Consider the two level system shown below einsteins

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Consider the two-level system shown below. Einstein's postulates 21 are embodied in a set of rate equations for the level populations- viz. 21 Using our semiclassical theory of matter-radiation interaction, we can easily show that B ba = π ℘ 2 3 ε 0 h 2 , 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.
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T HE I NTERACTION OF R ADIATION AND M ATTER : S EMICLASSICAL T HEORY P AGE 45 R. Victor Jones, March 9, 2000 dN b dt = N a A ab - N b B ba W ϖ r ( 29 + N a B ab W ϖ r ( 29 dN a dt = - N a A ab + N b B ba W ϖ r ( 29 - N a B ab W ϖ r ( 29 [ IV-11 ] It particularly useful to examine the results of these equations under conditions -- viz. General equilibrium or steady-state condition: N a A ab - N b B ba W ϖ r ( 29 + N a B ab W ϖ r ( 29 = 0 [ IV-12a ] Thermal equilibrium condition: W T ϖ ( 29 = A ab N b N a ( 29 B ba - B ab [ ] - 1 [ IV-12b ] Of course, in the latter case we use fact that the level populations N a and N b at thermal equilibrium are related by Boltzmann's law -- i.e. N b N a = g b exp E b ( 29 [ ] g a exp E a ( 29 [ ] = g b g a ( 29 exp β h ϖ ab ( 29 [ IV-13 ] so that W T ϖ ( 29 = A ab g b g a ( 29 exp β h ϖ ( 29 B ba - B ab [ ] - 1 [ IV-14 ]
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T HE I NTERACTION OF R ADIATION AND M ATTER : S EMICLASSICAL T HEORY P AGE 46 R. Victor Jones, March 9, 2000 Thus, by comparison with Equation [ IV-10 ], we see that g b g a ( 29 B ba = B ab [ IV-15a ] and h ϖ 3 π 2 c 3 B ab = A ab [ IV-15b ] and, by comparison with Equation [ IV-9 ], we see that B ab W T ϖ ( 29 = A ab n T [ IV-16a ] or The rate of spontaneous emission The rate of stimulated emission = A ab B ab W T ϖ ( 29 = 1 n T = exp β h ϖ ( 29 - 1 [ IV-16b ]
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