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- 1. THE INTERACTION OF RADIATION AND MATTER: SEMICLASSICAL THEORY PAGE 38IV. REVIEW OF B ASIC QUANTUM MECHANICS : T HE T HERMAL RADIATION F IELD P LANCK S RADIATION LAW FOR THERMAL S OURCES :16 To set the stage for subsequent discussions of laser physics and quantization of the electromagnetic field we briefly explore the earliest, seminal notions in the quantum theory of light. As we all remember, in 1900 Planck found that he could account for the measured spectral distribution of radiation from a thermal source by postulating that the energies of a c e r t a i n s e t of harmonic oscillators are quantized! Drawing on Plancks success, Einstein in 1905 was able to show that the extraordinary features of photoelectric effect could be explained by hypothesizing the corpuscularity of the electromagnetic field. The crowning triumph of early quantum optics is Einsteins amazingly simple, phenomenological theory of 1917 which provided a quantitative basis for analyzing the absorption and emission of light by atoms. In treating thermal sources the basic assumption is that emitted radiation is a sample of the total electromagnetic field -- viz. all of the modes of a resonator -- in thermal equilibrium with its material environment -- viz. the walls of the resonator. In the traditional treatment of the theory of black-body radiation, a particular thermodynamic system is assumed - - viz., a hollow resonator which is a cube of length L with has perfectly conducting walls. 17 Thus, to satisfy boundary conditions at the walls, the electric field associated with a particular mode of the cavity is given by16 This section and some parts of the following section draw heavily upon discussions in Rodney Loudons The Quantum Theory of Light (2nd edition), Oxford (1983).17 The detail results of black-body radiation theory are indeed sensitive to the assumed character of the thermodynamic systemas will be demonstration presently. R. Victor Jones, March 9, 2000
- 2. THE INTERACTION OF RADIATION AND MATTER: SEMICLASSICAL THEORY PAGE 39 ν , νy , νz Ex x r ( r , t) = E 0 x (k νx , νy , νz ) cos ( π ν x ( x L) sin ( π ν y y L) sin ( π ν z z L) exp i ω νx νy νz t ) ν , νy , νz Ey x r ( r , t) = E 0 y (k νx , νy , νz ) sin ( π ν x x L) cos ( π ν y y L) sin ( π ν z z L) exp ( i ω νx νy νz t) [ IV-1 ] ν , νy , νz Ez x r ( r , t) = E 0 z (k νx , νy , νz ) sin ( π ν x x L) sin ( π ν y y L) cos ( π ν z z L) exp ( i ω νx νy νz t) where {ν x , ν y ,ν z} is a set of positive integers. To satisfy the homogeneous Helmhotz equation we must have kνx νy νz = (π ν x L) + ( π ν y L) + ( π ν z L) = ω νx νy νz c ( ) 2 2 2 2 2 [ IV-2 ] The cycle-averaged value of the stored energy density associated with the particular mode is given by ( W kνx νy νz = ) 1 4V ∫ cavity 0 r r 0 r ε E νx , νy , νz ( r ,t) 2 +µ H νx , νy, νz ( r , t) 2 dV r [ IV-3 ] ∫ r r ε E νx , νy, νz ( r , t) 2 dV = 1 ε E 0 k ( ) 2 1 = r 2V 0 16 0 νx νy νz cavity For a thermal source, the most significant experimentally measurable object is the noise spectrum -- i.e., the frequency distribution of the stored energy density. To obtain this distribution, we take the energy density in the frequency range between ω and ω+ dω -- viz. W (ω) dω = ∑ { ν x , ν y , ν z} ( W kνx νy νz ) ⇐ {with k νx νy νz between ω c and (ω+ dω) c } [ IV-4 ] = W kνx νy νz( ) Number of modes with frequencies × between ω and ω+ dω R. Victor Jones, March 9, 2000
- 3. THE INTERACTION OF RADIATION AND MATTER: SEMICLASSICAL THEORY PAGE 40 Thus the d e n s i t y o f s t a t e s is defined as Number of modes with frequencies [ IV-5a ] ρ k dk ≡ between ω and ω+ dω In the frequency range where λ << 2L , the following k-space argument holds: Two polarization Fraction of shell Volume in shell r ρ k dk ≈ r = Number of k states within shell states per k state with vaid states Volume of state 1 4π k 2 dk ω 2 dω [ IV-5b ] ≈ 2× =V ≡ ρ ωdω 8 ( π L) 3 c3 π 2 This extremely important density-of-states construction may be visualized most elegantly in two dimension.1818 For use in later discussions, we note here that by this same argument the 2D density-of-states is found to be ρ ω(2D ) = Aω π c 2 and the 1D value ρ ω(1D) = L π c . R. Victor Jones, March 9, 2000
