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The fall of Classical Physics

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Classical physics: Fundamental Models

Particle Model (particles, bodies) Motion in 3 dimension; for each time t, position and speed

are known (they are well-defined numbers, regardless we know them). Mass is known.

Systems and rigid objects Extension of particle model

Wave Model (light, sound, …) Generalization of the particle model: energy is transported,

which can be spread (de-localized) Interference

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Classical physics at the end of XIX Century Scientists are convinced that the particle and wave model can describe the

evolution of the Universe, when folded with Newton’s laws (dynamics) Description of forces

Maxwell’s equations Law of gravity. …

We live in a 3-d world, and motion happens in an absolute time. Time and space (distances) intervals are absolute.

The Universe is homogeneous and isotropical; time is homogeneous. Relativity

The physics entities can be described either in the particle or in the wave model.

Natura non facit saltus (the variables involved in the description are continuous).

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Something is wrongRelativity, continuity, wave/particle (I)

Maxwell equations are not relativistically covariant!

Moreover, a series of experiments seems to indicate that the speed of light is constant (Michelson-Morley, …) A speed!

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Something is wrong Relativity, continuity, wave/particle (IIa)

In the beginning of the XX century, it was known that atoms were made of a heavy nucleus, with positive charge, and by light negative electrons Electrostatics like gravity:

planetary model All orbits allowed

But: electrons, being accelerated, should radiate and eventually fall into the nucleus

s10

41

41

32

10

2

2

0

23

2

0

mre

mFa

dtdEa

ceW

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Something is wrong Relativity, continuity, wave/particle (IIb)

If atoms emit energy in the form of photons due to level transitions, and if color is a measure of energy, they should emit at all wavelengths – but they don’t

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Something is wrong Relativity, continuity, wave/particle (III)

Radiation has a particle-like behaviour, sometimes

Particles display a wave-like behaviour, sometimes

=> In summary, something wrong involving the foundations: Relativity Continuity Wave/Particle duality

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Need for a new physics A reformulation of physics was needed

This is fascinating!!! Involved philosophy, logics, contacts with civilizations far away from us…

A charming story in the evolution of mankind

But… just a moment… I leaved up to now with classical physics, and nothing bad happened to me! Because classical physics fails at very small scales, comparable with

the atom’s dimensions, 10-10 m, or at speeds comparable with the speed of light, c ~ 3 108 m/s

Under usual conditions, classical physics makes a good job.

Warning: What follows is logically correct, although sometimes historically inappropriate.

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ILight behaves like a particle,

sometimes

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1) Photoelectric Effect The photoelectric effect

occurs when light incident on certain metallic surfaces causes electrons to be emitted from those surfaces The emitted electrons

are called photoelectrons

When the system is kept in the dark, the ammeter reads zero

When plate E is illuminated, a current is detected by the ammeter

The current arises from photoelectrons emitted from the negative plate (E) and collected at the positive plate (C)

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Photoelectric Effect, Interpretation

Electrons are trapped in the metal, by a potential V > Ve

Light might give to the electrons enough energy E to escape

Electrons ejected possess a kinetic energyK = E - eV

Kmax = E – = eVe is called the work function The work function represents the minimum energy with which

an electron is bound in the metal Typically, ~ 4 eV

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At large values of V, the current reaches a maximum value All the electrons emitted

at E are collected at C The maximum current

increases as the intensity of the incident light increases

When V is negative, the current drops

When V is equal to or more negative than Vs, the current is zero

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Photoelectric Effect Feature 1 Dependence of photoelectron

kinetic energy on light intensity Classical Prediction

Electrons should absorb energy continually from the electromagnetic waves

As the light intensity incident on the metal is increased, the electrons should be ejected with more kinetic energy

