Albert Einstein - ferenc.physics.ucdavis.eduferenc.physics.ucdavis.edu/9D/lectures.pdf · •Lorentz Transformations and 4-vectors (1.11) •The Doppler Effect (1.6) x1 x2 . Lorentz

Albert Einstein

1905

How Fast

is “Fast”?

How Small

is “Small”?

Fast => v~c c= the velocity of light

Small => e.g. atomic size

Our First Topic

v<c

Einstein’s Postulates of Relativity:

(Albert Einstein, 1905)

Postulated in an attempt to explain the laws of Classical

Electromagnetism

(Maxwell’s Equations, the constancy of the speed of light)

There are many-many experiments that

prove the consequences of the two

postulates of relativity

and

no experiments that disprove, so far,


• Michelson- Morley Experiment, light soucre, and the medium (Discussion sessions, see Appendix A)

• Definition of an “Event”

• Some Consequences of Einstein’s Postulates:

1. Relative Simultaneity

2. Time Dilation

3. Length Contraction

1st Postulate:

The form of each physical law is the same in all

inertial frames of reference

INERTIAL FRAME = ?

The reference frame in which an

object, experiencing zero net force – a “free” object

– moves at constant velocity

v = constant

2nd Postulate:

Light moves with the same speed (c)

relative to all observers

Bob measures:

Speed of light = c

and not v+c

Anna measures:

Speed of light = c

?


• Light Souce, Medium and Michelson- Morley Experiment (Discussion sessions, see Appendix A)


• Consequences of Einstein’s Postulates:


2. Time Dilation



• Light Souce, Medium and Michelson- Morley Experiment (Discussion sessions, see Appendix A)




2. Time Dilation


Simultaneous Flash

Consequence 1:

Relative Simultaneity, or The absence of absolute simultaneity

Simultaneous Arrival

=> AN EVENT

The same EVENT For Bob as well!


AN EVENT For Bob as well!


THEN:

Simultaneous

Emission is

impossible

for Bob

2nd Postulate

AN EVENT For Bob as well!


BUT THEN:

Simultaneous

Emission is

impossible

for Bob

2nd Postulate

Both emission

and arrival are

simultaneous Arrival is simultaneous

but EMISSION is not

simultaneous

QUESTIONS ?

Consequence 2:

Time Dilation, or The absence of absolute time

The time for the light to return:

t = 2H/c

The time for the light to return:

T > 2H/c

Longer

Path + 2nd

Postulate

The time for the

light to ‘return’:

T > 2H/c

Longer Path (L) +

2nd Postulate

L2 = H2 + (v T/2)2

L

t = 2H/c

T = 2 L/c

T = t (1 – (v/c)2 )-1/2

PRECISELY =>

Light Clock

Light Clock

“MUON”

Consequence 3:

Lorentz Length Contraction, or The absence of absolute distance

L = L0 (1 – (v/c)2 )1/2

Consequence 3:


WHAT IS WRONG IN THIS

CARTOON?

There are many-many

experiments that prove the

consequences of the two

postulates of relativity

Muon Lifetime

Muons are created abundantly in

elementary particle showers in the

atmosphere, initiated by energetic cosmic

rays (photons, particles and nuclei).

Muons originate from decays of particles

called “pions” (p) that are the primary

products in these showers

Muon Lifetime

Muon (m) is an elementary

particle

similar to electron, but heavier

(will learn more in Chapter 11)

Muon Lifetime

Earth

Cosmic Ray

Neutrinos

Atmosphere

Particle shower

CONCLUSION: According to Classical Physics,

muons should not be able to reach the ground!

BUT THEY DO

Consequence 3:


L = L0 (1 – (v/c)2 )1/2

CLOUD CHAMBER

QUESTIONS ?

The distance between the

creation and detection points

Light Clock

“MUON”

The time for the

light to return:

T > 2H/c

Longer Path (L) +

2nd Postulate

L2 = H2 + (v T/2)2

L

t = 2H/c

T = 2 L/c

T = t (1 – (v/c)2 )-1/2

PRECISELY =>


• Michelson- Morley Experiment – NO AETHER !



2. Time Dilation


SUMMARY

QUESTIONS

1. Twin Paradox

2. Lorentz Transformations

3. Examples

New Topics:

Viewed by Carl:

- Bob moves towards right at v = +0.98 c

- Anna moves towards left, at v = -0.98 c

- Carl sends laser pulses towards Bob

- Bob reflects those pulses towards Anna

=> What is the velocity of each laser pulse

received by Anna, when measured by Anna?

