The fall of Classical Physics

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

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

10. 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)

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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 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 – f

18. 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

19. 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

20. 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

21. 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 • Wavelengths greater than lc incident on a material having a work function f do not result in the emission of photoelectrons

22. 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…

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

24. 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

25. 3) 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

26. 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

27. 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

28. Planck’s hypothesis • Only the oscillation modes for which E = hf are allowed…

29. 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

30. Which lamp emits e.m. radiation ? 1) A 2) B 3) A & B 4) None

31. 4) Particle-like behavior of light:now smoking guns… • The reaction has been recorded millions of times…

32. 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.

33. 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

34. IIParticles behave like waves, sometimes

35. 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

36. 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

37. Diffraction of electrons • Davisson & Germer 1925: Electrons display diffraction patterns !!!

38. 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)

39. Atomic spectra • Why atoms emit according to a discrete energy spectrum? Balmer • Something must be there...

40. 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…)

41. 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 !

42. 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:

43. 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]

44. Level transitions and energy quanta • We obtain Balmer’s relation!

45. 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…

46. 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…

47. 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.

48. 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).

49. L’equazione di Schroedinger

50. Proprieta’ della funzione d’onda