Lecture 4. Particle properties of waves

1. v/c Relativistic mechanics, El.-Mag. (1905) Relativistic quantum mechanics (1927-) Lecture 4. Particle properties of waves Classical physics Quantum mechanics (1920’s-) h/s S – the action=momentumdistance, units kgm2/s • Outline: • Light: waves vs. particles • Photons • Photoelectric effect

2. Historical Development Newton (Opticks, 1704): light as a stream of (classical) particles (corpuscles). Descartes (1637), Huygens, Young, Fresnel (1821), Maxwell: by mid-19th century, the wave nature of light was firmly established (interference and diffraction, transverse nature of e.-m. waves). Physics of the 19th century - mostly investigation of light waves; physics of the 20th century – interaction of light with matter. One of the challenges – understanding the “blackbody spectrum” of thermal radiation (to be considered later in the course). Planck (1900) suggested a solution based a revolutionary new idea: emission and absorption of electromagnetic radiation by matter has quantum nature. The energy of a quantum of e.-m. radiation emitted or absorbed by a harmonic oscillator with the frequency fis given by the famous Planck’s formula: h is the Planck’s constant - at odds with the “classical” tradition, where energy was always associated with amplitude, not frequency In terms of the angular frequency: where

3. Historical Development (cont’d) This progress leads to the concept of photons as quanta of the electromagnetic field. However, Planck thought that the quantum nature of light reveals itself only in the processes of interaction with matter. Otherwise, he thought, “classical” Maxwell’s equations adequately describe all e.-m. processes. Einstein (1905) put forward even more radical idea: light itself consists of a collection of particles with wave-like properties. He is purported to have said, “I know that beer comes in pint bottles” (in referring to the quantum features of blackbody radiation). “What I want to know is whether all beer comes in pint bottles” (that is, whether the “quantum character” of the electromagnetic field has its origins in the atoms or as an intrinsic property of the field). H.J. Kimble, Strong Interactions of Single Atoms and Photons in Cavity QED, Physica Scripta T76, 127 (1998) (posted on our Webpage). Even when light propagates in free space, it can be thought of as beam of quantum particles (not “old” classical particles, as Newton thought). “Light” – a shorthand notation for any e.-m. radiation ( from 0 to ).

4. t = 0 t = t0  A0 x vt0 x -A0  Waves Wave equation in one dimension for any quantity : Solution: a plane wave traveling in the negative (positive) direction x with velocity v: v – the phase velocity Harmonic plane wave traveling in the positive direction x:  angular frequency wave number A0 Electromagnetic waves: (transverse in free space) t -A0 T

5. Photons According to the quantum theory of radiation, photons are massless particles of spin 1 (in units ħ). They move with the speed of light : Quantum character of this equation is illustrated by the fact that the energy is associated with the frequency of oscillations rather than their amplitude. The phase is a Lorentz-invariant quantity, the (scalar) product of two 4-vectors: Particle properties of light Wave properties of light - both the time-like and space-like components of these 4-vectors should transform under L.Tr. in a similar way Thus, if Planck’s idea E=ħ is correct, than we must conclude that

6. Some numbers Visible light:  = 0.4 ↔ 0.8 micrometers violet red for comparison, the momentum of a (non-relativistic) electron with K =3.1eV: more than two orders of magnitude greater than for a photon with this energy! (this difference, however, is diminished when one considers an ultra-relativistic electron and equally energetic photon).

7. Photoelectric Effect Historical Note: The photoelectric effect was accidentally discovered by Heinrich Hertz in 1887 during the course of the experiment that discovered radio waves. Hertz died (at age 36) before the first Nobel Prize was awarded. Observation: when a negatively charged body was illuminated with UV light, its charge was diminished. J.J. Thomson (Nobel 1906) and P. Lenard (Nobel 1905) determined the ration e/m for the particles emitted by the body under illumination – the same as for electrons. The effect remained unexplained until 1905 when Albert Einstein postulated the existence of quanta of light -- photons -- which, when absorbed by an electron near the surface of a material, could give the electron enough energy to escape from the material. Robert Milliken carried out a careful set of experiments, extending over ten years, that verified the predictions of Einstein’s photon theory of light. Einstein was awarded the 1921 Nobel Prize in physics: "For his services to Theoretical Physics, and especially for his discovery of the law of the photoelectric effect." Milliken received the Prize in 1923 for his work on the elementary charge of electricity (the oil drop experiment) and on the photoelectric effect.

8. light e- I _ + V Photoelectric Effect (cont’d) Parameters: intensity (S) and frequency (f) of light, applied voltage (V), measured photocurrent (I) • Observations: • For a given material of the cathode, the “stopping” voltage does not depend on the light intensity. • The saturation current is proportional to the intensity of light at f =const. • Material-specific “red boundary”f0 exists: no photocurrent at f < f0. • Practically instantaneous response – no delay between the light pulse and the photocurrent pulse (many applications are based on this property). I stopping voltage V0 V intensity of light increases, f =const retarding I cesium calcium stopping voltage V V0(f2) V0(f1) f intensity = const, f increases “red boundary” f2 >f1 f0

9. Attempts to explain Ph.E. using the wave approach(and the classical model of matter) fail ! 1. Classical equation of motion of an electron in the light wave Observation: Kmax (and, thus, the stopping potential) does not depend on the intensity S, and is proportional to . The photocurrent for sodium can be observed at S as low as 10-6 W/m2. Number of electrons in the volume 1m1m  1 monolayer ~ 1019. Thus, for a single electron, 2. Observation: almost instantaneous response

10. Photon - based model of Ph. E. Absorption of a photon by an electronin metal (inelastic collision between these particles) after before However, we’ve concluded that a free electron cannot absorb a photon! the rest RF of an electron after the collision after before What’s wrong? The electron is not “free”, it is embedded in metal, and the chunk of metal is the third “body” that participates in the collision: energy conservation momentum conservation (see Slide 6) energy conservation Thus, while the electron is still inside metal momentum conservation The photon energy is absorbed by an electron (the energy absorbed by metal is negligibly small), but the momentum exchange between electron and metal is crucial for momentum conservation.

