Notes_ Quantum Theory of Light and Matter Waves

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  Hwa Chong Institution – H3 Physics 2009 kwh May 19, 2009 Page 1 of 34 Quantum Theory of Light and Matter Waves Learning Outcomes Quantum Theory of Light  Candidates should be able to:i) show an understanding of what is meant by ideal blackbody radiation.ii) apply Wien’s displacement law in related situations or to solve problems.iii) discuss qualitatively the failure of classical theory to explain blackbody radiation at highfrequencies.iv) describe qualitatively Planck’s hypothesis of blackbody radiation.v) derive, using relativistic mechanics, the equation p = h/λ for a photon and use theexpression in related situations or to solve show an understanding of the Compton shift effect and how it supports the concept of thephoton.vii) derive, using relativistic mechanics, the Compton shift equation and use the equation inrelated situations or to solve problems.viii) show an understanding of discrete electron energy levels in isolated atoms (Bohr theory)and use the terms absorption, spontaneous emission and stimulated emission to describephoton processes involving electron transitions between energy levels, including the typicallifetime of such transitions.ix) recall and use the equation hf = E1 – E2 to solve problems.x) apply the relation Nx = Noexp(–(Ex – Eo)/kT) to explain and solve problems on thepopulation distribution of atoms with energy Ex .xi) describe the process of population inversion and explain why this cannot be achieved with just two energy levels.xii) use population inversion and stimulated emission to explain the action of a laser, using theHe-Ne laser as a specific example. (Details on the structure and operation of the laser arenot required.) Matter Waves i) show a qualitative understanding of the concept of wave-particle duality and the principle of complementarity.ii) show an understanding that a particle can be described using a wave function Ψ and give asimple mathematical form of the free particle wave function.iii) discuss qualitatively the probabilistic interpretation of the wave function and state that thesquare of the wave function amplitude IΨ I 2 gives the probability density.iv) show an understanding of the normalisation of a wave function.v) show an understanding that the resolving power of optical instruments is determined by thewavelength of the radiation and use this to explain how electron microscopes can achievehigher resolution than normal optical describe the use of X-ray diffraction to probe crystal structures.vii) recall and apply Bragg’s equation nλ = 2d sinθ to solve problems.viii) extend the use of Bragg’s equation to electron diffraction in probing the surfaces of solids.(Only non-relativistic cases are considered.)  Hwa Chong Institution – H3 Physics 2009 kwh May 19, 2009 Page 2 of 34 A Brief History Throughout scientific history, light has been viewed alternately as either waves or particles .Dutch scientist Christiaan Huygens perceived light as waves propagating in a medium known asthe ether  . While it is able to explain the wave behavior of light (interference and diffraction), iteventually failed as ether is not detected. Then Isaac Newton came up with the corpusculartheory of light which imagined light as consisting of small particles, while it can explain reflectionof light, it failed to account for interference and diffraction. Furthermore, English scientistThomas Young conducted a series of experiments, including the famous Young's double-slitsexperiment that supported the wave nature of light. And when James Clark Maxwell formulatedthe electromagnetic theory, establishing that light is not a disturbance of ether, but ratherfluctuations in the electromagnetic field, the scientific community had more or less embracedthe wave theory of light. However, by then there are new observations which theelectromagnetic theory was unable to explain, two phenomena that have puzzled scientists atthe end of the 19 th century: the radiation spectrum of a   blackbody and the photoelectric   effect . Phillip Lenard observed that electrons emitted from a metal surface when light is incident on it,depended on the frequency of light, not the intensity of the light as predicted by classical physics.Indeed, no electrons were emitted from the surface if the frequency of the incident radiation wasless than some threshold value that depended on the metal.The shape of this blackbody spectrum had already been determined experimentally by the end of the nineteenth century. However, a satisfactory theory of blackbody radiation should  provide a precise mathematical expression for the blackbody spectrum. 1900 – the history of quantum mechanics false starts with Max Planck  From 1895 onwards Planck tried to find a way to derive the blackbody radiation law afterpartially successful attempts by Wien and Lord Kelvin.Planck’s model went through successive refinements in his attempt to obtain a perfect matchbetween theory and experiment. He eventually succeeded, but only at the cost of incorporating‘energy elements’ in his model. In this model, the total energy of all the oscillators in a blackbodyis divided into a finite (but very large) number of equal (but tiny) parts, determined by a constantof nature, which he labelled h . This became known as Planck’s constant. In a letter to R. W. Wood,Planck called his limited postulate “ an act of desperation .” 