Transcript Lezioni di Fisica

The fall of Classical Physics
1
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
2
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).
3
Something is wrong
Relativity, 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 (MichelsonMorley, …)
A speed!
4
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
2  1  e2 2
dE
 3 a  
W  
3  4 0  c
dt
F  1  e2
 2
a   
m  4 0  m r
   1010 s
5
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
6
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
7
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.
8
I
Light behaves like a particle,
sometimes
9
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)
10
Photoelectric Effect, Interpretation




Electrons are trapped in the metal, by a potential V >
Ve
Light might give to the electrons enough energy Eg to
escape
Electrons ejected possess a kinetic energy
K = Eg - eV
Kmax = Eg – f



f = eVe is called the work function
The work function represents the minimum energy with
which an electron is bound in the metal
Typically, f ~ 4 eV
11

At large values of DV, 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 DV is negative, the
current drops
When DV is equal to or more
negative than DVs, the current
is zero
12
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
13
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
14
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
15
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
16
Cutoff Frequency


The lines show the linear
relationship between K and ƒ
The slope of each line is
independent of the metal
h ~ 6.6 10-34 Js


The absolute value of the yintercept is the work function
The x-intercept is the cutoff
frequency

This is the frequency below
which no photoelectrons are
emitted
Kmax = hƒ – f
17
Photoelectric Effect Features
and 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
E  hf  

h 

 

2 

h is called the Planck constant, and plays a
fundamental role in Quantum Physics
18
Photon Model Explanation

Dependence of photoelectron kinetic energy on light intensity




Time interval between incidence of light and ejection of the
photoelectron


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
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
19
Photon Model Explanation of the
Photoelectric Effect, final

Dependence of photoelectron kinetic energy on light
frequency

Since Kmax = hƒ – f, 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
20
Cutoff Frequency and Wavelength


The cutoff frequency is related to the work function
through ƒc = f / h
The cutoff frequency corresponds to a cutoff
wavelength
c
hc
lc 

ƒc
f

Wavelengths greater than lc incident on a material
having a work function f do not result in the emission
of photoelectrons
21
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…
uE   0 E 2 / 2  0 0 1/ c2
2


u

u

u


E
E
B
0
uB  B 2 / 20
Maxwell: pressione di radiazione
p u

V c
22

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, l‘> l0, is caused by the
scattering of free electrons
l ' lo 

h
1 cos 
mec
This is called the Compton shift equation
(wait the relativity week…)
23
Compton Effect, Explanation

The results could be explained, again, by treating the
photons as point-like particles having



Assume the energy and momentum of the isolated
system of the colliding photon-electron are conserved



energy hƒ
momentum hƒ / c
Adopted a particle model for a well-known wave
The unshifted wavelength, lo, is caused by x-rays
scattered from the electrons that are tightly bound to
the target atoms
The shifted peak, l', is caused by x-rays scattered
from free electrons in the target
24
3) Blackbody radiation

Every object at T > 0 radiates electromagnetically, and
absorbes radiation as well
W 

4
I   T   ~ 5.7 10
Stefan-Boltzmann law:
2
4 
m
K


8

Blackbody: the
perfect absorber/emitter
“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 25
Experimental findings
& classical calculation

Wien’s law: the emission
peaks at
2.9  m
lmax 
T /1000 K
Example: for Sun T ~ 6000K

But the classical calculation
(Rayleigh-Jeans) gives a
completely different result…

Ultraviolet catastrophe
26
Experimental findings
& classical calculation

Classical calculation (RaileighJeans): the blackbody is a set of
oscillators which can absorb any
frequency, and in level transition
emit/absorb quanta of energy:
dI
kBT
dI dI d l
 8 4 

 E2
dl
l
dE d l dE
No maximum; a ultraviolet
catastrophe should absorb all
energy
Experiment
27
Planck’s hypothesis

