Table 1: Approximation of the energy levels of the harmonic oscillator in . Here, represents the energy state, is the exact solution for the energy, is the upper bound for obtained by the variational analysis, with the eigenvalues of H minimized over the box size, and is the optimal value obtained. The table shows the energies of the first states, .


0 1 1.0000000000 6.86
1 3 3.0000000000 7.55
2 5 5.0000000000 7.09
3 7 7.0000000000 7.14
4 9 9.0000000000 7.61
5 11 11.0000000000 7.49
6 13 13.0000000000 6.85
7 15 15.0000000000 7.07
8 17 17.0000000000 7.27
9 19 19.0000000000 7.43
10 21 21.0000000003 7.46
11 23 23.0000000017 7.49