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

Table 8

Approximation of the energy levels of a confined hydrogenic atom in dimension 𝑑=3. The angular-momentum quantum number is ℓ, 𝑛 is 1 plus the number of nodes of the radial wave function, 𝑏 is the radius of the box, 𝐸 is the exact value of the energy, 𝐸_𝑓 is the expression in floating point arithmetic, and 𝜀 is the variational approximation obtained from the matrix H of dimension 𝑁=250.

ℓ	𝑛	𝑏	𝐸	𝐸𝑓	𝜀
0	1	4	−1/16	−0.06250000000	−0.0624999668
1	1	12	−1/36	−0.02777777778	−0.0277777498
2	1	24	−1/64	−0.01562500000	−0.0156250000
3	1	40	−1/100	−0.01000000000	−0.0100000000
0	1	3(3−√3)	−1/36	−0.02777777778	−0.0277777466
	2	3(3+√3)	−1/36	−0.02777777778	−0.0277775785
1	1	4(5−√5)	−1/64	−0.01562500000	−0.0156249729
	2	4(5+√5)	−1/64	−0.01562500000	−0.0156248833
2	1	5(7−√7)	−1/100	−0.01000000000	−0.0100000000
	2	5(7+√7)	−1/100	−0.01000000000	−0.00999999997
3	1	36	−1/144	−0.006944444444	−0.006944444438
	2	72	−1/144	−0.006944444444	−0.006944444431