Only the oscillation modes for which
E = hf
are allowed…
28
Interpretation



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)
E
n
4hf
4
3hf
3
2hf
2
hf
1
The classical calculation is accurate for large
wavelengths, and is the limit for h -> 0
dI
hc
hc
kBT
 2 c 5 hc / lk T

 2 c 4
h 0  2 c 5
B
dl
l  hc / l kBT 
l
l e
 1 o l 
29
Which lamp emits e.m. radiation ?
1) A
2) B
3) A & B
4) None
30
4) Particle-like behavior of light:
now smoking guns…

The reaction
g e e
 
has been
recorded
millions of
times…
31
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.
32
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
E  hf  
h ~ 6.6 10-34 Js is called the Planck’s constant
33
II
Particles behave like waves,
sometimes
34
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
E  hf  
h ~ 6.6 10-34 Js is called the Planck’s constant
35
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
h
l
p
36
Diffraction of electrons

Davisson & Germer 1925:
Electrons display diffraction patterns !!!
37
de Broglie’s wavelength

What is the wavelength associated to a particle?
de Broglie’s wavelength:

h
l  p k
p
Explains quantitatively the diffraction by Davisson and Germer……
Note the symmetry
E  ω


p  k
What is the wavelength of an electron moving at 107 m/s ?
6.63 1034 Js 

h
11
l


7.28

10
m
31
7
mv  9.1110 kg 10 m/s 
(smaller than an atomic length; note the dependence on m)
38
Atomic spectra

Why atoms emit according to a discrete energy spectrum?
1 
 1
 RH  2  2 
l
m n 
RH legata "numerologicamente" a h
Per l'idrogeno
1
Balmer
m  n interi

Something must
be there...
39
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=nl
n=1,2,3,…
h
nh
l   2 r   pr  L  n
p
p
=> Angular momentum is quantized (Bohr
postulated it…)
40
Hydrogen (Z=1)
Ek
mv 2
e2
e2
F
2
 ke 2  Ek  ke
r
r
r
2r
e2
e2
E p   ke
 Ek  E p  E  ke
r
2r
v
m
r
F
NB:
• In SI, ke = (1/40) ~ 9 x 109 SI units
• Total energy < 0 (bound state)
• <Ek> = -<Ep/2> (true in general for bound states, virial theorem)
L  n  mvr
2
2
2 2
k
e
m
n
n


e
 r
 rn
m 2
e2 

 
2
2  mr 
2r
ke me
Ek 
v  ke
2
2r
Only special values are possible for the radius !
41
Energy levels

The radius can only assume
values

rn 
2
ke me 2
The smallest radius (Bohr’s radius) is r1 
2
ke me
2

e2
Radius and energy are related: E  ke
2r

And thus energy is quantized:
n2
 .0529 nm  a0
ke e 2 1
e2
13.6 eV
En   ke



2rn
2a0 n 2
n2
42
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]
43
Level transitions and energy quanta
f 
Ei  E f
h
e2  1
1 
 ke
 2 2
2a0 h  n f
ni 
 1
f
e2  1
1 
1 
  ke
 2  2   RH  2  2 
n

l c
2a0 hc  n f
ni 
n
f
i


1

We obtain Balmer’s relation!
44
E  ω

Limitations
Semiclassical models wave-particle duality can
explain phenomena, but the thing is still insatisfactory,





p  k
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…
45
Wavefunction



Change the basic model!
We can describe the position of a particle
through a wavefunction y(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…
46
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,
 2
dE   ( r , t ) dV
but of probability
 2
dP   ( r , t ) dV

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.
47
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
E  hf  

Wave-like property of particles: particles behave in certain condistions as
waves with wavenumber
p  h/l  k


( E , p)  ( , k )
Role of Planck’s constant, h ~ 6.6 10-34 Js
Concepts of wave and particle need to be unified: wavefunction
y (r,t).
48
L’equazione di Schroedinger
49
Proprieta’ della funzione d’onda
50
L’equazione di S.
51
Laboratorio virtuale
Origini della Meccanica Quantistica

Radiazione termica del corpo nero

Diffrazione degli elettroni
52