IMPORTANT: space and time coordinates mix

together!

(Not the case in Classical Physics)

Another way of expressing

the Lorentz Transformation Equations

4-vectors

x’x’ + y’y’ + z’z’ xx + yy + zz

The LENGTH

is not INVARIANT,

i.e. it is not conserved

under Lorentz

transformations

x’x’ + y’y’ + z’z’ xx + yy + zz

Is there anything

that IS INVARIANT

under Lorentz

Transformations?

x’x’ + y’y’ + z’z’ – (ct’)(ct’) = xx + yy + zz – (ct)(ct)

4-dimensional “LENGTH”



4-vectors

Energy-Momentum

Another Example:

Another Example:

In the “Classical Limit”

?

“Classical Limit”:

v << c


v << c

<< 1


v << c

<< 1

~ 1

~1

~1 ~1

~1 ~0

~0

~0

~0

x’ ~ x t’ ~ t


First case: v = 0, g = 1

~1

~1 ~1

~1 ~0

~0

x’ ~ x – vt t’ ~ t


General case: v << c, g ~ 1

OK

What Follows:

• Lorentz transformations of distance and time intervals

• The Twin Paradox, revisited

• Velocity Transformation (1.7)

HOMEWORK:

• Lorentz Transformations and 4-vectors (1.11)

• The Doppler Effect (1.6)

x1

x2

Lorentz Transformation of

Distances and Time Intervals

IMPORTANT: space and time distances mix together

This time using Lorentz Transformations



0 =

0

0



0 =

-2 m =

-2 m

-2 m

0

0


• Michelson- Morley Experiment – NO AETHER !



2. Time Dilation


SUMMARY

EXPERIMENT

Next:

• Velocity Transformation (1.7)

• Is causality absolute?

• Energy and mass

HOMEWORK:

• Lorentz Transformations and 4-vectors (1.11)

• The Doppler Effect (1.6)

Velocity Transformation

u u’

?


u u’

u v + u’

What about y and z coordinates?

(x - direction of motion)

Lorentz Transformations

NOTE that coordinates

orthogonal to the direction

of motion stay the same


NOTE that coordinates

orthogonal to the direction

of motion stay the same

NO (Lorentz) Length Contraction in directions

other than along the direction of relative

motion


Relativistic velocity Transformations

BUT ALL Components of the

Velocity Vector Transform!




Velocity Vector Transform! WHY ?




Velocity Vector Transform! WHY ?

Because:

The time transforms independently

of the direction of motion,

coordinates do not,

and velocity combines both


u u’

u v + u’

dtdt /

u

d

d

Parallel to the

Direction of

Relative motion

Orthogonal to the

Direction of

Relative motion

Classical Limit?

(both u and v << c)

O.K.

u and/or v ~ c ?

Please

try to figure out

yourself

ux = c

ux’ = c uy’= 0 uz’ = 0

Is there Absolute Causality?

Might cause precede effect in one reference

frame but effect precede cause in different

reference frame(s)?

e.g. can someone see you

first die, and then

see you get born?

Benjamin Button

Let’s assume that the order of events is

changed in some reference frame S’

0'

0

>

t

t

Is that Possible?

t and t’ are the time intervals between

the same two events observed in

S and S’, respectively

=

=

1'

'

2

2

t

x

c

vtt

txc

vt

v

v

g

g

Using Lorentz transformations….

0'

0

>

t

tif then

=

=

1'

'

2

2

t

x

c

vtt

txc

vt

v

v

g

g


0'

0

>

t

tif then

tv

cx

t

x

c

v

>

2

201

=

=

1'

'

2

2

t

x

c

vtt

txc

vt

v

v

g

g


0'

0

>

t

tif then

tv

cx

t

x

c

v

>

2

201

Impossible

Is there Absolute Causality?

Might cause precede effect in one reference

frame but effect precede cause in different

reference frame(s)?

e.g. can someone see you

first die, and then

see you get born?

Viewed by Carl:

- Bob moves towards right at v = +0.98 c in his 10 m long spaceship. He

wears a clock that blinks every 10 seconds.

