Download - 19 January 2011Modern Physics III Lecture 21 Modern Physics for Frommies III A Universe of Leptons, Quarks and Bosons; the Standard Model of Elementary

19 January 2011 Modern Physics III Lecture 2 1

Modern Physics for Frommies IIIA Universe of Leptons, Quarks and

Bosons; the Standard Model of Elementary Particles

Lecture 2

Fromm Institute for Lifelong Learning, University of San Francisco


Agenda• Administrative Matters• Clouds on the Horizon• Relativity• Quantum Mechanics• Marrying Relativity and Quantum Mechanics • Quantum Field Theories

– Relativistic Wave Equations• Fermions and Bosons

– Second or Field Quantization• Quantum Electrodynamics (QED)


Administrative Matters• George Coyne S. J. – Director Emeritus, Vatican Observatory

– USF Physics and Astronomy Colloquium of 3 November 2010 The Dance of the Fertile Universe: Chance and Destiny Embrace

• http://www.usfca.edu/coyne/

• First colloquium will be 2 February• Professor E. Commins, UC Berkeley, Eighty Five Years of

Electron Spin.• Full schedule of colloquia will be provided as soon as it is

available. It will also be posted in Fromm Hall.

http://www.usfca.edu/coyne/


Clouds on the Horizon“ Beauty and clearness of theory… Overshadowed by two clouds…”

Lord Kelvin

Baltimore Lectures

Johns Hopkins University

1900

The two clouds:

Failure of the Michelson – Morley experiment→ Einstein’s Relativity

Failure of classical electrodynamics to describe thermal radiation → Quantum Mechanics


Albert Michelson

(1852-1931)

1st American Nobel Prize in Physics

1st Physics Dept. Chair at U of Chicago

Edward Morley

(1838-1923)


Troubles with ElectrodynamicsGalilean Relativity: Addition of velocities

Lab runner car car Labv u v


Inertial reference frame: A frame in which Newton’s 1st law is valid. A frame which moves with constant velocity (w.r.t. what?).

v = const.

A frame which moves with constant velocity w.r.t. an inertial frame is also an inertial frame.

There aren’t any true inertial frames but some are close enough to be treated as though they were, e.g. the Earth.

Relativity principle: The basic laws of physics are the same in all inertial frames.


tim

e

Note: The path followed by the object is different as viewed from different frames. However the same laws of motion and the same law of gravitation apply in both frames.


Where is the luminiferous ether?

What’s waving?

Ocean waves H2O

Sound waves air, liquid, solid

What supports the vibration of electromagnetic waves?

Postulate the existence of an invisible, weightless, tasteless etc. substance which permeates all of space. Referred to as the luminiferous ether.

It is assumed that the propagation velocity is measured w.r.t the ether.

8

0 0

13 10 m/secc


Galilean Relativity and Maxwell’s Equations

1 2

x’2

vy2

z2

x1

y1

z1

v is velocity of frame 2 w.r.t. frame 1(x-direction). v = const.

2 1

2 1

2 1

1 2

1 2

1 2

Inverse

x x vt

y y

z z

x x vt

y y

z z

2 1 1 2 v v v v v v

This works fine in classical mechanics.Newton’s 3 laws are invariant under these kind of Galilean transformations.


What about Maxwell’s elegant electromagnetic theory?

The equations predict the velocity of light in vacuo to be

If this is true in Frame 1 (see previous slide) then we would predict that the velocity in Frame 2 would be c-v.

However, Maxwell’s equations do not differentiate between the 2 systems.

Proposed a “preferred inertial frame” in which velocity is c. But this is just the old luminiferous ether frame.

The Michelson-Morley experimental results can be interpreted as saying the velocity of light is c in any inertial frame.

0 0

1c


Use of the Lorentz-Fitzgerald postulate can make Maxwell’s equations invariant. Strictly an ad hoc or empirical fix.

Albert Einstein (1905) presented an explanation that was much more fundamental. It required a complete rethinking of our conceptions of space and time.

