Coulomb problem regardless of sign of the potential is in all text books. You are speaking about continuous spectrum. In quantum mechanics it needs special care.

Regards,

Eugene.

Dear Prof Eugene,

Thanks for the help.

But, if my understanding of the textbook problem is correct, then I think the sign is important. When one attempts to find the solution for the radial part of the H-atom problem (for ex.), because of the sign he/she can identify it with the Lagaurre Eqn.

Dipankar

You are on the right way, Dipankar.

Regards,

Eugene.

The problem of solving the Schrödinger equation for the repulsive 1/r potential belongs to the category of scattering problems, where the requirement of normalisation of states is left out. Here one deals with a given constant flux of incoming particles, described by plane waves, and deals with the problem of the scattering these particles/waves undergo by the mediation of the potential in question. The scattered particles/waves are described by plane waves in combination with some other waves that differ from plane waves. Describing the scattered waves in the near-field region is relatively complicated, however not in the far-field region where the contribution of the evanescent waves is exponentially small; in this region to leading order the scattering is characterised by the so-called scattering amplitude. Good books on the subject matter are Quantum Mechanics, by Leonard I. Schiff, 3rd edition, and Modern Quantum Mechanics,  revised edition, by J.J. Sakurai. Conform the earlier statement on this page by Eugene, the spectrum of the problem (the scattering problem) forms a continuum.

For completeness, there is a mathematical way of solving the Schrödinger equation for the repulsive 1/r potential in the usual manner, which consists of placing the 1/r potential on an infinite lattice (that is, periodically extending this repulsive potential) and solving the problem under the Born-von Karman periodic boundary condition. Following this, by increasing the lattice constant(s) of the lattice, one achieves one's goal. The basic idea is that for sufficiently large lattice constant(s) the overlap of the tails of the 1/r potentials are sufficiently small that within a region surrounding each lattice point a particle "feels" only one repulsive 1/r potential. The approach is one of 'compactification'.

Dear Behnam,

This is precisely the reason why I want to know if there is an exact solution available. I want to study Rutherford scattering via partial wave analysis. Now, the assumption that the scattering potential becomes ineffective asymptotically breaks down for 1/r potential. This is the reason why people either place a hard cut-off or use a exponentially dying Yukawa prefactor. But the problem is, when we take the limit for exact coulomb potential, the partial wave amplitude blows up logarithmically. I was wondering if there is any way to avoid this -- it is a central force problem, and can be, in principle, dealt with partial wave technique. Btw, I am aware of the two textbooks that you referred and additionally i have also consulted the book on Scattering theory by Taylor.

The last paragraph of your response is very helpful. I will try to see if I can get analytic expressions for the wave functions.

You caught up the problem, Dipankar. Logarithmical phases. Continuous spectrum in QM is the problem as a whole. To my mind You are trying to solve Coulomb scatering problem in nuclear physics. There are many literature from fifties. To my mind Coulomb wave functions in continuous spectrum are tabulated. May be Abramovitz and Stegun.

Regards,

Eugene.

Dear Dipankar, three main points are in order.

First and foremost, Eugene is absolutely right with regard to the Coulomb wave function. As he indicates, the problem is covered already by Abramowitz & Stegun: see Chapter 14 in their Handbook of Mathematical Functions (Dover, 1970). Importantly, the parameter η, in 14.1.1, whose sign determines whether the interaction is attractive or repulsive, can take any real value between minus infinity and plus infinity.

Second, the Coulomb problem, both attractive and repulsive, is extensively discussed by Philip G. Burke in his voluminous book R-Matrix Theory of Atomic Collisions (Springer, 2011). The problem was originally solved independently by Walter Gordon (of the Klein-Gordon equation) and G. Temple in 1928, the links to the relevant publications I attach below (the former is in German and the latter in English).