- Anna stands next to Carl in her 20 m long spaceship, and doesn’t

move. She wears a clock that blinks every 10 seconds, too.

a. What is the length of Bob’s spaceship, when measured by Anna?

b. What is the length of Anna’s spaceship when measured by Bob?

c. What is the time interval between two blinks of Bob’s clock, when

measured by Carl?

d. What is the time interval between two blinks of Anna’s clock, when

measured by Bob?

e. What is the time interval between two blinks of Anna’s clock, when

measured by Carl?

Light Cone

New topics:

1. Relativistic Dynamics:

2. General Theory of Relativity – skipped

END of Relativistic Physics

(Next time – Quantum Physics)

2mcE =



4-vectors

x’x’ + y’y’ + z’z’ xx + yy + zz

The LENGTH

is not INVARIANT,

i.e. it is not conserved

under Lorentz

transformations

x’x’ + y’y’ + z’z’ xx + yy + zz

Is there anything

that IS INVARIANT

under Lorentz

Transformations?

x’x’ + y’y’ + z’z’ – (ct’)(ct’) = xx + yy + zz – (ct)(ct)

4-dimensional “LENGTH”



4-vectors

Energy-Momentum

Another Example:

Another Example:

22222222222222

422222222

'''' cpcpcpEcpcpcpE

cmINVARIANTcpcpcpE

zyxzyx

zyx

=

==

22222222222222

422222222

'''' cpcpcpEcpcpcpE

cmINVARIANTcpcpcpE

zyxzyx

zyx

=

==

2222

2222

2

2222

2

)(

)(

)(

mccpE

mccpE

mccpE

=

=

=

E = TOTAL

ENERGY

Revolutionary Concept

What about

2mcE =

?

2222

)(mccpE =

2mcE =

?

2222

)(mccpE =

E = TOTAL

ENERGY

E = INTERNAL

ENERGY

(when p = 0)

2mcEINTERNAL =

2222

)(mccpE =

E = TOTAL

ENERGY

E = INTERNAL

ENERGY

Expressions for (total) Energy

and Momentum of a particle of mass m,

moving at velocity u

Classical Limit

1

1

ump

2

22 mu

mcE

NEW

FAMILIAR kinetic energy

Kinetic Energy = KE

Energy Matter

Energy Matter

Our Experiment:

Muons are CREATED

From the Energy

of Cosmic Rays

Energy Matter

Atomic Bomb (Chapter 10):

Energy is CREATED

From the Mass of

Nuclei

(Internal energy is transformed

into kinetic energy)

ug ug

ug ug

General Theory of Relativity

Acceleration

Gravitational Force

Profound

Link

Free-falling Physicist

Cannon

Dog

Cannonball Trajectory:

?

Cannonball Trajectory:

•Relative to the Earth

= Parabola

•Relative to the Falling

Physicist

= Straight Line

In any small, freely falling reference

frame

anywhere in our real, gravity-endowed

Universe,

the laws of physics must be the same

as they are

in an inertial reference frame in an

idealized, gravity-free universe.

Einstein’s Principle of Equivalence

Special Theory of Relativity The two postulates:

earth BUT: Accellerating

frames

Special Theory of Relativity:

General Theory of Relativity:

locally

Special Theory of Relativity:

General Theory of Relativity:

locally

Deals exclusively with

globally

INERTIAL FRAMES -

v = constant

Deals also with

Accelerating -

LOCALLY INERTIAL FRAMES

Curved Space-time:

(Intercontinental flights)

Experimental verification of

General Relativity:

Precise Measurements of Orbits

QUESTIONS

Lecture Today

1. Black Body Radiation (Max Planck 1900)

2. The Photoelectric Effect (Albert Einstein 1905)

3. The Production of X-Rays (Wilhelm Roentgen 1901)

4. The Compton Effect (Arthur Compton 1927)

5. Particle-Antiparticle Pair Production (Carl Anderson 1932)

6. Discussion: a Wave or a Particle (?) –> “Duality”

Electromagnetic Waves

behaving like

Particles

(Chapter 3 (or 2-old book))

Black Body Radiation

(Max Planck 1900)

(More in Appendix C)

The Energy (E) in

the electromagnetic radiation

at a given frequency (f)

may take on values restricted to

E = n h f where:

n = an integer (1, 2, 3,…)

h = a constant

(“Planck Constant”)

The Planck’s Black-Body Radiation Law:

Experimental Fact:

E = nhf

BUT, why should the energy of an

Electromagnetic wave be

“Quantized”?

(n= integer)

No explanation

until 1905

AWave is a

Continuous

Phenomenon

Experimental Fact:

E = nhf

BUT Why should the energy of an

Electromagnetic wave be

“Quantized”?

