(1) The WKB method (also known as the JWKB method or rarely as the
Langer transformation) gives a correct asymptotic representation for
the large eigenvalues of QM anharmonic oscillators. Maybe you should
compare your work to the WKB results.
(2) The file share/orthopoly/variational_method.dem has two
variational methods for QM anharmonic oscillators, including one
method that is due to Akihiro Ogura (see "Post-Gaussian variational
method for quantum anharmonic oscillator"). Maybe you could compare
your result to these values.
Incidentally, the WKB + JWKB + Langer transformation methods owe credit
to Ernst Kummer, ``Uber die hypergeometrische Reihe,'' J. fur Math, 15
(1836). There is nothing new under our sun.
Barton
-----maxima-bounces at math.utexas.edu wrote: -----
>To:?Maxima at math.utexas.edu
>From:?Richard?Hennessy?<richhen2008 at gmail.com>
>Sent?by:?maxima-bounces at math.utexas.edu
>Date:?07/21/2009?04:58PM
>Subject:?[Maxima]?Quantum?number?and?energy?of?electron
>
>I?computed?the?solution?to?Shroedinger's?equation?for?an?electron?in
>potential?V(x)?=?mu?x^4?and?get?that?the?electron?energy?is
>Energy(qn):=7.0146168103584261*10^-21?*?qn^1.341209441374369?-
>6.3914598560847341*10^-21
>
>for?mu?=?10^15
>Curiously?the?exponent?does?not?depend?on?mu?(or?hbar?and?m)?and?is?always
>1.34120944137?to?twelve?places?yet?when?I?Google?this?number?or?the?first
>part?of?it?I?do?not?get?any?hits.??Am?I?missing?something?
>
>Rich
>
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