Einstein’s Theories of Relativity

Albert Einstein

(1879-1955)


Einstein’s Postulates

(1) Relativity Principle:

The laws of physics have the same form in all inertial reference frames.

Inertial frame one in which Newton’s laws are valid. i.e. one in which an object subject to no external force moves in a straight line with constant velocity.

(2) Constancy of the speed of light:

Observers in all inertial frames measure the same value for the speed of light in a vacuum.

Light propagates through empty space with a definite speed, c, independent of the velocity of source or observer.


Postulate (1) is the same as Galilean relativity extended to include not only the laws of mechanics but those of the rest of physics (in particular electricity and magnetism).

Postulate (2) violates our commonsense notions.

No ether, just E and B feeding off each other and propagating through empty space.

Not so bad. After all the ether proved impossible to detect so we have no proof of its existence. Fails to meet the testability requirement.

Speed of light in a vacuum is always measured to be c regardless of the relative speeds of the source and the observer.


0.5c

source light beam

c

observer

0.5c

According to Galilean relativity, the observer should measure the velocity of the light beam to be 2c.

1 2

x2

v v y2

x1

y1

z1

1 2v v v

In fact, the observer measures the velocity to be c.

Something is wrong with our velocity addition rule.

123

v v

3 1 2 2v v v v v


Our commonsense is based on a lifetime of experience in which we deal with velocities that are very small in comparison to c.

83 10 m/sec 300,000,000 m/secc

Walking 0.9 m/sec

Running 6.7 m/sec

Automobile 26.8 m/sec

Airliner 268.2 m/sec

Earth escape velocity 11,272.2 m/sec

Light travels approximately 1 foot in 1 nsec (1 x 10-9 sec).


Consequences of Einstein’s Postulates(for more detail see Wiki for course I)

•The rules for transforming from one reference frame to another change.

•Spatial and temporal measurements are linked as space-time

•Lorentz transforms: locations and velocities

•Moving rulers shrink

•Moving clocks slow down

•Velocities add such that light speed is c in all frames•Energy and momentum are redefined to keep them conserved

•Energy and momentum are linked

•Energy and mass are related


is called the Lorentz factor.

is called the speedv

c

2

2

1

1vc

Lorentz contraction2

0 02

11

vL L L

c

0t t Time dilation


Example: Velocity Addition

Find speed of 2 w.r.t. Earth

Earth is frame F. 2 is frame F’. x, x’ are along flight

'

'2 2

0.60 0.60 1.200.88

(0.60 ) 0.60 1.361 1

xx

x

v u c c cv c

uv c cc c

Galilean transform → vx = 1.20c

There is no addition of velocities that will result in v > c

TRY IT


Example: Constancy of c

Find speed of light w.r.t. Earth

Earth is frame F. 1 is frame F’. x, x’ are along flight

'

'2 2

0.60 1.600.60 1.601 1

xx

x

v u c c cv c

uv c cc c

Try it for 1 traveling at 0.99c w.r.t. Earth

Replace 2 with light from 1’s headlight in previous example

c w.r.t. 1

'

'2 2

0.99 1.990.99 1.991 1

xx

x

v u c c cv c

uv c cc c


Experiments and observations

Muon decay: e±e

Radioactive decay law:

2/1

)2(ln

0t

t

eNNwhere t1/2 = 1.52 x 10-6 seconds

2000 m

v = 0.98c

Count for some period of time

Top = 1000 ± 31.6

Bottom = 540 ± 23.2

mountain


Classically, a particle moving at 0.98c covers 2000 m in 6.8 x 10-6 sec.

Decay law only 45 should survive the trip

Relativistically, we realize that the quoted half-life of 1.52 x 10-6 sec. is that of a at rest. At = 0.98, time dilation is significant.

An observer in the lab will perceive that a clock moving with the to be slowed by a factor of .

= 0.98 = 5

Decay law corrected 538 should survive the trip. Agreement with experiment.


Alternatively, examine the problem from the point of view of an observer traveling with the

This observer sees the 2000 m flight path as length contracted by a factor of 1/ to 400 m.

The time to travel this contracted difference is thus reduced by a factor of 1/5.

