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If I have potential which are very well-known like, square barrier, or square well, or step potential, What I do is to set the boundary conditions in Schrödinger's equations. Sometime, the ground state can be done using variational principles using trial wave function.

Now for example I have a different shape of potential which is $$V= \begin{cases} -e^2/x & \text{for } x>0 \\ \phantom M \infty & \text{for } x<0 \end{cases} $$

The potential is more like compared to three dimension hydrogen atom problem where the potential is $-e^2/r$. Do you think it will be any help to find the energy eigenvalue for the given potentials?

Or, more precisely, my intention is to find the energy eigen value of one dimensional hydrogen atom potential using the known Three dimension.

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  • $\begingroup$ Eigenvalue is one word, not two (same with eigenvector and eigenfunction). $\endgroup$ – Kyle Kanos Aug 13 '19 at 18:36
  • $\begingroup$ Is there some relationship between eigenvalues in different numbers of dimensions? There obviously is when the potential is proportional to $r^2$, because this is $x^2+y^2+z^2$, but for $1/r$ the three dimensions do not contribute additively. $\endgroup$ – G. Smith Aug 13 '19 at 18:43
  • $\begingroup$ Please describe the 1D charge distribution that gives rise to this potential. $\endgroup$ – my2cts Aug 13 '19 at 19:07
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    $\begingroup$ Consider the Schrodinger problem with $\ell=0$ and $\chi(r)=r\psi(r)$ $\endgroup$ – ZeroTheHero Aug 13 '19 at 19:29
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Its not really clear what you are asking, but the $-e^2/x$ potential has quite a long history: See, for example Larry K. Haines, David H. Roberts, "One-Dimensional Hydrogen Atom" American Journal of Physics 37, 1145 (1969); https://doi.org/10.1119/1.1975232.

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  • $\begingroup$ My intention was to find the energy Eigen value of one dimensional Hydrogen atom potential using the known Three dimension. $\endgroup$ – user193422 Aug 13 '19 at 18:24
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    $\begingroup$ If I remember there is a route to this via the path integral. I think C R Hagen was an author on the paper. $\endgroup$ – mike stone Aug 13 '19 at 18:25
  • $\begingroup$ My bad. It was Hagen Kleinert and Duru who looked at path integral for 3d hydrogen, but I think there was a bit about 1d hydrogen in that paper. $\endgroup$ – mike stone Aug 14 '19 at 11:26
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    $\begingroup$ Here is another reference: "The mapping of the Coulomb problem into the oscillator" by David S. Bateman, Clinton Boyd, and Binayak Dutta-Roy, American journal of physics 60 (1992) pp 833-836 $\endgroup$ – mike stone Aug 14 '19 at 14:46

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