Experimental Result The maximum kinetic energy is

independent of light intensity The current goes to zero at the same

negative voltage for all intensity curves

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Photoelectric Effect Feature 2 Time interval between incidence of light and ejection

of photoelectrons Classical Prediction

For very weak light, a measurable time interval should pass between the instant the light is turned on and the time an electron is ejected from the metal

This time interval is required for the electron to absorb the incident radiation before it acquires enough energy to escape from the metal

Experimental Result Electrons are emitted almost instantaneously, even at very low light

intensities Less than 10-9 s

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Photoelectric Effect Feature 3 Dependence of ejection of electrons on light frequency

Classical Prediction Electrons should be ejected at any frequency as long as the light

intensity is high enough Experimental Result

No electrons are emitted if the incident light falls below some cutoff frequency, ƒc

The cutoff frequency is characteristic of the material being illuminated No electrons are ejected below the cutoff frequency regardless of

intensity

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Photoelectric Effect Feature 4 Dependence of photoelectron

kinetic energy on light frequency Classical Prediction

There should be no relationship between the frequency of the light and the electron maximum kinetic energy

The kinetic energy should be related to the intensity of the light

Experimental Result The maximum kinetic energy of the

photoelectrons increases with increasing light frequency

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Cutoff Frequency The lines show the linear

relationship between K and ƒ The slope of each line is

independent of the metalh ~ 6.6 10-34 Js

The absolute value of the y-intercept is the work function

The x-intercept is the cutoff frequency This is the frequency below

which no photoelectrons are emitted

Kmax = hƒ –

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Photoelectric Effect Featuresand Photon Model explanation

The experimental results contradict all four classical predictions

Einstein interpretation: All electromagnetic radiation can be considered a stream of quanta, called photons

A photon of incident light gives all its energy hƒ to a single electron in the metal

2hhfE

h is called the Planck constant, and plays a fundamental role in Quantum Physics

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Photon Model Explanation Dependence of photoelectron kinetic energy on light intensity

Kmax is independent of light intensity K depends on the light frequency and the work function The intensity will change the number of photoelectrons being emitted,

but not the energy of an individual electron Time interval between incidence of light and ejection of the

photoelectron Each photon can have enough energy to eject an electron immediately

Dependence of ejection of electrons on light frequency There is a failure to observe photoelectric effect below a certain cutoff

frequency, which indicates the photon must have more energy than the work function in order to eject an electron

Without enough energy, an electron cannot be ejected, regardless of the light intensity

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Photon Model Explanation of the Photoelectric Effect, final

Dependence of photoelectron kinetic energy on light frequency Since Kmax = hƒ – , as the frequency increases, the

maximum kinetic energy will increase Once the energy of the work function is exceeded

There is a linear relationship between the kinetic energy and the frequency

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Cutoff Frequency and Wavelength The cutoff frequency is related to the work function

through ƒc = / h The cutoff frequency corresponds to a cutoff

wavelength

Wavelengths greater than c incident on a material having a work function do not result in the emission of photoelectrons

ƒcc

c hc

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2) The Compton Effect

Compton dealt with Einstein’s idea of photon momentum Einstein: a photon with energy E carries a

momentum of E/c = hƒ / c According to the classical theory,

electromagnetic waves of frequency ƒo incident on electrons should scatter, keeping the same frequency – they scatter the electron as well…

20 0

21/ 20

020

/ 2/ 2

pMaxwell: pressione di radiazione V

cEE B

B

u Eu u u E

u Buc

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Compton’s experiment showed that, at any given angle, only one frequency of radiation is observed The graphs show the scattered x-ray for various

angles Again, treating the photon as a particle of

energy hf explains the phenomenon. The shifted peak, ‘> , is caused by the scattering of free electrons

This is called the Compton shift equation (wait the relativity week…)

' 1 cosoe

hm c

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Compton Effect, Explanation The results could be explained, again, by treating the

photons as point-like particles having energy hƒ momentum hƒ / c

Assume the energy and momentum of the isolated system of the colliding photon-electron are conserved Adopted a particle model for a well-known wave

