Table 4: Approximation of the energy levels of the harmonic oscillator in dimensions . represents the energy state, is the exact solution for the energy given by (12), and is the upper bound to obtained using the variational analysis. The eigenvalues of H are minimized over . This table shows specific examples of energy values for the quantum numbers and energy states . Note that the approximation error has diminished compared with those of Table 3. In the worst case it is of , while in others cases the upper bounds are almost exact.


3
0
1 3 3.00000000 5.4
5 19 19.00000000 7.4
10 39 39.00000001 8.7
1
1 5 5.000000119 5.4
5 21 21.00000234 6.9
10 41 41.00001129 8.4
2
1 7 7.000000002 6.0
5 23 23.00000000 9.0
10 43 43.00000001 9.0
3
1 9 9.000000003 6.0
5 25 25.00000000 8.1
10 45 45.00000000 10.0

4
0
1 4 4.000009387 4.7
5 20 20.00008831 6.5
10 40 40.00027064 8.1
1
1 6 6.000000003 7.7
5 22 22.00000002 7.4
10 42 42.00000014 8.8
2
1 8 8.000000002 6.0
5 24 24.00000001 7.4
10 44 44.00000000 10.0
3
1 10 10.00000000 6.6
5 26 26.00000000 7.6
10 46 46.00000000 10.0

5
0
1 5 5.000000119 5.4
5 21 21.00000234 6.9
10 41 41.00001129 8.4
1
1 7 7.000000002 6.0
5 23 43.00000001 9.0
10 43 43.00000001 9.25
2
1 9 9.000000003 6.0
5 25 25.00000000 8.1
10 45 45.00000000 10.0
3
1 11 1.00000000 6.5
5 27 27.00000000 7.6
10 47 46.99999999 10.3