1905 – The photoelectric effect  Lenard’s experiment was explained by Einstein in 1905 when he modeled light as photons withenergy hf  E  = . It was Einstein who made the bold assumption that light energy could also bedelivered as ‘packets’ we now call ‘photons’ and gave physical meaning to Planck’s constant.  Hwa Chong Institution – H3 Physics 2009 kwh May 19, 2009 Page 3 of 34 1913 – Bohr explains why atoms don’t implode Ernest Rutherford had earlier proposed an atomic model in which he envisaged the atom ashaving a central positive nucleus surrounded by negative orbiting electrons. Unfortunately,Rutherford's model faced a very fundamental problem. Maxwell's electromagnetic theorypredicted that a charge undergoing acceleration will radiate EM waves, losing energy in theprocess. This means that the orbiting electrons, which undergo centripetal acceleration, will loseenergy through EM radiation and rapidly spiral into the nucleus.The Rutherford-Bohr model of the atom resolved the issue by making some new postulates.Bohr speculated that each allowed orbit only had ‘room’ for a certain number of electrons sothat electrons further out from the nucleus cannot jump inwards if the inner orbits are alreadyfull. The electrons closest to the nucleus were simply forbidden to jump right into the centre of the atom. Although it is not yet the full quantum model, Bohr’s model was credible because itpredicted the positions of spectral lines for hydrogen – confirmed by observation. 1923 – Compton, the light quantum has not just energy, but momentum as well  In 1923 Arthur Compton (1892-1962) was scattering x-rays off graphite. He found that some of the scattered radiation has smaller frequencies than the incident radiation which is dependenton the angle of scattering. He could only explain his observations if he treated x-ray photons asparticles obeying the conservation of momentum and energy in their collisions with stationaryelectrons. So his experiment and explanation confirmed that x-ray photons carry not only‘energy’ but also ‘momentum’ of  chf  . 1924 - Louis de Broglie, whose PhD earned him a Nobel prize de Broglie combined two equations for the photon ( hf  E  = from the photoelectric effect and  pc E  = from relativity) and expressed the result in terms of frequency h pc f  = . He couldsubstitute this into λ= f c , re-arranged in terms of   ph pchc f c ===λ .What de Broglie’s equation tells us is that everything has dual wave-particle character butbecause Planck’s constant is so small, the ‘waviness’ of an everyday object is so utterly tiny thatit can never be detected. Niels Bohr summarized the situation in his  principle of complementarity  , The wave and particle models are complementary; if a measurement proves the wave character of radiation or matter, then it is impossible to prove the particle character  in the samemeasurement   , and vice versa.    Hwa Chong Institution – H3 Physics 2009 kwh May 19, 2009 Page 4 of 34 Section One : Blackbody Radiation 1.1 Blackbody Radiation All bodies absorb and radiate thermal energy. Isolated atoms (gases) produce discrete spectra(emission or absorption) which arise from electronic transitions between discrete energy levels.Dense bodies such as liquids and solids radiate continuous spectra (of any form and shape), inwhich all frequencies are present, due to mutual interactions between the atoms in closeproximity with one another (Band Theory).Different bodies have different rates of emission and absorption. A perfect   blackbody can bevisualized as an ideal body that absorbs all electromagnetic (EM) radiation landing upon it,regardless of frequency. More importantly, a blackbody is also the best emitter of radiation, andthe rate of emission ( total energy per unit time per unit area) is a function of the absolutetemperature alone, as expressed in the Stefan-Boltzmann’s Law : 4  T  σ   I  =  where 4-28 KmW10675 - . σ  ×= is called the Stefan-Boltzmann constant.A perfect blackbody can be modeled by a hollow body with only a small hole that allows entryinto the cavity inside. All EM radiation that enters the hole will be trapped inside the cavity andwill be absorbed. At the same time, the blackbody will also emit EM radiation of all possiblefrequencies, which is characteristic of the radiating system only and not dependent upon thetype of radiation which is incident upon it. This is why blackbody radiation is also known as cavity radiation . This model is especially useful in the attempt to formulate a theory forblackbody radiation. The radiated energy can be considered to be produced by standing waves  or resonant modes of the cavity (see Fig. 1.2).The spectrums of the radiation emitted from a blackbody at a few different temperatures areshown in Fig. 1.1. Radiation of  all wavelengths is present in the spectrums, but certainwavelengths tend to dominate, denoted by the “peak” of the curve. The position of the “peak”depends solely on the absolute temperature T  of the body. The peak wavelength  λ max  can befound using the Wien’s Displacement Law : Km10x2.898  -3max = T  λ   As T  increases, the spectrum shifts towards shorter wavelengths.We can therefore deduce the temperature of a body hot enough to be luminous by observingthe dominant colour of the radiation. For the body in this case, at 5000 K, the peak happens tobe in the visible range, around the yellow-orange region. Our more yellowish sun has a (surface)temperature of about 6000 K, whereas the cool blue of the distant star Vega indicates a much
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