Quanta of Light (4 lectures) Electromagnetic Waves Spectrum and generation Two slit intereference

1. Syllabus for 1AMQ, Atoms, Molecules and Quanta,Dr. P.H. Regan, 31BC04, x6783, p.regan@surrey.ac.uk Spring Semester 2001Books, Modern Physics, K. Krane, WileyQuantum Physics, Eisberg & Resnick, Wiley • Quanta of Light (4 lectures) • Electromagnetic Waves • Spectrum and generation • Two slit intereference • Single slit diffraction • Black Body Radiation • Energy Quanta and Planck’s Hypothesis • Photoelectric effect and Einstein’s Equation • Compton Effect See Krane, Chap. 3 & ERChaps. 1 and 2. 1AMQ P.H. Regan, Spring 2001

2. Introduction • `Classical Physics’ -> before ~ 1900 • Modern (quantum) Physics, after 1900 • New theories arose from the ability to do better measurements….ie. better technology • Allowed the exploration of 3 EXTREMES of nature, ie • very fast - special relativity replaces Newtonian mechanics • very small- Quantum mechanics replaces Newtonian mechanics • very large- General relativity replaces Newtonian gravitation. (note I. Newton, b. 25 Dec. 1642) 1AMQ P.H. Regan, Spring 2001

3. New experimentsNew theoriesNew concepts Relativity New concepts of time and space Measurements of the speed of light Quantum mechanics Spectrum of light, from (a) hot, glowing objects, (b) from electrical breakdown in gases (atoms) New ideas about determinism and measurement Modern physics theories are refinements of the old, classical ideas, but are CONCEPTUALLY RADICAL. Classical theories still work (as good approximations) at everyday speeds and sizes. The new ideas were discovered using advanced technology, therefore, become more important at extremes physical conditions. (key experiments were to do with light (very fast, c=3x108 ms-1 !) and atoms (very small, r~10-10m). 1AMQ P.H. Regan, Spring 2001

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5. Electromagnetic Waves Light often behaves like an electromagnetic wave, travelling with speed, c (in vacuum), predicted by Maxwell’s equations and exhibiting interference and diffraction effects. However, as we shall see, in some circumstances, the predictions of wave theory are wrong and it was the study of those cases which led to the development of the quantum theory. The Intereference Theory of Light was a success for wave theory. The two slit experiment of Thomas Young (1803) shows wave-like intereference for light. Condition for minima (destructive interference) is that: dsinq = l/2, 3l/2, 5l/2, 7l/2, nl/2 dsinq = path difference 1AMQ P.H. Regan, Spring 2001

6. From simple trigonometry, (a/2) sinq = (l/2), thus, asinq = l Further successes of wave theory, diffraction from a single slit………. If the size of the slit, a, is comparable with the wavelength of the light, l, then a diffraction pattern is observed, rather than a sharp image. There is a central maximum, the width of which is defined by the first minima on either side. ie. the wave-like nature of light was well established. 1AMQ P.H. Regan, Spring 2001

7. Quanta of Light Studying the speed of light led to the theory of special relativity. Studying interference, diffraction and refraction of light showed its wave-behaviour. These phenomena can not be understood by a particle or `corpuscular’ model of light. However….. At the atomic level, some phenomena can NOT be understood if light acts as a wave! …..but can be understood if we take light to be a stream of particles with mass mo=0 and speed, v=c (ie. b=1). Note, mo=0 if and only if v=c, since E=pc=mvc and E=mc2 Einstein (Nobel prize, 1921). 1AMQ P.H. Regan, Spring 2001

8. The study of `black body radiation’ gave the first clues to the breakdown of classical laws which led to quantum theory. • Thermal radiation: heated objects emit e-m radiation as they cool. • Hot coals glow red, very hot surfaces eg. Solar surface, incandescent filaments glow white. • The wavelengths (colour): • Depends on temperature, T. • As T increases, l decreases. red hot -> white hot -> blue hot, details don’t really depends on the actual material being heated. • Have a continous spectrum • Expts. give Wien’s Law (Nobel Prize 1911) • I • lmax (metres) = 2.9 x 10-3 / T(K) • where lmax is the peak wavelength and T is the absolute temperature of the surface. Black Body Radiation 1AMQ P.H. Regan, Spring 2001