(n= integer)

No Explanation

until 1905

Albert Einstein

The Photoelectric Effect

A wave is a

Continuous

Phenomenon

metal


(Albert Einstein)

metal


(Albert Einstein 1905)


(Max Planck 1900)

(More in Appendix C)

metal



Phenomenon observed long time before Einstein,

and something very strange was observed:

metal



Even With Very strong light of low frequency

NO

electrons

ejected

metal




NO

electrons

ejected

metal




NO

electrons

ejected



Even With Very-Very weak light intensity,

but of high enough frequency

Electrons

ejected

metal




NO

electrons

ejected

Albert Einstein

1905

Planck’s Law

( E = n h f )

Albert Einstein proposed:

Photoelectric Effect

(Threshold frequency)

The light is behaving as a collection of particles

called “photons” each of them having energy

E = h f



Even With Very-Very weak light intensity,

but of high enough frequency

Electrons

ejected nhfE

hfE

beam

photon

=

=

What happens

is that

1 PHOTON ejects 1 ELECTRON


hfEphoton =

= hfKEmax

To free an electron from the metal, one has to

“pay” a certain amount of energy

the Work Function

Also known at that time:

QUIZ 3:

a. At what speed will the kinetic

energy of a particle of mass m =

0.001 g (gram) be equal to its

internal energy ?

b. What is the ratio of that kinetic

energy and the kinetic energy of the

same particle moving at v = 100

km/h ?

2mcEINTERNAL =

2222

)(mccpE =

E = TOTAL

ENERGY

E = INTERNAL

ENERGY

Expressions for (total) Energy

and Momentum of a particle of mass m,

moving at velocity u

Classical Limit

1

1

ump

2

22 mu

mcE

NEW

FAMILIAR kinetic energy

Kinetic Energy = KE

2mcEINTERNAL =

2222

)(mccpE =

E = TOTAL

ENERGY

E = INTERNAL

ENERGY

Photon’s Mass = 0


hfEphoton =

= hfKEmax








behaving like

Particles

(Chapter 2)

The Production of X-Rays

(Wilhelm Roentgen 1901)

(The “reverse” of the Photoelectric Effect)

Bremsstrahlung CLASSICAL physics:

Radiation covers entire spectrum

SURPRISE:

Experiments indicate a cutoff wavelength:

Frequency f , Energy E=hf

1 photon 1 electron

(?) 1 electron 1 photon (?)


(Wilhelm Roentgen 1901)

(The “reverse” of the Photoelectric Effect)

Bremsstrahlung CLASSICAL physics:

Radiation covers entire spectrum

Experiments indicate a cutoff wavelength:

INDEED:

Frequency f , Energy E=hf

Frequency f

INDEED:

1 electron 1 photon

2mcEINTERNAL =

2222

)(mccpE =

E = TOTAL

ENERGY

E = INTERNAL

ENERGY

2mcEINTERNAL =

2222

)(mccpE =

E = TOTAL

ENERGY

E = INTERNAL

ENERGY

Photon’s Mass = 0


hfEphoton =

= hfKEmax

The Compton effect (Arthur Compton 1927)

Hypothesis: Experiment?

momentum

energy

Energy and Momentum Conservation

The Compton Effect

Photons carry momentum like particles

and scatter individually with other particles

QUESTIONS ?








behaving like

Particles

(Chapter 3, or 2-old book)

Particle-Antiparticle Pair

Creation

Bubble Chamber

Previous Lectures








behaving like

Particles

(Chapter 3, or 2 in old book)

“PHOTONS”

QUESTIONS ?





behaving like

Particles

(Chapter 2)

“PHOTONS”

Particle-Antiparticle Pair Production

PHOTONS

E = hf

The Compton Effect PHOTONS

p = hf/c = h/l

Double-slit Diffraction

Experiment

light

INDIVIDUAL

PHOTON

HITS

Although diffraction of light is

a wave phenomenon,

there is no smooth distribution of light

in the diffraction pattern,

but

the pattern is rather formed of many

individual hits of particles – the photons

A single photon

DOES NOT

get “disintegrated” in the

Diffraction process

to make a smooth

diffraction pattern

QUESTIONS ?

Electromagnetic

waves (light)

Particles

(photons)

Massive

Particles Waves

?

