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I read paper of Friedmann and Hagen Quantum Mechanical Derivation of the Wallis Formula for $\pi$

I am not a physicist but I know how to solve Schrödinger's differential equation for the hydrogen atom but in this article I don't understand why $$\psi_{\alpha l m}=r^le^{-\alpha r^2}Y^m_l(\theta,\phi)$$ and how to prove that $$\langle H \rangle_{\alpha l} \equiv \frac{\langle \psi_{\alpha l m}|H(r)| \psi_{\alpha l m}\rangle} {\langle \psi_{\alpha l m} |\psi_{\alpha l m}\rangle } =\frac{\hbar^2}{2m}\bigg(l+\frac{3}{2}\bigg)2\alpha-e^2 \frac{\Gamma(l+1)}{\Gamma(l+\frac{3}{2})}\sqrt{2 \alpha} $$ sorry for my English

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closed as off-topic by ACuriousMind, Kyle Kanos, HDE 226868, Gert, user36790 Dec 7 '15 at 2:51

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  • $\begingroup$ Related: physics.stackexchange.com/q/219133/2451 $\endgroup$ – Qmechanic Dec 6 '15 at 20:26
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    $\begingroup$ This is a variational approach. The wave function that is given is a test wave function used to estimate the ground-state energy with $\alpha$ some parameter that is tuned to give the smallest possible energy for that family of wave functions. You can check for yourself that $\langle H \rangle_{\alpha l} > -e^2/2a_0$ $\endgroup$ – Praan Dec 7 '15 at 22:12
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    $\begingroup$ Variational method in quantum mechanics. $\endgroup$ – Praan Dec 7 '15 at 22:22
  • $\begingroup$ I try to calculate $\langle H \rangle_{\alpha l}$, but Mathematica says that integral does not converge on ${[- \infty ,\infty]}$.. $\langle \psi_{\alpha l m}|H(r)| \psi_{\alpha l m}\rangle=\int_{-\infty}^{\infty}\big[\frac{-\hbar^2}{2m} \psi(r)^* \frac{d^2}{dx^2}\psi(r) +V(r)|\psi(r)|^2\big]dr, $ and $\langle \psi_{\alpha l m}| \psi_{\alpha l m}\rangle=\int_{-\infty}^{\infty}|\psi(r)|^2dr$. Am I right? $\endgroup$ – vito Dec 8 '15 at 15:12
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Do you know Spherical harmonics? Those Y are spherical harmonics and are often used in physics to display solutions. Try to look those up. They have many relations which are inserted for the second problem.

Actually I don't really know how much you know and if you only solved for the ground state like it is done in many schools.

Usually you use an seperation Ansatz for the wave function which is $$ \Psi = R(r)\Phi(\phi)\Theta(\theta) $$ and then you use this ansatz to solve the differential equation and it just pops up. Do you know how to do seperation ansatz?

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  • $\begingroup$ I use "separation of variable" $\psi(r,\theta,\varphi)=R(r)F(\theta,\varphi)$, then solve radial equation $R(r)=r^l L^{2l+1}_{n-l-1}(2br)e^{-br} $ and the angular equation $F(\theta,\varphi)=Y^m_l(\theta,\varphi)$. set of solutions are given by $\psi_{n,\theta,\varphi}(r,\theta,\varphi)=Nr^l L^{2l+1}_{n-l-1}(2br)e^{-br}Y^m_l(\theta,\varphi)$. but what is $\psi_{\alpha l m}=r^le^{-\alpha r^2}Y^m_l(\theta,\phi)$? $\endgroup$ – vito Dec 7 '15 at 9:36

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