9. Since c=nl (where n= frequency), nmax(Hz) = 1.03 x 1011 x T(K) Note that the power emitted (ie. energy radiated per unit time) increasesrapidly withT Power emitted per unit area is given by Stefan-Boltzman Law. S = s T 4Wm-2, with s =5.67 x 10-8Wm2K-4 S is the AREA under the spectral function, Sl . Note that the area under the curve a T 4. 1AMQ P.H. Regan, Spring 2001

10. The origin of these electromagnetic waves is the thermal motion (vibration) of the charged consituents of the atoms in the material. A Blackbody is an idealised perfect absorber and perfect emitterof thermal radiation. (The surface does not affect the radiation, and the spectrum of the radiation only depends on T). • Examples of Blackbodies ? • A lump of coal, which absorbs all incident light (ie. is apparently black in colour) • Tbe sun(see spectrum)….note blackbodies are not black in colour when they are hot! • Uniformly heated cavity with small exit/entrance hole. (see later) 1AMQ P.H. Regan, Spring 2001

11. The Ultra-Violet Catastrophe !!! aka `The Problem with Classical BB Theory Under the classical wave theory, if a cavity of dimension, a, is filled with e-m radiation, (note the hole rather than the cavity is the BB here), the number of standing waves of frequency, n, is given by (see ER, p11) N(n) dn = p . (2a/c)3 . n2 dn The average energy of each standing wave in the box is given by the classical equipartition law, (k = Boltzmann’s const.) eav = kT Result is that Sn is proportional to n2,ie, infinite at large n (small l corresponds to the UV regime). Conclusions wrong! UV Catastrophe 1AMQ P.H. Regan, Spring 2001

12. Note, in this fig. the y-axis is the TOTAL energy emitted per second, per unit volume, at a fixed frequency, n As the above figure (from ER p13) shows, although the classical theory works (approx) in the low frequency (long wavelength) region, it fails dramaticallyat higher frequencies. • Max Planck (Nobel Prize, 1918) showed that the `mistake’ was in the assumption that the average energy,eav was a constant. (Note, in this example, the frequency corresponds to that of the vibration of the atoms in the walls). • This was derived from the assumptions: • The energies followed a Boltzmann distribution and • The range of possible energies was continuous. 1AMQ P.H. Regan, Spring 2001

13. 2 new ideas were needed: • assume the energy allowed per (atom) oscillator is NOT CONTINUOUS (ie energy is to have DISCRETE values. • Assume that the gaps between allowed values are larger for higher frequency (atomic) oscillators. 1AMQ P.H. Regan, Spring 2001

14. The classical theory, known and Raleigh-Jeans prediction, clearly fails at long (uv) wavelengths. Planck guessed that the gaps between allowed values of oscillator frequency, increased with frequency, ie De=hn, where h=const and that each oscillator can only emit or absosb energy is discrete amounts given by, e=nhn, where n= integer. 1AMQ P.H. Regan, Spring 2001

15. Result is that eav(n) -> 0 as n ->  Planck’s suggestion was that the average energy per oscillator at a given temperature was a function of oscillator frequency such that, Since in the high frequency limit, In the long-wavelength (UV) limit, 1AMQ P.H. Regan, Spring 2001

16. The result is that the spectral function,which corresponds to the product of the average emitted energy at a given frequency times the number of oscillators at that frequency, is given by: This fits the data perfectly for a value of h=6.63x10-34 Js (Planck’s constant) Physical Picture of Planck’s Hypothesis The physical background behind Planck’s proposal was that the atomic oscillators behave like simple (quantum) harmonic oscillators, which have a potential energy given by x = displacement k = constant 1AMQ P.H. Regan, Spring 2001

