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

Table 4

Approximation of the energy levels of the harmonic oscillator in dimensions 𝑑=3,4,5. 𝑛 represents the energy state, 𝐸 is the exact solution for the energy given by (12), and 𝜀 is the upper bound to 𝐸 obtained using the variational analysis. The eigenvalues 𝜀_𝑛 of H are minimized over 𝐿. This table shows specific examples of energy values for the quantum numbers ℓ=0,1,2,3 and energy states 𝑛=1,5,10. Note that the approximation error has diminished compared with those of Table 3. In the worst case it is of 10⁻⁵, while in others cases the upper bounds are almost exact.

𝑑	ℓ	𝑛	𝐸	𝜀	𝐿
3	0	1	3	3.00000000	5.4
		5	19	19.00000000	7.4
		10	39	39.00000001	8.7
	1	1	5	5.000000119	5.4
		5	21	21.00000234	6.9
		10	41	41.00001129	8.4
	2	1	7	7.000000002	6.0
		5	23	23.00000000	9.0
		10	43	43.00000001	9.0
	3	1	9	9.000000003	6.0
		5	25	25.00000000	8.1
		10	45	45.00000000	10.0
4	0	1	4	4.000009387	4.7
		5	20	20.00008831	6.5
		10	40	40.00027064	8.1
	1	1	6	6.000000003	7.7
		5	22	22.00000002	7.4
		10	42	42.00000014	8.8
	2	1	8	8.000000002	6.0
		5	24	24.00000001	7.4
		10	44	44.00000000	10.0
	3	1	10	10.00000000	6.6
		5	26	26.00000000	7.6
		10	46	46.00000000	10.0
5	0	1	5	5.000000119	5.4
		5	21	21.00000234	6.9
		10	41	41.00001129	8.4
	1	1	7	7.000000002	6.0
		5	23	43.00000001	9.0
		10	43	43.00000001	9.25
	2	1	9	9.000000003	6.0
		5	25	25.00000000	8.1
		10	45	45.00000000	10.0
	3	1	11	1.00000000	6.5
		5	27	27.00000000	7.6
		10	47	46.99999999	10.3