Decay law corrected 538 surviving s.

Identical result, in agreement with experiment, is obtained by using either time dilation or space contraction.


Relativistic Kinematics and Dynamics

Relativistic momentum:

Newton’s 2nd law, pF

t

, is invariant under Gallilean transforms

What about Lorentz transforms?

If we use our “accepted” definiition of linear momentum,

p mv

2nd law no limit to velocity attained under the application of an external force.

Either the 2nd law or the definition of momentum needs modification at high velocity.

And, what about conservation of momentum?


This is disconcerting. Momentum conservation is one of the “rocks upon which the church was founded” 1.

To avoid “throwing out the baby with the bath water” it was less disturbing to define relativistic momentum as:

With this definition, momentum is conserved.

Note that for v<<c the relativistic momentum becomes the usual old momentum. (

p mv

It can be shown that for relativistic collisions the classical momentum is not conserved.


Be careful

rel

uF m

t

is not true

The correct, relativistic formulation of the 2nd law is

mvF

t

Relativistic energy:

Classically21

2K mv

Use our relativistic momentum, the work-energy theorem and some integration and algebra and we get the relativistic kunetic energy

21K mc

2 2E mc K mc

And the total energy of a “free” (no potential energy) isRest energy


Binding Energy:

Consider a bound state and its constituents

M ∑ m2 2

2 2

B ii

i Di

Mc E m c

Mc m c E

stable

unstable

Example #1, Nuclear Fission:235 141 92 200U Ba Kr MeV

Example #2, Thermonuclear Fusion (D-T reaction):

2 3 4H H He n several MeV


Example #3, Combustion:

2 2 22 2H O H O eV

Mass is just another form of energy

Energy-Momentum2 2 2 2 4E p c m c

2 2 2, , and c m E p c are Lorentz invariants


More Troubles with Electrodynamics(→ Quantum Mechanics)

Black Body Radiation:Stefan–Boltzman law:

P/A = T4

= 5.67 x 10 –8 W/m2 K4

Wien’s displacement law:peakpeak = k / T = k / T

k= 2.9 x 10k= 2.9 x 10-3 -3 K-mK-m

Max Planck (1858-1947)


The ultraviolet catastrophe

Treating the black body as a collection of oscillators (atoms) and applying Maxwell’s equations.

I as f gets large ( gets small)

We know this doesn’t happen

The quantum hypothesis:

Max Planck (ca. 1900); Oscillators are not allowed to emit radiation continuously. They are only allowed to emit (and absorb) energy in discrete packets.

E = hf where, h = 6.63 x 10-34 J·sec (Planck’s constant)


When this condition is imposed the experimental distribution is matched and the U. V. catastrophe is averted.

Often see this written as E= ħ, where ħ = h/2

2

5 /

23

2 1 Planck's radiation law: ,

1

1.38 10 is the Boltzmann constant

hc kT

c hI T

e

k x J K


Planck was applying a technique invented by Boltzmann.

He hoped to avoid the infinity by imposing quantization and then to eventually take the limit as h 0.

Planck spent several years trying to keep agreement with experiment while letting h 0.

He failed.

The value of h can be experimentally determined (see P.E. effect) to be 6.6261x10-34 J sec


The Great Rent Strike at the Black Body Arms1

A large number (essentially infinite) number of tenants occupy a large, cold warehouse.

If the thermostat is set to 50º F everyone pays landlord $50, 55º F everyone pays $55, etc.

Since there are an infinite number of tenants the landlord’s payment is infinite.

This is a financial ultraviolet catastrophe.

1. Brian Greene, The Elegant Universe, Norton Press (1999), ISBN 0-393-04688-5


Now, change the payment rules:

The landlord doesn’t give change

Those who can pay exactly do so, otherwise they pay only as much as they can without requiring change

Organize the wealth:One person holds all the penniesOne holds all the nickelsAll the dimesAll the quartersAll $1 bills, etc.


Set the temperature to 80º F

8,000 pennies1,600 nickels

etc.

