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

Table 2

The energy levels for the quartic anharmonic oscillator 𝐻=−Δ+𝑥²+𝑥⁴ in dimension 𝑅. 𝑛 represents the energy state, 𝐸 represents the accurate energy values obtained in [1], and 𝜀 is the upper bound to 𝐸 obtained using the present variational analysis in a basis of size 𝑁=20. The eigenvalues 𝜀_𝑛 of H were minimized over the box size, and 𝐿 is the optimal value obtained. The table shows the first 6 states, 𝑛=0,1,…,5.

𝑛	𝐸	𝜀	𝐿
0	1.3923516415	1.3923516415	3.4
1	4.6488127042	4.6488127042	3.4
2	8.6550499577	8.6550499586	3.4
3	13.1568038980	13.1568038994	3.7
4	18.0575574363	18.0575574558	3.4
5	23.2974414512	23.2974415625	3.4