17. The quantum energy hypothesis means that only certain amplitudes are allowed and there is a non-zero minimum. If the atom is not given enough energy in collisions with its neighbours, it will not oscillate at all. Higher frequencies need a greater amplitude to start vibrating. They carry more energy, but vibrate less often, with the result that as eav(n)->0 as n-> infinity. 1AMQ P.H. Regan, Spring 2001

18. The Photoelectric Effect This is a quantum effect involving light and electrons. Light shining on a (metal) surface can cause electrons to be emitted. Experiments should study the effect systematically and highlight the important features. A suitable apparatus is shown below. 1AMQ P.H. Regan, Spring 2001

19. There are 5 main features to explain, namely: • The number of electrons ejected per unit time is proportional to the intensity of light. • The electrons are emitted with velocities up to a maximum velocity (Vstop<0). • The maximum kinetic energy does NOT depend on intensity of light. (Vstop is the same regardless of intensity BUT increases linearly with frequency of light, n). • There is a threshold frequency, no such that there is no emission for n<no. (Note no depends on the metal). • There is no measurable time delay between the light striking the metal and the electron emission. • The simple Wave Theory of light, (energy transmitted per unit time is proportional to Eo2) has the following problems….. • Intensity dependence is predicted incorrectly • Threshold effect is NOT predicted • Time delay should easily be several seconds. • No electron KE dependence on light intensity. 1AMQ P.H. Regan, Spring 2001

20. where h = Planck’s constant and nis the classical, wave model light frequency. E=hn Einstein’s Photon Hypothesis Einstein proposed that light (em-radiation) consists of particle-like packets of energy, called photons. Each photon carries an energy, This extends Planck’s ideas regarding emission/absorption so that they also apply to radiation as it is transmitted. Quantum Theory of the Photoelectric Effect (Einstein, Nobel prize, 1921, theory 1905). The emission of electrons is caused by single photons which are completely absorbed by individual electrons. The electrons are initially energetically bound to the metal and need some minimum initial energy to overcome this binding. 1AMQ P.H. Regan, Spring 2001

21. The minimum energy required for the electrons to escape is called the work function, f. The maximum kinetic energy for electrons is then given by Kmax = hn - f (Note, K<Kmax for more tightly bound e-s or from e- collisions after emission.) Note that Kmax=Vstope where e is the electron charge. Therefore, plotting Vstop versus n has a slope of h/e. Thus h can be measured and compared with the value obtained from the Black Body spectrum. h = 6.626 x 10-34 Js = 4.136 x 10-15 eV Recalling that 1eV = 1.6x10-19J 1AMQ P.H. Regan, Spring 2001

22. The Einstein model accounts for all 5 • features of the photoelectric effect, ie. • The intensity dependence.Since the intensity is equal to the energy deposited per unit area per unit time, this means that the intensity is proportional to the number of incident photons. • Proportionality of Kmax to n, • T he non-zero maximum velocity. • The existence of n o. All arise directly fromEinstein’s equation. • No measurable time delay. Photons can be absorbed instantaneously. 1AMQ P.H. Regan, Spring 2001

23. The Compton Effect The Compton effect refers to collisions between photons and electrons. Arthur Compton’s experiments (performed 18 years after Einstein’s PE theory) showed a definite, particle-like (photon) behaviour for X-rays. 1AMQ P.H. Regan, Spring 2001

24. Compton scattering occurs in addition to the classical process of Thomson scattering (where l(q)=l(0o), ie absorption followed by re-radiation). The Compton shift, Dl , has a clear angular dependence, but does not depend on the material used for the scatterer. This suggests that the photons are colliding with something found in all materials, such as electrons…. Comparing the x-ray photon energies with e- energies from the pe effect. For an x-ray photon with l=0.07 nm (as used by Compton), from E=hc/l, E=17.7 keV, ie >> than e- bind. ene. Simplify situation by considering a collision between a photon and a free (unbound) electron, initially at rest For the incoming photon, the momentum is given by E=pc (mo=0 since v=c for photon). Cons. of linear momentum, mass energy and the energy-momentum relationship can then calculate the scattered energies for any incident photon energy and scattering angle . 1AMQ P.H. Regan, Spring 2001

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