Einstein Coefficients - Laser Beams

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SUMMARY

The discussion focuses on the derivation of Einstein coefficients in a two-level laser system with no degeneracy, specifically demonstrating that \(B_{12} = B_{21}\) under the condition \(N_2 + N_1 = \text{constant}\). The equations governing the population dynamics are presented, including \(\frac{\partial N_1}{\partial t} = -B_{12}\rho(v_{21}N_1\) and \(\frac{\partial N_2}{\partial t} = -B_{21}\rho(v_{21}N_2\). The participant expresses confusion regarding the signs in the equations and seeks clarification on LaTeX formatting.

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  • Understanding of Einstein coefficients in laser physics
  • Familiarity with differential equations
  • Knowledge of population inversion in two-level systems
  • Proficiency in LaTeX for mathematical expressions
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  • Study the derivation of Einstein coefficients in detail
  • Learn about population dynamics in laser systems
  • Explore LaTeX formatting for complex equations
  • Investigate the implications of non-degenerate energy levels in lasers
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Students and researchers in laser physics, particularly those studying two-level laser systems and Einstein coefficients, as well as anyone needing assistance with LaTeX formatting for scientific documentation.

Ivor Denham
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Homework Statement


Assume a two level laser system with no degeneracy (g1 = g2 = 1).
If N_2 + N1 = \text{constant}[\tex], show that B12 = B21[\tex].<br /> <br /> <h2>Homework Equations</h2><br /> \frac{\partial N_1}{\partial t}=-B_{12}\rho(v_{21}N_1[\tex]<br /> \frac{\partial N_2}{\partial t}=-B_{21}\rho(v_{21}N_2[\tex]<br /> <br /> <h2>The Attempt at a Solution</h2> <br /> I think there might be something up with the signs in the relevant equation, a convention I am not observing.<br /> <br /> N_1+N_2=c[\tex]<br /> <b>\frac{\partial N_1}{\partial t}+\frac{\partial N_2}{\partial t}=\frac{\partial c}{\partial t}[\tex]</b><br /> <b><b><b>\frac{\partial N_1}{\partial t}+\frac{\partial N_2}{\partial t}=0[\tex]</b></b><br /> <b><b><b><b><b><b>-B_{12}\rho(v_{21}N_1+-B_{21}\rho(v_{21}N_2=0[\tex]</b></b></b></b><br /> <b><b><b><b><b><b><b><b><b><b><b><b>-B_{12}N_1+-B_{21}N_2=0[\tex]</b></b></b></b></b></b></b></b><br /> <b><b><b><b><b><b><b><b><br /> From here I am stuck, where have I gone wrong?</b></b></b></b></b></b></b></b></b></b></b></b></b></b></b>[/B]
 
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Also could someone tell me how to fix the latex?
 

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