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The fall of Classical Physics. 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
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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
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).
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!
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
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
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
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.
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 • h is called the Planck constant, and plays a fundamental role in Quantum Physics
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…
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
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
Blackbody radiation • Every object at T > 0 radiates electromagnetically, and absorbes radiation as well Stefan-Boltzmann law: • 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
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
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 Experiment
Planck’s hypothesis • Only the oscillation modes for which E = hf are allowed…
Interpretation n E 4 4hf 3 3hf 2 2hf 1 hf • 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) • The classical calculation is accurate for large wavelengths, and is the limit for h -> 0
Which lamp emits e.m. radiation ? 1) A 2) B 3) A & B 4) None
Particle-like behavior of light:now smoking guns… • The reaction has been recorded millions of times…
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.
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
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
Diffraction of electrons • Davisson & Germer 1925: Electrons display diffraction patterns !!!
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)
Atomic spectra • Why atoms emit according to a discrete energy spectrum? Balmer • Something must be there...
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,… => Angular momentum is quantized (Bohr postulated it…)
Hydrogen (Z=1) v m r 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) Only special values are possible for the radius !
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:
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]
Level transitions and energy quanta • We obtain Balmer’s relation!
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…
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…
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.
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).