Third, with regard to the range of the Coulomb potential, the problem to which you refer is one of the Born series expansion, which already in the second order logarithmically diverges for the Coulomb potential. With the Born series the range of the interaction potential is a critical characteristic to be mindful about (the long range of the Coulomb interaction gives rise to similar problems in any kind of finite-order series expansion; the problem shows up in the calculation of the second virial coefficient, to name but one specific example). The problem was discussed by L.R.B. Elton in 1954, the link to whose relevant publication I also attach below.

http://link.springer.com/article/10.1007%2FBF01351302#page-1

http://rspa.royalsocietypublishing.org/content/121/788/673

http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=2046468

Thanks Eugene for your inputs. Thank you Behnam for the history and those relevant links. I will consult those.

One thing I don't understand is where this perturbative Born expansion enter my analysis. My idea is simply this :

\psi = \psi_inc(plane wave) + f(\theta) e^ikr/r  ------ (1)

... this is the starting point. Now, for a central potential, we can always decompose f(\theta) in terms of partial waves and from the unitarity of the S matrix each partial wave amplitude is bounded ---- THIS SHOULD NOT BLOW UP !!

Now what people usually do is that they place a hard cut-off ... the Coulomb potential is sharply cut at r=\rho (say) --- and for r >>\rho they essentially drop the coulomb term --- and solve the SE --- They get an expression for \psi which they plug into the LHS of Eq. (1) .. Now, since everything is known .. one can extract each partial wave amplitude from Eq.(1) and they involve Log(\rho) ... Now when we take \rho -->infinity, the thing blows up ... But this is incorrect, right?? ... When we take the limit \rho --> infinity; the very assumption on which our asymptotic solution is based on, breaks down -- So the previous solution for the LHS of Eq.(1) is no longer valid.

But, if we can find the exact asymptotic solution for the repulsive Coulomb potential without any assumption (I understand that the solution will not be normalizable, etc ... but you see, the plane wave solution for the free particle is also similar ... But that doesn't mean that it's partial wave expansion co-effs will blow up) ... Then we can plug this wave function into the LHS of eq.(1) ... Note that, this is the exact solution, not a a first- or second order approximation. ... Now again, we can find the partial wave amplitudes that constitute f(\theta) ... The question is, since we are using the exact solution, would the partial amplitude still diverge?? If yes, I would like to understand WHY .. because from the sacred principle of S-matrix unitarity, I wouldn't normally expect them to blow up. As Debmalya mentioned, I'm sure that many people have stumbled over this issue ... and most likely, there is an answer which I still don't know.

I hope that this description makes sense and I will be waiting for further clarifications.

Thanks again to all of you for your illuminations.

Dear Dipankar, before expanding the discussion, I join Debmalya in inviting you to consult Gottfried (or Gottfried-Yan if you consider the 2nd edition of Gottfried's original Quantum Mechanics). He expands (along with others, like Burke) on the issue that for when the scattering potential V(r) vanishes for r approaching infinity, however r V(r) does not (as is the case for the 1/r Coulomb potential), the leading asymptotic contribution to the scattered wave function is not a plane wave (see section 3.6b of Gottfried-Yan). This observation is in fact apparent from the explicit expression in Eq. (7.13.14), p. 437, of Sakurai's Modern Quantum Mechanics, which concerns the Coulomb potential. Note that since the logarithmically-divergent term occurs in the argument of the exponential function, it gives rise to an anomalous power-law behaviour of the scattered wave function for r approaching infinity. This anomalous exponent of the power law cannot be captured by means of a finite-order perturbation theory (on general grounds, one can show that the perturbation series must contain unbounded terms;* only an infinite re-summation of these terms is capable of reproducing the anomalous exponent).

______

* Consult Appendix C of my paper 'Some rigorous results concerning ...', beginning on page 35.

Not only is the solution of the Schrodinger wave equation with a Coulomb potential well known - everything from the lowest lying bound states to continuum states - just recently, the Logarithmic Schrodinger equation has been solved for a Coulomb potential for lowest lying bound states (albeit with zero-angular momentum so far). This is in a recent J. Phys. Commun. publication about to come out.