I am too old too remember, and too lazy to search in the library ... ;-) Actually, the matter is probably treated in "old" math phys books, such as Courant & Hilbert or Morse & Feshbach. The chapter where this is dealt with is on Sturm Liouville theory (http://en.wikipedia.org/wiki/Sturm-Liouville_theory ). You do not need to dig very much in math. All I know I learnt why I wrote a solver for the 1D Schroedinger equation, which is the ancestor of the solvers presently included in the QE distribution ... ("practice is the Mother of learning") Cheers - SB
On Aug 11, 2008, at 10:43 PM, Prasenjit Ghosh wrote: > Dear Stefano, > > Thanks for the nice explanation. > Can you please suggest some "old" quantum mechanics books or some > other reading materials where they deal with such issues? > > With regards, > Prasenjit. > > 2008/8/11 Stefano Baroni <baroni at sissa.it> > Dear Wenmei: > > besides the very sound suggestion by Hande, I would also recommend > that you take a glance at any (possibly, old) textbook in elementary > quantum mechanics. Why old? Because modern texts often given > elementary stuff for granted, which usually it is not. > > Coming to the specific issue that you raise. The number of nodes in > the solution of a 1D Schroedinger equation depends on the energy > parameter. The lower the energy, the fewer the nodes. Of course, for > an arbitrary energy parameter the resulting wavefunction is not > normalizable, that is why it has to be discarded, but for very > special, discrete, values, and energy is quantized. It is interesting > to understand "intuitively" how quantization of energy comes about. As > you rise the energy from any given value, the outer node moves out > furthhr and futher from the origin, but the number of nodes stays > constant if you do not change the energy too much. If the energy is > raised too much, a further node will appear. For some critical value > of the energy, the "extra" node will be at infinity: that's when an > energy eigenvalue is crossed. > > I do not expect that, if you have never heard about similar > arguments, you will understand mine, which have been stated rather > sloppily. Actually, if you decide to study this matter in a serious > book, your ability to understand my "sloppy arguments" will be a check > of your actual understanding of the serious arguments expounded in the > book. > > Hope this helps, a bit > Stefano > > On Aug 11, 2008, at 6:36 PM, Hande Ustunel wrote: > > > Dear Wenmei, > > > > I usually look at the output of the all-electron calculation and > > choose an > > energy close to the ae orbital energy. Once I manage to generate the > > pseudopotential I go as far away as possible from that reference > > without > > upsetting the code. > > > > Hope this helps... > > Hande > > > > On Tue, 12 Aug 2008, [gb2312] ??? wrote: > > > >> > >> Dear all: > >> > >> I want to generate my own RRKJ type ultrasoft pseudopotential > >> with atomic code in PWSCF, > >> I encountered a very confusing situation: > >> > >> when I chose -0.20Ry as the second energy used to pseudize the 5s > >> orbit, the ld1.x always complains : > >> ================================================= > >> from gener_pseudo : error # 1 too many nodes > >> ================================================= > >> however if I change the energy to be -0.50Ry, the corresponding > >> complain disappeared. > >> > >> How should I choose these energy parameters to generate a good > >> pseudopotential and how should I have a > >> correct understanding of these parameters? > >> > >> Thanks very much . > >> > >> Wenmei > >> > >> My &inputp file is also listed as : > >> > >> 64D 3 2 10 0.00 1.00 2.004D 3 2 0 -0.20 1.00 > >> 2.005S 1 0 2 0.00 2.00 2.505S 1 0 0 -0.50 2.00 > >> 2.505P 2 1 3 0.00 2.66 2.985P 2 1 0 -0.20 2.66 > >> 2.98 > >> > >> > >> > >> > >> > >> _________________________________________________________________ > >> ??????????????? > >> http://cnweb.search.live.com/video/results.aspx?q=%E5%8F%AF%E7%88%B1%E7%8C%AB%E5%92%AA&Form=MEVHAA > > > > -- > > Hande Toffoli > > Department of Physics > > Office 439 > > Middle East Technical University > > Ankara 06531, Turkey > > Tel : +90 312 210 3264 > > http://www.physics.metu.edu.tr/~hande > > > > _______________________________________________ > > Pw_forum mailing list > > Pw_forum at pwscf.org > > http://www.democritos.it/mailman/listinfo/pw_forum > > _______________________________________________ > Pw_forum mailing list > Pw_forum at pwscf.org > http://www.democritos.it/mailman/listinfo/pw_forum > > > > -- > PRASENJIT GHOSH, > POST-DOC, > ROOM NO: 265, MAIN BUILDING, > CM SECTION, ICTP, > STRADA COSTERIA 11, > TRIESTE, 34104, > ITALY > PHONE: +39 040 2240 369 (O) > +39 3807528672 (M) > _______________________________________________ > Pw_forum mailing list > Pw_forum at pwscf.org > http://www.democritos.it/mailman/listinfo/pw_forum --- Stefano Baroni - SISSA & DEMOCRITOS National Simulation Center - Trieste http://www.sissa.it/~baroni / [+39] 040 3787 406 (tel) -528 (fax) / stefanobaroni (skype) La morale est une logique de l'action comme la logique est une morale de la pens?e - Jean Piaget Please, if possible, don't send me MS Word or PowerPoint attachments Why? See: http://www.gnu.org/philosophy/no-word-attachments.html -------------- next part -------------- An HTML attachment was scrubbed... URL: http://www.democritos.it/pipermail/pw_forum/attachments/20080812/7e5229fb/attachment-0001.htm
