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A. La Rosa Lecture Notes APPLIED OPTICS Lecture-3 QUANTUM THEORY of LIGHT ________________________________________________________________ 3.1 Planck’s Hypothesis to calculate the atom’s average energy W To bring the theoretical prediction closer to the experimental results Planck considered the possibility of a violation of the law of equipartition of energy described above. The starting expression would be expression (39) I (ω) 1 ω 2 W , with the average 2 2 3 c energy of the oscillator W no to have a constant value (as the equipartition theorem predicted) but rather being a function of the frequency, W (ω) , with the following requirements, 2 W () 0 ω0 and (51) 2 W () 0 ω For the statistical calculation of W , Planck did not question the classical Boltzmann’s Statistics described in the section 2.1.E above; That procedure would still be considered valid. (52) Planck realized that he could obtain the desired behavior expressed in (51) if, rather than treating the energy of the oscillator as a continuous variable, that is the energy states of the oscillator should take only discrete values: 0 , , 2, 3, … (53) the energy steps would be different for each frequency = () (54) 1 where the specific dependence of in terms of to be determined Incident radiation q Planck postulated that the energy of the oscillator is quantized Fig. 11 An atom receiving radiation of frequency , can be excited only by discrete values of energy 0 , , 2, 3, … According to Planck, the classical integral expression to calculate the E / kBT average energy Wclassical E g ( E )dE Ee E '/ k T g ( E ' )dE ' e 0 would be B 0 replaced by a discrete summation, E e n WPlanckl (ω) E En / kBT n0 E / k T e n B Planck (55) n0 where En = n() ; n= 1 2, 3, … A graphic illustration can help understand why this hypothesis could indeed work: First we show how classical physics evaluate the average energy. E classical E [ P( E )] dE = area under the curve EP(E ) 0 2 E P(E) P(E) Classic calculation: <E>= continuum addition =Area (integral) Boltzmann distribution Energy E Energy E Fig. 12 Schematic representations to calculate the average energy of the oscillator under the classical approaches Using Plank’s hypothesis E Planck E n P( E n ) n 0 Case: Low frequency values of ≈ For this case, Planck assumed P(E) small value (56) E P(E) <E>= Quantum calculation: =Area Boltzmann distribution discrete addition Energy E ~kB T Energy E Notice, the value of E P( E ) n will be very close to the n n 0 classical value. It is indeed desirable that Planck’s results agree with the classic results at low frequencies, since the classical predictions and the experimental results agree well at low frequencies (see Fig. 12.) Case: High frequency values of For this case, Planck assumed 3 ≈ big value (57) P(E) E P(E) <E>= Quantum calculation: =Area Boltzmann distribution discrete addition Energy E ~kB T Energy E Notice, the value of E En P( En ) will be very different that Planck n 0 the classical result (area under the curve.) In fact, E P(E ) tends n n n 0 to zero as the step is chosen larger. This result is desirable, since the experimental result indicate that E tends to zero at high frequencies. It appears then that the Planck’s model (55) could work. Now, which function () could be chosen such that: it is small at low , and large at large ? An obvious simple choice is to assume a linear relationship () = (58) where is a constant of proportionality to be determined (when fitting the model prediction to the experimental results.). Replacing (58) in (55) one obtains, WPlanckl (ω) E En e n0 e En / kBT n0 Let En / kBT ne n / k T B n0 e n / k T B n0 kT 4 kT WPlanckl (ω) ne n n0 kT n0 e n e n e n n0 n0 Since d [ ln(u) ] /dx = (1/u)du/dx WPlanckl (ω) k T ln [ If we define en ] n0 x e , then en 1 x x 2 x3... n0 This series is equal to 1/(1-x) = 1 1 = 1 x 1 e 1 WPlanckl (ω) k T ln[ e n ] k T ln[ ] 1 e n 0 ln[1 e ] e 1 kT k T 1 e e 1 kT WPlanckl (ω) 1 (59) e kT 1 kT Notice: WPlanckl (ω) 0 WPlanckl (ω) 0 With this result, expression (39) becomes I (ω) 1 ω 2 WPlanck 2 2 3 c (60) 5 I() Experimental results Particle’s and wave’s energy quantization Historically. Planck initially (1900) postulated only that the energy of the oscillating particle (electrons in the walls of the blackbody) is quantized. The electromagnetic energy, once radiated, would spread as a continuous. It was not until later that Plank accepted that the oscillating electromagnetic waves were themselves quantized. The latter hypothesis was introduced by Einstein (1905) in the context of explaining the photoelectric effect, which was corroborated later by Millikan (1914). 6