1ea. $50 bill, 2 would require changehigher denomination tenants don’t pay at all

Landlord only gets $690

The tenants have introduced a denomination cutoff. Currency denominations above this cutoff are not paid.


Planck introduced a frequency cutoff in his black body calculation.

The energy carried by an E.M. wave, like money, comes in

lumps

is an integer i.e., no change is given.

E nhf

n


The Photoelectric Effect:

e-

Light frequency f

A

V

I

Light falling on the photocathode -- causes emission of electrons.

Electrons are collected – and a current is measured.

Studying the measured photocurrent as a function of V, the light intensity and the light frequency yields results which have striking inconsistencies with those expected from the classical wave picture of E. M. radiation.


Photoelectric Effect Observations and Comparisons to Classical (wave) Theory

(1) Kinetic energy of photoelectrons is independent of light intensity. For a given intensity there is a maximum photocurrent

For a light wave, intensity ~ E2 Ke should increase with the light intensity

CLASSICAL THEORY FAILS

(2) The maximum kinetic energy of the photo electrons for a given material depends only on the frequency

For a light wave Ke should continue to increase with the light intensity. There should be no maximum. Why should frequency matter?



(3) A given material has associated with it a work function, , which is the energy necessary for the photoelectrons to just escape from the material. The smaller the value of , the smaller is the threshold frequency, f0, required for the ejection of photo electrons. Below f0 no electrons are ejected regardless of the intensity.

Classical theory cannot explain the existence of a threshold frequency


(4) When photoelectrons are produced their number is proportional to the intensity of the light

Classical theory is successful here. ABOUT TIME

(5) The photoelectrons are emitted almost instantly, independent of the intensity of the light.


Classical: A weak light beam should take longer to excite an electron to the point of emission.


SEASON RECORD: 1 – 4

Enter Albert Einstein and the Quantum (ca. 1905)

Einstein thought about Planck’s quantization of absorption and emission of radiation.

What if this is not just a mathematical “trick”? Suppose the radiation field itself is quantized.Energy of a beam of light is not continuously distributed in space but consists of a finite number of localized lumps (photons). Each photon carries an energy

E = hf


These lumps propagate without dividing and can only be emitted or absorbed as complete units (“no change given”).

Conservation of energy then yields

ehf K Einstein’s theory successfully explains all five of the photoelectric effect observations above.


Wave Particle Duality

We seem to have developed a rather schizophrenic description or set of descriptions for E.M. radiation.

P. E. effect, Compton scattering etc. => a particle theory of light.

Interference and diffraction experiments of Young and others => a wave theory of light.

Incompatible but both shown to have validity

Nature of light is dual, more complex than a simple wave or a simple beam of particles


Principle of complementarity:

Niels Bohr (1885-1962)

To understand any given experiment, we must use either the wave or the photon theory, but not both.

Duality cannot be “visualized”. The 2 aspects of light are different faces that light shows to experimenters.

Difficulty arises from thought process:

In the mundane world we see energy propagated by the two methods


However, we cannot see directly whether light is a wave or a particle, so we do indirect experiments.

To explain the experiments we apply either the wave or the particle model.

There is no reason why light should conform to these models (visual images) taken from the mundane world.

The “true” nature of light, what ever that means, is not possible to visualize. The best we can do is to realize that we are limited to indirect experiments and that in mundane terms, light reveals both wave and particle properties.


de Broglie’s Hypothesis

Major symmetry fan:

Waves sometimes act like particles (given)

Particles sometimes act like waves (hypothesis)

h

p Sometimes called the de Broglie

wavelength of a particle

Sounds nuts, but remember h is very small ( 10-34 J·sec)

A couple of examples may restore your gullibility.

Louis de Broglie (1892-1987)


Wavelength of a ball:

0.20 kg moving with speed of 15 m/sec

34

346.6 10 J sec2.2 10 m

0.20 kg 15 m/sec

h h xx

p mv

Very small, something like 20 Planck lengths

of any ordinary object is much to small to be detected.

