Wide Effectiveness of a Sine Basis for Quantum-Mechanical Problems in 𝑑 Dimensions

Table 1

Approximation of the energy levels of the harmonic oscillator 𝐻=−Δ+𝑥² in 𝑅. Here, 𝑛 represents the energy state, 𝐸 is the exact solution for the energy, 𝜀 is the upper bound for 𝐸 obtained by the variational analysis, with the eigenvalues 𝜀_𝑛 of H minimized over the box size, and 𝐿 is the optimal value obtained. The table shows the energies of the first 12 states, 𝑛=0,1,…,11.

𝑛	𝐸	𝜀	𝐿
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