The unshifted wavelength, o, is caused by x-rays scattered from the electrons that are tightly bound to the target atoms

The shifted peak, ', is caused by x-rays scattered from free electrons in the target

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Every object at T > 0 radiates electromagnetically, and absorbes radiation as wellStefan-Boltzmann law:

Blackbody: the perfect absorber/emitter

3) Blackbody radiation

“Black” body

Classical interpretation: atoms in the object vibrate; since <E> ~ kT, the hotter the object, the more energetic the vibration, the higher the frequency The nature of the radiation leaving the cavity through the hole depends only on the temperature of the cavity walls

842 4~ 5.7 10 WI Tm K

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Experimental findings & classical calculation Wien’s law: the emission peaks at

Example: for Sun T ~ 6000K

But the classical calculation (Rayleigh-Jeans) gives a completely different result…

Ultraviolet catastrophe

max2.9/1000

mT K

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Experimental findings & classical calculation Classical calculation (Raileigh-Jeans):

the blackbody is a set of oscillators which can absorb any frequency, and in level transition emit/absorb quanta of energy:

No maximum; a ultraviolet catastrophe should absorb all energy4

28 BdI dI dIk T d Ed dE d dE

Experiment

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Planck’s hypothesis Only the oscillation modes for which

E = hfare allowed…

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Interpretation

The classical calculation is accurate for large wavelengths, and is the limit for h -> 0

0 5 4/5o

2 2 2/1B

Bhc k hT

B

hc hc k TdI c ch

cd c k Te

Elementary oscillators can have only quantized energies, which satisfy E=nhf (h is an universal constant, n is an integer –quantum- number)

Transitions are accompanied by the emission of quanta of energy (photons)

n4

3

2

1

E

4hf

3hf

2hf

hf

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Which lamp emits e.m. radiation ?

1) A2) B3) A & B4) None

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4) Particle-like behavior of light:now smoking guns…

The reaction

has been recorded millions of times…

e e

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Bremsstrahlung "Bremsstrahlung" means in German

"braking radiation“; it is the radiation emitted when electrons are decelerated or "braked" when they are fired at a metal target. Accelerated charges give off electromagnetic radiation, and when the energy of the bombarding electrons is high enough, that radiation is in the x-ray region of the electromagnetic spectrum. It is characterized by a continuous distribution of radiation which becomes more intense and shifts toward higher frequencies when the energy of the bombarding electrons is increased.

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Summary The wave model cannot explain the behavior of light in certain

conditions Photoelectric effect Compton effect Blackbody radiation Gamma conversion/Bremsstrahlung

Light behaves like a particle, and has to be considered in some conditions as made by single particles (photons) each with energy

h ~ 6.6 10-34 Js is called the Planck’s constant

E hf

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IIParticles behave like waves,

sometimes

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Summary of last lecture The wave model cannot explain the behavior of light in certain

conditions Photoelectric effect Compton effect Blackbody radiation Gamma conversion

Light behaves like a particle, and has to be considered in some conditions as made by single particles (photons) each with energy

h ~ 6.6 10-34 Js is called the Planck’s constant

E hf

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Should, symmetrically, particles display radiation-like properties?

The key is a diffraction experiment: do particles show interference?

A small cloud of Ne atoms was cooled down to T~0. It was then released and fell with zero initial velocity onto a plate pierced with two parallel slits of width 2 m, separated by a distance of d=6 m. The plate was located H=3.5 cm below the center of the laser trap. The atoms were detected when they reached a screen located D=85 cm below the plane of the two slits. This screen registered the impacts of the atoms: each dot represents a single impact. The distance between two maxima, y, is 1mm.

The diffraction pattern is consistent with the diffraction of waves with

ph

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Diffraction of electrons

Davisson & Germer 1925:Electrons display diffraction patterns !!!

