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Atomic and Condensed Matter
Transition dipole moment - polarized absorption
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[QUOTE="Nemanja989, post: 5508494, member: 259712"] Hi again, To me it seems that there is a problem with the previous derivation. Let us for simplicity consider a 2D case. [tex]B_{01}=\frac{\pi e^2}{\epsilon\hbar^2\omega^2_{01} m^2}|\langle\Psi_{S_1}|\vec{e}\cdot\vec{p}|\Psi_{S_0}\rangle|^2 [/tex] and define [tex] \begin{split} D_r & =\langle\Psi_{S_1}|\vec{e}\cdot\vec{p}|\Psi_{S_0}\rangle \\ & =\frac{i\Delta E}{\hbar}(\cos\phi\langle\Psi_{S_1}|x|\Psi_{S_0}\rangle+\sin\phi\langle\Psi_{S_1}|y|\Psi_{S_0}\rangle)\\ & \propto\cos\phi D_x+\sin\phi D_y\\ & =\vec{e}\cdot\vec{D}\\ &=|\vec{D}|\cos\delta \end{split} [/tex] Since being bound states, ## |\Psi_{S_1}\rangle ## and ## |\Psi_{S_0}\rangle ## are real functions normalized with a multiplicative constant, which could be a complex number. Therefore, ## D_x ## and ## D_y ## must be real values multiplied by the same complex number, which is later taken care of with the modulus squared. Namely, ## \begin{split} \Psi_{S_0}=e^{i\alpha}|A|\psi_{S_0} \\ \Psi_{S_1}=e^{i\beta}|B|\psi_{S_1} \end{split} ## Here ## \psi_{S_0} ## and ## \psi_{S_1} ## are real functions and ## e^{i\alpha}|A| ## and ## e^{i\beta}|B| ## complex normalization constants. ## \begin{split} D_x=e^{i(\alpha-\beta)}|A||B|\langle\psi_{S_1}|x|\psi_{S_0}\rangle\\ D_y=e^{i(\alpha-\beta)}|A||B|\langle\psi_{S_1}|y|\psi_{S_0}\rangle \end{split} ## ## D_r=e^{i(\alpha-\beta)}|A||B|(\cos\phi\langle\psi_{S_1}|x|\psi_{S_0}\rangle+\sin\phi\langle\psi_{S_1}|y|\psi_{S_0}\rangle) ## ## \begin{split} |D_r|^2&=|A|^2|B|^2|\cos\phi\langle\psi_{S_1}|x|\psi_{S_0}\rangle+\sin\phi\langle\psi_{S_1}|y|\psi_{S_0}\rangle|^2\\ &=|A|^2|B|^2|\cos\phi D_x+\sin\phi D_y|^2 \end{split} ## and further, ## \begin{split} |D_r|^2&=|A|^2|B|^2|R|^2|\cos(\phi-\theta)|^2\\ &=|A|^2|B|^2R^2\cos^2(\phi-\theta) \end{split} ## Where ## R^2=D_x^2+D_y^2## , and ## tg(\theta)=\frac{Dy}{Dx}##. Where depending on values of ## D_x## and ##D_y## we have ##\theta\in [0,2\pi]##. Although ##D_x## and ##D_y## are known up to a complex multiplicative constant, their ratio is well defined. What makes me suspicious of this result comes from a consideration of a symmetric molecule, where wavefunctions are either odd or even. In this case there is absolutely no argument that light absorption should be more pronounced along the angle ##\theta## than ##-\theta##. And in order to have a symmetric absorption, either ##D_x## or ##D_y## need to be zero. I assume that this might be some bad math from my side, but so far I did not see it. Later I was searching through the literature which deals with this subject, and found in paper ( Don L. Peterson, William T. Simpson, Polarized Electronic Absorption Spectrum of Amides with Assignments of Transitions, J. Am. Chem. Soc. 79 (1957) 2375-2382) the following argument: [I][B]"Crystal spectra must be understood as involving absorption of energy out of two independent beams along the principal directions, or, equivalently, as requiring that the light be represented as a statistical ensemble having parts polarized along the two principal directions. The weights of the two streams of photons, oppositely polarized, are given by the cosine squared law. It is believed that this phenomenon is an example of a disturbance due to the possibility of there having been a “measurement” (absorption of a photon by a crystal oscillator) thus leading to the reduction of the wave function of the light."[/B][/I] Results from this paper are very well fitted with this argument and therefore it seems to be experimentally validated. There are some newer results which use the same idea that a photon is being absorbed by one or the other absorption axis. Although this model would fit my results very well, I am a bit concerned about this argument. Namely, I am not an expert about the collapsion of the wavefunction to the basis functions, and hence I would like to ask some of you here who know much more about it to share it with me and everyone else interested in this topic. [/QUOTE]
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