Very strange

Very strange

Like a

Wave

Massive

Particles Waves

?

electrons

X-ray photons

or

electrons

Massive

Particles Waves

electrons

Properties of

Matter Waves

Wavelength p

h=l

De Broglie (1929)

Frequency h

Ef =

~1905 ~1930

Properties of

Matter Waves

Amplitude =

function of (x,t) ),( tx

Probability

Amplitude

Amplitude =

function of (x,t)

),( tx

Probability

Amplitude

The probability density

to find

a particle

at coordinate x, at time t

2|),(| tx

2|),(| txdxP =

The probability

to find

a particle

in an interval x

Integrate over x

x

t 2|)(| xdxP =

Why l is so

Small?

Because h is very-very

Small

p

h=l

QUESTIONS ?

l

p2k

fT

pp

22

=

p2

h

Wave Number

Angular Frequency

DEFINITIONS

l

p2k f

Tp

p 2

2=

p2

h

kh

p ==l

== hfE

2|)(| xdxP =

The probability

to find

a particle

in an interval x

Integrate over x

How does the Probability Wave Move?

Equation of Motion for y ?

The Free-Particle

Schrodinger Wave

Equation

),( txy

2* |),(|),(),( txtxtx yyy =

Probability Wave Function

Probability

Density

a complex function ),( txy

),(),(),( 21 txitxtx yyy

Probability Density = ?

),(),(),( 21

* txitxtx yyy

Complex Conjugate

1i

12 =i

),(),(),( 21 txitxtx yyy

Probability Density

),(),(),( 21

* txitxtx yyy 1=i

== ),(),(|),(| *2 txtxtx yyy

== 12212211 yyyyyyyy iiii

2211 yyyy =

),(),(|),(|2

2

2

1

2 txtxtx yyy =

The Plane Wave

Is the Plane Wave

a solution of the Schrodinger Equation?

Is the Plane Wave


Is the Plane Wave


Is the Plane Wave


Is the Plane Wave


22

)( 22 mv

m

mvE ==

The Magnitude of a Plane Wave

Constant in space and time!

Constant Probability Density

= constant (x,t)

Kinetic

energy

Plane

wave

The Uncertainty

Principle

The Uncertainty Relations

in 3 Dimensions

The Uncertainty Relations and the

Fourier Transform

Any wave may be expressed mathematically as a

superposition of plane waves of different

wavelengths and amplitudes

A Single-Slit

Spectral Content

Gaussian Wave form

barpx x = h2

1= minimum uncertainty

Chapter 3 –

The End

Bound States -

Some Simple Cases

1. A Review of Classical Bound States (4.4)

2. The Schrodinger Equation for Interacting Particles (4.1) 3. Stationary States (4.2) 4. Well-Behaved Functions and Normalization (4.3) 5. Case I: Particle in a Box – Infinite Potential Well (4.5) 6. Expectation Values, Uncertainties and Operators (4.6) 7. Case II: The Finite Potential Well (4.7) 8. Case III: The Simple Harmonic Oscillator (4.8)

The Schrodinger Equation

for Interacting Particles

A Particle Interacting

With What?

F=mg



A Particle Interacting

With What?

Simplification:

The Concept of Potential (replaces all individual

particle-particle interactions

with a single smooth potential)



Try to add potential energy U(x)

For free

particles



and for

Stationary Potentials

)(

)(

tUU

xUU

=

Key Assumption:

Factorization of the wave function

Spatial Part Temporal Part

What happens with the Schrodinger equation?

t and x are independent

Temporal part

Total wave

function

Temporal part

Total wave

function

The probability density is

time-independent Stationary

States

The spatial part of y(x,t)

The time-independent

Schrodinger equation:

y(x) is Real,

but y(x,t) is Complex, because

(t)=e-it

NOTE:

Spatial part

Normalization of y(x,t)

Smoothness of y(x,t)

Well-behaved wave

functions


The particle must be

somewhere in the universe

at any time

(the total probability should be = 1)


1. Continuity of y(x,t)

2. Continuity of (dy(x)/dx)


Discontinuity

in y(x)

x

)(xy


Discontinuity

in y(x) dxexk ikx

= )(2

1)(~ y

py

x

)(xy

Extremely large k (or short l) >

> Infinite Momentum impossible

Spatial part

Temporal part

Total wave

function



Summary

Case I: Particle in a box –

Infinite Potential well

Case I: Particle in a box –

Infinite Potential well



General solution

for region I

positive

y

xF = D = 0

C = G = 0

A = ?