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de Broglie’s wavelength What is the wavelength associated to a particle?

de Broglie’s wavelength:

Explains quantitatively the diffraction by Davisson and Germer……

Note the symmetry

What is the wavelength of an electron moving at 107 m/s ?

(smaller than an atomic length; note the dependence on m)

h p kp

3411

31 7

6.63 10 Js7.28 10 m

9.11 10 kg 10 m/shmv

kpωE

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Atomic spectra Why atoms emit according to a discrete energy spectrum?

2 2

1 1 1Per l'idrogeno interi

legata "numerologicamente" a h

H

H

R m nm n

R

Something must be

there...

Balmer

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Electrons in atoms: a semiclassical model Similar to waves on a cord, let’s imagine that

the only possible stable waves are stationary…

2 r = n n=1,2,3,…

2h nhp p

nr pr L

=> Angular momentum is quantized (Bohr postulated it…)

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2

2

2 2

2

2

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2

ke k e

p e ek p

Emv e eF k E kr r r r

eE k E Er

eE kr

v

rm

F

NB:

• In SI, ke = (1/40) ~ 9 x 109 SI units

• Total energy < 0 (bound state)

• <Ek> = -<Ep/2> (true in general for bound states, virial theorem)

2 22 22

2 22 22 2

e

k en

e

L n mvrk em n

m e mr rE v kr

nr rk me

Only special values are possible for the radius !

Hydrogen (Z=1)

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Energy levels The radius can only assume values

The smallest radius (Bohr’s radius) is

Radius and energy are related:

And thus energy is quantized:

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2ne

r nk me

2

2eeE kr

22

2 20

1 13.6 eV2 2

en e

n

k eeE kr a n n

2

1 02 .0529e

r nm ak me

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Transitions An electron, passing from an orbit of energy Ei

to an orbit with Ef < Ei, emits energy [a photon such that f = (Ei-Ef)/h]

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Level transitions and energy quanta

0

2

2 2

1 12

i fe

f i

E E ef kh a h n n

0

2

2 2 2 2

1 1 1 1 12e H

f i f i

f ek Rc a hc n n n n

We obtain Balmer’s relation!

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Limitations Semiclassical models wave-particle duality can

explain phenomena, but the thing is still insatisfactory, When do particles behave as particles, when do they behave

as waves? Why is the atom stable, contrary to Maxwell’s equations?

We need to rewrite the fundamental models, rebuilding the foundations of physics…

kpωE

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Wavefunction

Change the basic model! We can describe the position of a particle

through a wavefunction (r,t). This can account for the concepts of wave and particle (extension and simplification).

Can we simply use the D’Alembert waves, real waves? No…

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Wavefunction - II

We want a new kind of “waves” which can account for particles, old waves, and obey to F=ma. And they should reproduce the characteristics of “real” particles: a

particle can display interference corresponding to a size of 10-7 m, but have a radius smaller than 10-10 m

Waves of what, then? No more of energy,

but of probability

The square of the wavefunction is the intensity, and it gives the probability to find the particle in a given time in a given place.

Waves such that F=ma? We’ll see that they cannot be a function in R, but that C is the minimum space needed for the model.

dVtrdE 2),(

dVtrdP 2),(

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SUMMARY Close to the beginning of the XX century, people thought that physics was understood. Two models (waves, particles). But:

Quantization at atomic level became experimentally evident Particle-like behavior of radiation: radiation can be considered in some conditions as a set of particles (photons) each with energy

Wave-like property of particles: particles behave in certain condistions as waves with wavenumber

Role of Planck’s constant, h ~ 6.6 10-34 Js Concepts of wave and particle need to be unified: wavefunction (r,t).

E hf

/p h k ( , ) ( , )E p k

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L’equazione di Schroedinger

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Proprieta’ della funzione d’onda

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L’equazione di S.

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Laboratorio virtualeOrigini della Meccanica Quantistica

Radiazione termica del corpo nero

Diffrazione degli elettroni