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

Table 5

Approximation of the energy levels of the hydrogen atom in dimension 𝑑=3, for quantum numbers ℓ=0,1,2. Note that the variational parameter that minimizes the upper bound tends to be very large for all the states. The error is of the order of 10⁻⁴ in the “best” cases.

ℓ	𝑛	𝐸	𝜀	𝑏
0	1	−0.2500000000	−0.2499790730	17.5
	2	−0.06250000000	−0.06246859682	40.5
	3	−0.02777777778	−0.02773301831	70.5
	4	−0.01562500000	−0.01556528040	107
1	1	−0.06250000000	−0.06231120892	33
	2	−0.02777777778	−0.02747649731	60
	3	−0.01562500000	−0.01526320869	94
	4	−0.01000000000	−0.009656788911	143
2	1	−0.02777777778	−0.02777640178	75
	2	−0.01562500000	−0.01561644406	108
	3	−0.01000000000	−0.009970374676	146
	4	−0.006944444444	−0.006872824074	189