Step-up and step-down operators for the pseudo-harmonic potential V=12r2+B/2r2 in one and two dimensions

✍ Scribed by C.J. Ballhausen

• Publisher: Elsevier Science
• Year: 1988
• Tongue: English
• Weight: 216 KB
• Volume: 151
• Category: Article
• ISSN: 0009-2614
• DOI: 10.1016/0009-2614(88)85162-5

✦ Synopsis

In dimensionless units the step-up and step-down operators for the one-dimensional potential V= fx2+B/2x2, are A^? =ffxd/bx~$-f~2+f~,with~=-~ddl/dX2tV(~).WritingB~u(u-l),~+~~~[(n+l)(n+v+f)]"2~~+, arid..____ = [ n( n + v-l) ] "jv_, , corresponds to a raising and lowering of the energy by two harmonic quanta. In two dimensions V= ~P~+~/~~~.P~=~~+Y~.A~ =~fpafap~~-~p2+f~,~~=-~az~ap2-~1~2p~a~ap-~1~2p2~ a'ia~z+v(~).HereA,~,.,= [(n+l)(ntatl)]'~*~~+,,,andA^_~~,~=[n(n+a)]"*~,_,,, witha=(Btm2)1'Z.FurtherE,.,=(2n+lta)fiw.