B = ? From Smoothness:

Only certain Energy Levels are Allowed

Normalization of y(x,t):


Infinite Potential Well:

01 E

Bound States Some Simple Cases (Chapter 5 – The End (old book 4))

• A Review of Classical Bound States (4.4)

• The Schrodinger Equation for Interacting Particles (4.1)

• Stationary States (4.2)

• Well-Behaved Functions and Normalization (4.3)

• Case I: Particle in a Box – Infinite Potential Well (4.5)

1. Case II: The Finite Potential Well (4.7)

2. Case III: The Simple Harmonic Oscillator (4.8)

3. Read at Home: Expectation Values, Uncertainties and

Operators (4.6)

Finite Potential Well:

Infinite Potential Well:

Case 2: The Finite Potential Well

A, B, C, G, En = ?



Requirement:

continuity at

x = 0

continuity at

x = L

continuity at

x = 0

continuity at

x = L

CBk

CA == ,

/(-)

/(-)

Transcendental Equation –

impossible to solve analytically

Transcendental Equation –

impossible to solve analytically, but

“NUMERICAL” Solution is possible

(e.g. “graphing solution”)

Graphing solution

Graphing solution

Graphing solution

Parabola

~ k2

Graphing solution

Parabola

~ k2

Graphing solution

Parabola

~ k2

Solutions

left=right

Graphing solution

Solutions

left=right

Graphing solution

Solutions

left=right

Graphing solution

U’

U’

Finite probability - penetration

Infinite Potential Well: Finite Potential Well

Infinite Potential Well: Finite Potential Well

Case 3: The Simple Harmonic Oscillator

The Importance of the Harmonic Oscillator

Unbound States (Chapter 6 (old book 5))

Unbound States

(Chapter 6 (old 5))

5.1 Obstacles and Tunneling

5.2 Decay and other applications

5.3 Particle-wave propagation

5.4 The Classical Limit @ home

@ home

Obstacles and Tunneling

Potential Wall

Potential Wall

E > U

Obstacles and Tunneling Potential Wall

Obstacles and Tunneling Potential Wall

< 0

incident

reflected

transmitted

incident reflected

incident

reflected

transmitted

Decay

90

protons

Decay

Only?

Decay

Hydrogen Atom

(Chapter 7 (old book - 6))

6.1 The Schrodinger Equation in 3 dimensions

6.2 The 3D Infinite Potential well

6.3 Toward the Hydrogen Atom

6.4 Central Forces

Schrodinger Equation in 3 Dimensions

KE

operator

In 3-D

Schrodinger Equation in 3 Dimensions

Probability Density in 3 Dimensions

Stationary States

f(r) f(t)

Stationary States

Time-

independent

Sch. equation

temporal part

Stationary States in a 3-D Box


“Factorization”

f(x) f(y) f(z) f(x,y,z)


The Solution


Solution

3,2,1: === zyx LLLexample


“Degeneracy”

Toward the Hydrogen Atom

proton

electron

Energy loss due to Bremsstrahlung

(because of centripetal acceleration)

The

Problem:


proton

electron

Stationary

state?

Towards the Hydrogen Atom


Spherical Polar

Coordinate

system


Toward the Hydrogen Atom:

Schrodinger equation in Spherical Polar

Coordinates

Summary: Hydrogen Atom

The meaning of l

Angular part of Schrodinger Eq.

Units of angular

momentum

L ~ Magnitude of Angular

Momentum

Quantized so far:

The projection of

angular momentum to

z- axis

The magnitude of

(orbital) angular momentum

ml =

MAGNETIC QUANTUM

NUMBER

l =

ORBITAL

QUANTUM NUMBER

The Uncertainty

Principle

Angular

Momentum

Quantization

Angular

Momentum

Quantization

Energy levels of a

hydrogen atom:

homework

Ground State

Radial

Distribution of

the Electron

Probability

Density in a

Hydrogen Atom

Bohr Radius

How Small is “Small”?

Quantum

numbers

“accidental” degeneracy Because of

1/r

Traditional naming

scheme

Lecture Today

Lecture Today

0)1( == llLs 2)1( == llLp

0)1( == hllL eGroundStat s p

Ground State:

The Electron is NOT

Orbiting around the

proton

0)1( == llL eGroundStat

Classical Physics:

The Electron is Orbiting

around the Proton

Spectral

Lines

New Topic

Spin and Atomic Physics

• The concept of “SPIN”

• The Stern-Gerlach Experiment

• Toward the Exclusion Principle for

“Fermions”

Orbiting in Classical Physics

Magnetic Dipole

Moment

Ground State -> l = 0 -> L = 0 -> F = 0 (???)

