I am following the derivation of the radial wavefunction and correlation length of a Cooper pair in Walter Greiner's book "Quantum Mechanics - Special Chapters". While I understand how to obtain the wavefunction, his method of estimating the correlation length has me puzzled. He uses the form of the wavefunction \begin{equation*} \psi(r)\simeq C\left(\frac{\cos k_Fr}{\alpha r^2}+\frac{\sin k_Fr}{\alpha^2r^3}\right) \end{equation*} where $C$ and $\alpha$ are constants, and says that by determining the first extremum we arrive at the relation $\alpha r_{max}\simeq 1$. I have tried differentiating the wavefunction and equating to zero, but I end up with a horrible transcendental equation in terms of $k_Fr$. Am I missing something, and is there an alternative way to obtain the extremum of this function, or do I need to solve a transcendental equation?

  • $\begingroup$ You can not find all solutions of the extremums of the function $\psi$, but you can apply some approximations to find the first extrema. I guess $k_{F}$ is large, so you're interested in small $k_{F}r$, or something like that. Would help if you provide some physical ground for the approximation... for instance, it sounds strange that your wave-function diverges at $r=0$... $\endgroup$ – FraSchelle Dec 19 '17 at 5:38
  • $\begingroup$ I thought about using a series expansion for the trigonometric functions, but I was not sure if this was justified, i.e. if $k_Fr$ could be considered small., especially as the form of the wavefunction is an approximation for large $r$! $\endgroup$ – dgwp Dec 19 '17 at 9:43
  • $\begingroup$ I should probably have mentioned that the wavefunction above is obtained using the first two terms in the series expansion for the sine and cosine integrals, si(x) and Ci(x), and is valid only for large $r$ (so the divergence at $r=0$ is not relevant). $\endgroup$ – dgwp Dec 20 '17 at 16:29

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