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

Table 6

Approximation of the energy levels of the harmonic oscillator 𝐻=−(1/2)Δ+(1/2)𝑥² in dimension 𝑑=1. For the state 𝑛, 𝐸_𝑛 is a highly accurate solution for the energy provided by Aguilera-Navarro et al. [2], 𝜀_𝑛 is the upper bound for 𝐸_𝑛 obtained using the present variational analysis with basis size 𝑁=250, and 𝐿=0.5. This table shows the energies of the first 12 states 𝑛=0,1,…,11.

𝑛	𝐸	𝜀
0	4.951123323264	4.951129323244
1	19.774534178560	19.774534179209
2	44.452073828864	44.452073829725
3	78.996921150976	78.996921150748
4	123.410710456832	123.410710456280
5	177.693843822080	177.693843818558
6	241.846458758144	241.846458765623
7	315.868612673536	315.868612686280
8	399.760332976128	399.760332979135
9	493.521634054144	493.521634068796
10	597.152524107776	597.152524136545
11	710.653008064512	710.653008103290