The variational principle in quantum mechanics gives an upper bound on the energy eigenvalue $E_k$ of the $k\mathrm{th}$ state if its trial wave function is orthogonal to the eigenfunctions of all the lower states. A lower bound on $E_k$ has been derived assuming that: (i) its upper bound be less than $E_{k+1\mathrm{}}$, and (ii) the energy fluctuation ${(〈H^2〉-{〈H〉}^2)}^{\frac{1}{2}}$ be less than $\frac{1}{2}(E_{k+1\mathrm{}}-E_k)$. An upper bound on the error in the wave function of the $k\mathrm{th}$ state has also been derived. The formulas for the bounds have been applied to calculate the accuracy of the energies and wave functions of the various $J$ states of the ground and excited $K=0\mathrm{}$ bands in ${}_{}{}^{20}{}_{}{}^{}\mathrm{Ne}$. The intrinsic wave functions of the $K=0\mathrm{}$ bands were taken from the earlier calculations performed by using the deformed Hartree-Fock and Tamm-Dancoff approximations. Techniques for calculating $〈H^2〉$ have also been discussed. Our results show that the energies and wave functions calculated by these approximations are fairly accurate for a number of states. The wave functions of the ground and first excited $J=0\mathrm{}$ states are accurate at least to the order of 92.5 and 87.3%, respectively.

• Received 8 October 1971

DOI:https://doi.org/10.1103/PhysRevC.6.179