Surprise: Real Experimental Result

SPIN

Like for L:

s – the quantum number of SPIN

Intrinsic property of a particle

For an electron: s = 1/2

Spin Orientation

All Elementary Particles: either

Bosons or Fermions

Bosons

Fermions

The Pauli Exclusion Principle

for Fermions

2

1=zS

2

1=zS

The Exclusion Principle for

Fermions

2

1=zS

2

1=zS


Fermions

2

1=zS

2

1=zS

2

1=zS


Fermions

2

1=zS

2

1=zS

2

1=zS

Let’s “create” multi-electron atoms

Our Toolkit:


Quantization of Energy Levels

The Exclusion Principle

“Progressive Occupation” of Energy Levels

Hydrogen

H

Helium

He

Lithium Li

(3 electrons)

Energy Levels

Hydrogen

H, ground state

Energy Levels

He 2-electrons

Energy Levels

Neon (Ne) 10-electrons

Neon

Ne

Energy Levels

Fluorine: 9-electrons

Fluorine

F

Like Ne with

one electron

(and proton)

less

Fluorine

F

Like Ne with

one electron

(and proton)

less

A fluorine atom would

gladly accept one more

electron, to

“look more like Ne”

That is why it is

chemically very reactive

What matters

for Chemical

Properties is the

state of the most

loose electrons

What maters

for Chemical

Properties is the

state of the most

loose electrons

The other, more

strongly bound

electrons are

merely passive

“placeholders”

The traditional naming

scheme

Principal

quantum

number “n” The

number of

electrons in

that

“subshell”

Noble gasses

Noble gasses

He + 1 electron

Li: 1s2 2s1

VALENCE

First Ionization Energy

9.1 When atoms come together

9.5 Energy Bands

9.6 Conductors, Insulators, Semiconductors

Bonding of Atoms

Chapter 10 (old book - 9)

2 Atoms Come Together

Electrons become “shared”

N Atoms Come Together

Formation of energy “Bands”

Molecules – Kovalent Bond

Molecules – Ionic Bond

Extremely

“Loose”

Molecules – Ionic Bond

“Ne”

“Xe”

Electrostatic Attraction: Cs(+) F(-)

Extremely

Loose

Cs: First Ionization Energy

Form

Energy Levels

to

Energy Bands

4- atom crystal

N- atomic States

Formation of ~continuous Bands

Band Structure

Central

role in

condensed

matter

physics !

Form

Energy Levels

to

Energy Bands

4- atom crystal

Fermi-Dirac

Distribution

Insulators, Conductors, Semiconductors

Semiconductors –

insulators with a

small energy gap

(Eg < 2 eV)

con

du

cto

r Band filling

Band filling co

nd

uct

or

sem

ico

nd

uct

or Energy gap-

no electrons

!

Electron-hole creation and conduction

Current conducted by electrons and holes

Semiconductor Doping

Doping elements: As, Al, Ga, B,…


Doping elements: As and B

n-type p-type


Extrinsic (doped)

semiconductor

Intrinsic

semiconductor

e.g. Si

n-type

p-type

Semiconductor Devices

Equilibrium – “depletion zone”

p-n Junction “Bias”

Current can flow through the diode

in only one direction

p-n Junction “Bias” and the DIODE

n-p-n structure and the TRANSISTOR

Signal

amplification

Very weak (low

doping)

The Light Emitting DIODE (LED)

photon

Recombination:

Electron-hole

“annihilation”

LASER

LASER

LED

MASSIVE

Recombination:

Very high doping level

needed,

and

REFLECTION

Mirrors needed

The Laser Diode

Laser

The Genealogy of Forces

Unified Theories

Physics & Cosmology

The forces

L

[MeV]

L

[MeV]

Fermions

Bosons:

How the elementary particles interact

Exchange of mediators (bosons)

The range of a force and the mass

of the mediator

Finite Mass Finite Range

Zero Mass Infinite Range

The range of the force and the

mass of the mediator

Photon - the mediator of the

Electromagnetic force

Mass = 0

Range = infinite (~1/r potential)

The range of a force and the mass

of the mediator

Finite Mass Finite Range

Zero Mass Infinite Range

mcxrange

mcmcctcx

mcEt

mcE

tE

1

22

2

2

2

2

m

Relativistic Treatment is Mandatory

Because particles

may be created

and annihilated

(internal energy

plays a role; mc2)