Interference and diffraction are significant only when the sizes of objects or slits are not much larger than


D 0.3 nm


Atomic Instability - Another UV CatastropheClassical Rutherford atom (1911)

e- are accelerated and should radiate.

e- lose energy and spiral into nucleus

Atoms should be unstable and the universe should only have lasted a small fraction (10-9) of a second

Oh, and there are other problems, line spectra.


Emission from rarified gasses Absorption in rarified gasses

Line spectra cannot be explained by a classical model


Neils Bohr (1885-1962)

Bohr’s Postulates

Electrons in atoms cannot lose energy continuously, but must do so in quantum “jumps”.

Electrons move about the nucleus in circular orbits, but only certain orbits are allowed.


An electron in an orbit has a definite energy and moves in the orbit without radiating energy. The allowed orbits are referred to as stationary states.

Emission and absorption of radiation can only occur in conjunction with a transition between 2 stationary states. This results in an emitted or absorbed photon of frequency such that

hf = E1 - E2

What makes an orbit allowed?

Maybe energy is not the only quantized quantity.

No

tice

d that

can be

shown to have un

Requir

its of angula

e that

r moment

where 1,2

2,

um

3,L n n

h


de Broglie Waves and Bohr’s Quantization

Bohr’s model was largely ad hoc. Assumptions were made so that theory would agree with experiment.

No reason why orbits should be quantized.

No reason why ground state should be stable

De Broglie proposed that an electron in a stable orbit is actually a circular standing matter wave.

h

mv


The circumference of the wave must contain an integral number of wavelengths

2 1, 2,3,

Bohr's condit

or

2

2

ion

n

n

n

r n n

nhr

mvnh

mvr n

Bohr published his model in 1913. de Broglie did not propose matter waves until 1923. Bohr tried many quantization conditions in attempting to explain the experimental data. Quantization of angular momentum worked.


Quantum Mechanics

Erwin Schrödinger

1887 - 1961

Werner Heisenberg

1901 - 1967de Broglie proposes matter waves in 1923

Less than two years later two comprehensive theories were independently developed by Schrödinger and Heisenberg.

2

2 , ,2

V E x y zm

2

2

x

hp x

hE t


Now recall the discussion of the double slit when we made the light beam very weak so that we were counting one photon at a time.

Over time, as we accumulate counts a distribution of detected photons builds up which is identical to the intensity distribution obtained from the wave picture. Thus, E2 is a measure of the probability of finding a photon at that location.

Now consider matter waves where h

p

The displacement is described by a wave function, x,t), as a function of time and position.

2

2

-

, Probability of finding particle in volume s

Normalization Condi

urrounding ,

tion: ,

.

1x t dV

x t dV dV x t


If we treat particles (including photons) as waves then or E or B) represents the wave amplitude.

If we treat them as particles we must do so on a probabalistic basis

2, Probability / volume of finding particle at ( , )x t x t

We cannot predict, or even follow, the path of a single particle through space and time


Heisenberg Uncertainty Principle

Expectation: By using more precise instrumentation, the uncertainty in a measurement can be made indefinitely small.

Quantum Mechanics: There is a limit to the accuracy of certain measurements. Not an instrumental restriction but an inherent fact of nature.-Ray Microscope and some estimates:

Try to measure the position of an e-

Requires the scattering of at least one , transferring some momentum to the e-.

Greater precision (smaller x) requires shorter Shorter => higher momentum and hence the higher the possible momentum transfer to the electron (higher p).

Feeling pool ball in the dark


The act of observing produces an uncertainty in either or both the position or momentum of the electron. Heisenberg 1927

Let’s make an estimate of the magnitude of this effect. x

Suppose the e- can be detected by a single having momentum h

p

Some or all of this momentum will be transferred, but we can’t tell beforehand how much. Therefore the final e- momentum is uncertain by

hp

x p h Multiplying we obtain


More careful calculations, e.g. using Fourier analysis show that the very best we can do is

2x

hx p

Another form of the uncertainty principle relates energy and time

2

hE t

x<x>

X

x x

Download - 19 January 2011Modern Physics III Lecture 21 Modern Physics for Frommies III A Universe of Leptons, Quarks and Bosons; the Standard Model of Elementary

Top Related