Chapter XX Quantum theory of light

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At the end of the 19-th century, physics was at its most confidence situation. Classical phyics, as formulated in Newton’s law of mechanics and Maxwell’s theory of electromagnetism, have proved very successful in solving every problem. →At that time there seemed to be no question for which physics could not provide an answer !!! But then it came as a great shock when some simple phenomena were observed which could not be explained by classical physics →a new theory, quantum theory, was developed at the beginning of the 20-th century...

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Nội dung Text: Chapter XX Quantum theory of light

GENERAL PHYSICS III Optics & Quantum Physics
Chapter XX Quantum theory of light §1. Blackbody radiation. Planck’s theory of radiation §2. Photoelectric effect. Einstein’s theory of light §3. Compton scattering
 At the end of the 19-th century, physics was at its most confidence situation. Classical phyics, as formulated in Newton’s law of mechanics and Maxwell’s theory of electromagnetism, have proved very successful in solving every problem. → At that time there seemed to be no question for which physics could not provide an answer !!!  But then it came as a great shock when some simple phenomena were observed which could not be explained by classical physics → a new theory, quantum theory, was developed at the beginning of the 20-th century  We begin our study of quantum physics by two following phenomena: • Blackbody radiation • Photoelectric effect We will see what were the failures of classical physics and how a new theory had been developed.
§1. Blackbody radiation. Planck’s theory of radiation: • Heat bodies emit electromagnetic radiation in the infra-red region of the spectrum (see the next slide). In this region the radiation is not visible. • As the temperature of a body is increased to any value, the body begins to glow red and then white, emitting visible electromagnetic radiation. (an example is the variation of the radiation of a filament of electric lamp when the electric current varies). • Observation of the spectrum emitted by a solid shows that the radiation extends over a continous range of frequency. Such a spectrum is called a continuum. 1.1 Experimental laws of blackbody radiation: 1.1.1 Stephan-Boltzmann law: • It was observed that the intensity of the radiation emitted from a body increases rapidly with increasing temperature of the body.
• The relation between intensity and temperature is given by the following formula: I =  T4 ,  where I → the average power of radiation per unit surface area, → a fundamental physics constant called the Stephan-Boltzmann constant = 5,67 x 10-8 W m-2 K-4  a dimensionless number (0 <  1) called the emissivity, which → < depends on the nature of the radiating surface. It is found that for any surface the absorption is the exact reverse of the emission process. It means that the ability of emission is proportional to that of absorption:   0, the surface has low emissivity and low ability of absorption →   → 1, the surface has high emissivity and high ability of absorption The idealized case, when  1, such a body is called absolute blackbody. = A blackbody absorbs completely all the incident radiation and gives no reflecting radiation (that is why the body is black !).
• Therefore we have for a blackbody: I = T4 This is the Stephan-Boltzmann law for a blackbody • Remark: For a blackbody the total intensity depends only on absolute temperature • A cavity with a small aperture is an example of a blackbody. Electromagnetic radiation entering the cavity is eventually absorbed after successive reflections → the cavity is a perfect absorber of e-m radiation. 1.1.2 Wien displacement law: Note that the intensity I in the Stephan-Boltzmann is the total intensity, that is, the radiation intensity for all wavelengths. Denote by I( dthe intensity corresponding to wavelengths in the ) interval  and  d  can write + we  I d I( ) 0
I( is the distribution function ) over all wavelengths. It is called the spectral emittance. • The form of I( depends on ) temperature. By experimental measures one has IT ( shown ) in the picture. • Each curve has a peak at  = m It is observed that as the temperature T increases, the peak grown larger, and shifts to shorter wavelengths. • By experiment Wien showed that  is inversely proportional m to T, and  T = 2.90 x 10-3 m.K m This is Wien displacement law
1.2 Rayleigh–Jeans formula . Ultraviolet catastrophe: • Rayleigh and Jeans attemped to explain the observed blackbody spectrum on the base of the concept that radiation is a e-m wave. They derived the following formula for the intensity distribution: 2ckT I (  )  4 • Comparing the Rayleigh-Jeans spectrum with experiment one can see that: The R-J formula agrees well with experiment at large  IT() But there is a serious disagreement at small  The unrealistic behavior of the Rayleigh-Jeans distribution at short wavelengths is known in physics as “ultraviolet catastrophe”.
A more impressive indication of the complete failure of the Rayleigh- Jeans spectrum is the result on the total intensity:   d  2ckT 1  I   d 2  4  I( ) ckT lim  3  0 0   3  0   It means that I →  , an unceptable result !!! The blackbody radiation spectrum could not be explained by classical physics 1.2 Planck’s theory of radiation: The discrepancy between experiment and theory was resolved in 1900 by Planck, by introducing a postulate which was revolutionary with respect to certain concepts of classical physics.
1.2.1 Planck’s postulate and radiation law: Planck’s postulate: “Electromagnetic radiation consists of simple harmonic oscillations which can possess only energies  nh (n = 0, 1, 2, 3, ….) = where n is the frequancy of the oscillation, and h is an universal constant” Energy level diagrams for classical sipmple harmonic oscillations and for that obeying Planck’s postulates. The constant h is called the Planck constant. It’s value was determined by fitting theoretical consequences with experiment data (see below).
