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Transcript of 1 The fall of Classical Physics. 2 Classical physics: Fundamental Models Particle Model (particles,...
1
The fall of Classical Physics
2
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
3
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).
4
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!
5
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
4
1
4
1
3
2
10
2
2
0
23
2
0
mr
e
m
Fa
dt
dEa
c
eW
6
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
7
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
8
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.
9
ILight behaves like a particle,
sometimes
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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
2
hhfE
h is called the Planck constant, and plays a fundamental role in Quantum Physics
11
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…
12
Compton’s experiment showed that, at any given angle, a different 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
This is called the Compton shift equation
' 1 cosoe
h
m c
13
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, 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
14
Every object at T > 0 radiates electromagnetically, and absorbes radiation as well
Stefan-Boltzmann law:
Blackbody: the
perfect absorber/emitter
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 10W
I Tm K
15
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
max
2.9
/1000
m
T K
16
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 energy
428 BdI dI dIk T dE
d dE d dE
Experiment
17
Planck’s hypothesis
Only the oscillation modes for which
E = hf
are allowed…
18
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 TdIc c
hc
d 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)
n
4
3
2
1
E
4hf
3hf
2hf
hf
19
Which lamp emits e.m. radiation ?
1) A2) B3) A & B4) None
20
Particle-like behavior of light:now smoking guns…
The reaction
has been recorded millions of times…
e e
21
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.
22
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
23
IIParticles behave like waves,
sometimes
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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 mm, separated by a distance of d=6 mm. 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
p
h
25
Diffraction of electrons
Davisson & Germer 1925:
Electrons display diffraction patterns !!!
26
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)
hp k
p
34
11
31 7
6.63 10 Js7.28 10 m
9.11 10 kg 10 m/s
h
mv
kpωE
27
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
28
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 nh
p pnr pr L
=> Angular momentum is quantized (Bohr postulated it…)
29
2
2
2 2
2
2
22
2
ke k e
p e ek p
Emv e eF k E k
r r r r
eE k E E
r
eE k
r
v
rm
F
NB:
• In SI, ke = (1/4pe0) ~ 9 x 109 SI units
• Total energy < 0 (bound state)
• <Ek> = -<Ep/2> (true in general for bound states, virial theorem)
2 22 2
22 22 2
2 2
e
k e
ne
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)
30
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:
22
2ne
r nk me
2
2ee
E kr
22
2 20
1 13.6 eV
2 2e
n en
k eeE k
r a n n
2
1 02.0529
e
r nm ak me
31
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]
32
Level transitions and energy quanta
0
2
2 2
1 1
2i f
e
f i
E E ef k
h a h n n
0
2
2 2 2 2
1 1 1 1 1
2e H
f i f i
f ek R
c a hc n n n n
We obtain Balmer’s relation!
33
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
34
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…
35
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.
dVtrdE2
),(
dVtrdP2
),(
36
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
y (r,t).
E hf
/p h k ( , ) ( , )E p k
37
L’equazione di Schroedinger
38
Proprieta’ della funzione d’onda
39
L’equazione di S.