11. Photon - based model of Ph. E. (cont’d) In the experiment, the electron is observed outside the metal. It takes some energy to escape: (consider an attraction between an electron and the positive “image” charge induced on the metal surface) metal q+ q- The “escape” energy: the work functionW (material-specific) Thus, for the electron outside metal “red” boundary of Ph. E. Planck’s constant measurements:

12. Millikan with Michelson, Kinsley, and Gale,circa 1910

13. Photon - based model of Ph. E. (cont’d) • Observations: • For a given material of the cathode, the “stopping” voltage does not depend on the light intensity • – the energy of photons is determined by the light frequency, not intensity • The saturation current is proportional to the intensity of light at f =const • – the saturation current is proportional to the number of photons, thus to the light intensity • Material-specific “red boundary”f0 exists: no photocurrent at f < f0 • – atf < f0(hf < W) the photon energy is insufficient to extract an electron from metal • Practically instantaneous response – no delay between the light pulse and the photocurrent pulse • – single act of e-ph collision metal vacuum  energy of a free electron in vacuum with Ke =0   

14. Well, not so simple... 1921: Einstein received the Nobel prize for his interpretation of Ph.E. However, even much later in his life, he considered the concept of the quanta as heuristic at best: incomplete and not fully compatible with his own physical picture of the world. The title of his paper: “On a heuristic point of view concerning the production and transformation of light”. 1969: Lamb and Scully in “The photoelectric effect without photons” (in Polarisation, Matiere et Rayonnment, Presses University de France, 1969) showed that one can account for the Ph.E without the concept of photons, and, thus, the Ph.E. does not provide proof of the existence of photons. The theory of Lamb and Scully treated atoms quantum-mechanically, but regarded light as being a purely classical electromagnetic wave (see the link on our Webpage). So what would give us a proof? The study of statistical properties of photons. “Although surely the correct description of the electromagnetic field is a quantum one, just as surely the vast majority of optical phenomena are equally well described by a semiclassical theory, with atoms quantized but with a classical field. ... The first experimental example of a manifestly quantum or nonclassical field was provided in 1977 with observation of photon anti-bunching for the fluorescent light from a single atom (PRL 39, 691 (1977))”. H.J. Kimble, Physica Scripta T76, 127 (1998). Excellent reading on this subject: G. Greenstein and A.G. Zajonc, “The Quantum Challenge” (Jones and Bartlett Publishers, 2005).

15. In hindsight ... We should not forget, however, that when Einstein suggested his model of photoelectric effect, quantum mechanics had not been invented yet. Even a solid foundation of the atomistic theory was brand-new at that time. Einstein (1900), after reading the work of Ludwig Boltzmann on the molecular theory of gases: “The Boltzmann is absolutely magnificent. I am firmly convinced of the correctness of the principles of his theory, i.e., I am convinced that in the case of gases we are really dealing with discrete particles of definite finite size which move accordingly to certain conditions”. Planck (1882): “Despite the great success that the atomic theory has so far enjoyed, ultimately it will have to be abandoned in favor of the assumption of continuous matter””. After Einstein published his theoretical explanation of Brownian motion in terms of atoms (1905), Jean Perrin did the experimental work to test and verify Einstein's predictions, thereby settling the century-long dispute about the physical reality of molecules. Perrin received the Nobel Prize in Physics in 1926 for his work on “the discontinuous structure of matter” (including the experimental finding of the Avogadro’s number). http://nobelprize.org/nobel_prizes/physics/laureates/1926/perrin-lecture.html

16. metal vacuum Problems 1. The work function of tungsten surface is 5.4eV. When the surface is illuminated by light of wavelength 175nm, the maximum photoelectron energy is 1.7eV. Find Planck’s constant from these data. • 2. The threshold wavelength for emission of electrons from a given metal surface is 380nm. • what will be the max kinetic energy of ejected electrons when  is changed to 240nm? • what is the maximum electron speed? • the loss of electrons due to the photoelectric effect will cause an isolated sphere of this metal to acquire a positive charge. Find the largest electric potential (in Volts) that could be achieved by this sphere for  = 240nm. (a) (c) (b) electron energy in the (repulsive) electric field

17. o Inverse Photoelectric Effect (production of X-rays) X-rays: -rays X-rays UV Production: The upper “cut-off” of the spectrum corresponds to the full conversion of the electron kinetic energy into the photon energy.

18. Problem (Midterm 1, 2008) (5) Light of wavelength =410-7 m is incident perpendicularly on a flat metal surface at the rate of 10-6 W/m2. How many photons are incident per square meter per second? (10) A metal surface illuminated by light of wavelength 3.510-7m emits electrons whose maximum kinetic energy is 0.60 eV. The same surface illuminated by light of wavelength 2.510-7 m emits electrons whose maximum kinetic energy is 2.02eV. From these data find Planck’s constant and the work function of the metal in eV (a) where Nph is the number of photons per square meter per second (b)