Schrodinger Equation NON-relativistic

Relativistic Quantum Mechanics

Schrodinger Equation NON-relativistic

Relativistic Quantum Mechanics:

The Klein-Gordon Wave Equation

The Dirac Equation (for Fermions)

The concept of SPIN follows naturally

The Atomic Nucleus

All have spin s = ½

Fermions Fermi-Dirac Statistics

Let’s “create” multi-nucleon nuclei

Our Toolkit:


Quantization of Energy Levels

The Exclusion Principle

“Progressive Occupation” of Energy Levels

The Helium Nucleus

2 protons + 2 neutrons

The Hydrogen “Isotopes”

Heavy Water: D2O

“Rutherford Experiment”

Rutherford Experiment

Back-scattering

Electrostatic

repulsion

Rutherford Back-scattering and

The measurement of the Size of a Nucleus

Kinetic Energy = Potential Energy at Closest Approach

r” = 0

No return -

the nuclei

collide

r

r’

The Radius of a Nucleus

From Rutherford back-scattering,

and other types of measurements,

the radius of a nucleus:

Atomic number 1.2 fm (femtometer)

1 fm = 10-15 m

The Volume of a Nucleus

The volume is

proportional to the

number of nucleons

(A)

Spherical Shape

Assumed

(an excellent

approximation)

The Density of Nuclear Matter

independent of A

~ everywhere inside a nucleus

The density is constant, and:

Constant Density of Nuclear

Matter

Nucleons are closely packed in

an ~incompressible liquid-like

“droplet”

Binding

“Strong Force”

Why would protons and neutrons

stay together in a nucleus?

Forces

The real Strong Force acts among

the constituents of a NUCLEON –

quarks and gluons

The potential well due to the (residual)

strong force

Nucleon-nucleon distance

Finite-range interaction Nucleons cannot

penetrate each other

Building a nucleus:

Deuteron

Deuteron is barely bound!

Building a nucleus:

Deuteron

Deuteron is barely bound!

Note: no

Coulomb

repulsion

Building a nucleus:

No Coulomb repulsion

Building a nucleus:

With Coulomb repulsion

Building a nucleus:

with Coulomb repulsion

Nucleons

are

Fermions !

Building a nucleus:

with Coulomb repulsion

Without Coulomb interaction

Binding Energy per Nucleon

Building a nucleus: Shell Structure

Nucleons

are

Fermions !

Building a nucleus: Shell Structure

Nucleons

are

Fermions !

Binding

energy per

(the least

bound)

electron in

an atom

(~10 eV)

Binding

energy per

nucleon in a

nucleus

(~ MeV)

Building a nucleus: Binding Energy

N > Z

Building a nucleus: Large Nuclei

Building a nucleus: the Curve of Stability

N > Z

Away from the Curve of Stability (?)

Radioactivity

• Alpha – He nucleus

• Beta - Electron

• Gamma – Photon

• Fission – breakup

into two nuclei

Beta-decay – first evidence for the

existence of “Neutrinos”

One more particle in a decay!

Spontaneous Fission

Neutron-induced

Fission

Fission

Neutron-induced Fission – Chain Reaction

Example: n + U

Applications: Nuclear Bombs and

Nuclear Power

Controlled Fission (Chain) Reaction

Reactors

Power reactors

Research Reactors

Converters

Fission Explosives

U-235 and Pu-239

Exponential increase in energy release of a

supercritical assembly

Without control Nuclear (“atomic”) Bomb

Fusion

Fusion

Nucleons like to

be together

Fusion – the creation of light

elements in Stars

He + He + He 12C

The composition of

Nucleons

3 Quarks (u = “up”, d = “down”)

Many Interesting Experiments

• Search for Proton Decay

• Cosmic Rays of unexpectedly high energies

(1020 eV)

• SUSY (Super Symmetric Particles),

annihilation in space etc.

• Distribution of the Cosmic Background Radiation

• The mass of the neutrinos

• What is Dark Matter? Dark Energy?

The End

Thank You!

It was a great pleasure

Albert Einstein - ferenc.physics.ucdavis.eduferenc.physics.ucdavis.edu/9D/lectures.pdf · •Lorentz Transformations and 4-vectors (1.11) •The Doppler Effect (1.6) x1 x2 . Lorentz

Documents