- 4. THE INTERACTION OF RADIATION AND MATTER: SEMICLASSICAL THEORY PAGE 41 Following the traditional (Rayleigh-Jeans) argument, we identify a resonator mode as a harmonic oscillator19 and take its average energy to be the classical thermodynamic value for a system with two degrees of freedom -- i.e. W kνx νx νx ( ) = kB T V Thus, we thermal obtain the famous Rayleigh-Jeans radiation law (which impels us to worry about an ultra-violet catastrophe.) ω 2 dω WT (ω) dω = kT 3 2 [ IV-6 ] c π Plancks quantization hypothesis: Following Planck we set ( ) ( ) [ = E n V = hω ( n + 1 ) V ] 2 1 W kνx νy νz = ε 0 E 0 kνx νy νz 2 [ IV-7 ] 16 where n = 0, 1, 2,…… Assuming that the resonator mode is in thermal equilibrium with its environment, we may use the Boltzmann probability factor exp (−β En ) = exp( −E n k B T ) to find the probability Pn that the mode (read oscillator) is thermally excited to an energy E n exp(−β E n ) exp (−β nh ω ) Pn = = = exp ( −β n h ω ){1 − exp( −β h ω )} [ IV-8 ] ∑ exp(−β En ) ∑ exp(−β nh ω ) n n The thermal mean value of n is obtained in a similar manner 20 -- viz.19 We justify this assertion in detail later: it suffices to note here, in anticipation of that later discussion, that a harmonic oscillator and an electromagnetic mode have analogous Hamiltonians. ∆n = + nT. 220 It is also easy to show that thermal mean-square fluctuation in n is given by n T T R. Victor Jones, March 9, 2000
- 5. THE INTERACTION OF RADIATION AND MATTER: SEMICLASSICAL THEORY PAGE 42 ∑ n exp(−β n h ω) n = ∑n P = n T n n ∑ exp(−β n hω) n = −{∂ ∂(β h ω)} ln ∑ exp(−β n hω) n [ IV-9 ] = −{∂ ∂(β h ω)} ln {1− exp(−β hω)} −1 1 = exp(β hω) − 1 which may be plotted R. Victor Jones, March 9, 2000
- 6. THE INTERACTION OF RADIATION AND MATTER: SEMICLASSICAL THEORY PAGE 43 Equation [ IV-9 ] taken in conjunction with Equations [ IV-5b ] and [ IV-7 ] leads then directly to the Planck radiation law h ω3 dω WT( ω ) dω = n + 1 c3 π 2 2 thermal [ IV-10 ] 1 (β h ω ) 3 1 = + 2 2 3 3 dω exp(β h ω ) −1 2 h π c β From this expression we see that the Rayleigh-Jeans radiation law is correct in the limit k B T >> h ω , but, to our great relief, Planck has staved off the ultraviolet catastrophe. We may then plot this famous and an exceedingly important result. (From the particular point view of laser physics, the Planck radiation law gives a measure of the background radiative thermal noise spectrum.) R. Victor Jones, March 9, 2000
- 7. THE INTERACTION OF RADIATION AND MATTER: SEMICLASSICAL THEORY PAGE 44 EINSTEIN S P HENOMENOLOGICAL THEORY OF RADIATIVE P ROCESSES -- Einsteins "A" and "B" coefficients: We return to the two-level system and, with Einstein, make some physically reasonable (early quantum mechanical) postulates concern the absorption and emission of light. Consider the two-level system shown below. Einsteins postulates21 are embodied in a set of rate equations for the level populations- viz. π ℘ 221 Using our semiclassical theory of matter-radiation interaction, we can easily show that Bba = , 3 ε 0 h2 but, of course, Einstein did not have that well-developed theory at hand. While the stimulated emission and absorption processes are reasonably intuitive, it is the introduction of the spontaneous emission process which flows from a profound understanding of the processes by which thermal equilibrium is achieved. R. Victor Jones, March 9, 2000
- 8. THE INTERACTION OF RADIATION AND MATTER: SEMICLASSICAL THEORY PAGE 45 dN b d t = N a Aab − N b Bba W (ω r ) + N a Bab W (ω r ) [ IV-11 ] dN a d t = − N a Aab + N b Bba W (ω r ) − N a Bab W (ω r ) It particularly useful to examine the results of these equations under conditions -- viz. General equilibrium or steady-state condition: N a Aab − N b Bba W (ω r ) + N a Bab W (ω r ) = 0 [ IV-12a ] Thermal equilibrium condition: WT (ω) = Aab [( N b N a ) Bba − Bab ] −1 [ IV-12b ] Of course, in the latter case we use fact that the level populations Na and Nb at thermal equilibrium are related by Boltzmanns law -- i.e. N b N a = [ gb exp(−β Eb )] [ ga exp(−β Ea )] = ( gb ga ) exp(β hω ab ) [ IV-13 ] so that WT (ω) = Aab [( gb ga ) exp(β hω) Bba − Bab ] −1 [ IV-14 ] R. Victor Jones, March 9, 2000
- 9. THE INTERACTION OF RADIATION AND MATTER: SEMICLASSICAL THEORY PAGE 46 Thus, by comparison with Equation [ IV-10 ], we see that (gb ga ) Bba = Bab [ IV-15a ] hω 3 and π 2 c 3 Bab = Aab [ IV-15b ] and, by comparison with Equation [ IV-9 ], we see that Bab W T(ω) = Aab n T [ IV-16a ] or The rate of spontaneous emission Aab 1 = = = exp(β h ω) − 1 [ IV-16b ] The rate of stimulated emission BabWT (ω) n T R. Victor Jones, March 9, 2000

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