With this postulate Planck derived the following spectrum distribution function 2 2hc I (   5 hc / kT Planck radiation law ) e  1  1.2.2 The agreement of the Planck law with experiment: The Wien displacement law could be derived from the Planck law: The transcendent equation dI (  0 is solved numerically, ) d hc 1   m . This coincides with the Wien displacement law, 4 . 965 k T if the value of h satisties the equation hc 2 . 9  3 m . K (*) 10 4 . 965 k The Stephan-Boltzmann law is also a consequence of Planck law:  25 k 4 4 → h must 2k 4 5 I   d  I( ) T  5   .67  W / m2K 4 (**) 10 8 0 2 3 15 c h satisfy 15c2h3
Equations (*), (**) and fits the Planck law curve with experiment leads to the same result for the value of the Planck’s constant: h = 6.626x10-34 J.s  Therefore, the experimental laws for blackbody radiation IT() could have a satisfactory explanation with the Planck’s theory. The root idea is that electromagnetic energy emitted by bodies can have only discrete values n.h The comparison of Planck’s spectrum → one say that energy is with experiment at T = 1646 0K quantized. The discrete nature of energy is the foundation of quantum theory. One more remark: The large wavelength limit of the Planck’s spectrum formula is the Rayleigh’s formula (recall e x ≈1 + x with small x). This means that at very long wavelengths (very small quanta energies), quantum effects become unimportant.
§2.Photoelectric effect. Einstein’s theory of light: 2.1 Photoelectric effect:  Consider a metal surface. Electrons in a metal are “bound” by the binding Binding potential energy. Introduce the potential quantity called “work function”. ,  is the minimum amount of energy an  induvidual electron has to gain to escape: • E <  the electron is confined inside metal → (E: the energy • E >  the electron can escape from metal. → of the electron ).  If you shine light on the metal surface, electrons absorb energy from the incident light, and have enough energy to escape. Eshtablish on the metal surface a electric field → the escaped electrons create an current that is called photoelectric current How will the photoelectric current depend on the intensity I and the frequency  the incident light ?  of
Incident Light  Experiment 1: Measure the maximum energy of ejected electrons (variable frequency  )  The electric field between the “collector” and the metal will Collector repel ejected electrons A  Increase negative voltage until flow of ejected electrons decreases to zero. electrons (Current = 0 at V = Vstop) + Metal Surface V  Measurement of Vstop tells the max vacuum kinetic energy of electrons Kmax = eVstop. The result: The “stopping voltage” is independent of light intensity I ! It means that increasing the intensity I does not increase kinetic energy of electrons !
 Experiment 2: Measure the maximum energy vs.  Incident Light (variable frequency  ) 3 Collector Vstop () 2 A 1 f0 0 0 0 5 10 15 electrons +  (x10 14 Hz) Metal Surface V vacuum The results:  Stopping voltage Vstop (and the maximum kinetic energy of electrons) decreases with decreasing (linear dependence).  Below a certain frequency  no electrons are emitted, even for intense o, light!
3 Vstop () h/e slope Collector 2 A 1  f0 0 1 0 0 5 10 15 electrons + (x1014 Hz) Metal Surface V vacuum Summary of results:  Energy of electrons emitted depends on frequency, not intensity  Electrons are not ejected for frequencies below 0  Electrons are emitted as soon as any light with     This results are hard to understand on the basis of classical physics ! (According to classical physics: if the light intensity increases, electrons gain more energy → have more chance to escape. Don’t understand why is there a limit frequency  ). 
2.2 Einstein’s quantum theory of light: To overcome the difficulty of classical physics, Albert Einstein introduced the quantum theory of light, and developed the correct analysis of the photoelectric effect in 1905. 2.2.1 Einstein’s postulates: By analogy to the earlier Planck’s theory of radiation, the postulates of Einstein’s theory is as follows:  Light consists of small “packages” of energy called photons or light quanta.  The energy  a photon is  h  of =  where h is the Plank constant. In vacuum  h hc/ = = 2.2.2 Analysis of the photoeffect by the quantum theory: A photon arriving at the surface is absorbed by an electron (one by one). After that the electron gets all the photon’s energy (h)
For an electron we can write the following equation: mv2 h   max  2 The energy of the electron The energy part The remaining part, the kinetic after absoption of photon for escaping from energy of the motion outside metal the metal surface Kinetic energy of the electron outside metal is telled by the value Vstop in the experiment: mv 2 K max  max eV stop eVstop   h  2 By this analysis it is understandable that • Energy of electrons emitted depends on frequency, not intensity • The limit frequency  for the photoeffect is determined by the 0 equation h =  0   Under this light frequency the electron has not enough energy to escape.
§3. Compton scattering: Compton scattering (Compton effect) is a phenomenon that provides additional direct confimation of the quantum nature of light, and particularly, of X-rays. 3.1 Experimental results: • The wavelength of the incoming monochromatic X-rays:  • The wavelength of the scattering X-rays:  ’ Experimental observations discovered a shift in wavelength:  ≠ (A.H. Compton 1923). ’ The wavelength shift of scattered X-rays is called the Compton effect. • It is found that the wavelength difference   -  varies with   ’ the scattering angle according to the equation