Direct solution of the Schrödinger equation for some muonic molecules Original Research Article

We use the hyperspherical harmonic method developed by us to directly solve the Schrödinger equation for some muonic molecules without recourse to the Born-Oppenheimer approximation. In our method, a kind of simplified potential harmonic representation derived by us is used, and analytic expressions for all matrix elements are obtained; the generalized Laguerre function expansion method is used in the solution of the coupled differential equations. The practical calculations yield very good results for the energy. For example, we obtain a ground-state energy of −0.055623 pau for ppμ, −0.029884 dau for ddμ, and −0.020521 tau for ttμ, in good agreement with the exact values of −0.055676 pau, −0.029921 dau, and −0.020553 tau. The calculated energy values for the first excited states of ppμ, ddμ, ttμ are also reported. It is shown that, the potential harmonic representation we used is simple and convenient, and the convergence of the basis functions is uniform and rather rapid.