon ends up at thisdetector.

Explore more paths to getmore arrows, a clearerpicture and greater accuracy.

The pattern is clear. Most ofthe amplitude comes from thepaths close to the path thattakes least time, only a littlefrom those far out.

near leasttime path

far fromleast timepath

D

D

D

WP

WP

WP

High kinetic energy

f = Ekh

S

D

At highfrequencies

3 chosenwaypointsgive arrowsthat curl up

Removing these paths doesnot have much effect on theprobabilitiy of finding aparticle at the detector

Small differences in triptime are enough to allowarrows to curl up

WP

WP

WP

Low kinetic energy

S

D

At lowfrequencies

the same 3waypointsgive arrowsthat line up

Removing these pathssignificantly affects theprobabilitiy of finding aparticle at the detector

Small differences in triptime mean arrows line up

Note: At higher kinetic energies we need only consider paths closer to the straight line

f = Ekh

Trying to pin down photons

Very wide slit

The photon has lots of space toexplore between x and y: as a resultits likely arrival places are not muchspread out.

scan detector to predictbrightness on a screen

S D

Only near the straight through pathdo the phasor arrows make a largeresultant.

barrier to restrictpaths explored

chance the photon endsup at each place

x

y

As the photon passes xy it hasonly a few paths to explore. Pathdifferences are small.

scan detector to predictbrightness on a screen

Phasor arrows add to a largeresultant at a wide spread ofplaces.

barrier to restrictpaths explored

chance the photonends up at each place

x

y

Wide slit

S D

D

As photon passes xy it has only onepath to explore: an infinitely thin slit!Now it could go anywhere!

The narrower slit the wider thespread.

barrier to restrictpaths explored

chance the photon endsup at each place

x

y

Very narrow slit

S

scan detector to predictbrigtness on a screen