Page 270 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

Problems 261

Use the variation trial function ö ˆ ø 1s (1 ‡ ëz), where ë is the variation
parameter, to estimate the ground-state energy for this system.
9.2 Apply the gaussian function
2
ö ˆ e ÿër =a 2 0
where ë is a parameter, as the variation trial function to estimate the energy of
the ground state of the hydrogen atom. What is the percent error?
9.3 Apply the variation trial function ö(x) ˆ x(a ÿ x)(a ÿ 2x) to estimate the energy
of a particle in a box with V(x) ˆ 0 for 0 < x < a, V(x) ˆ1 for x , 0, x . a.
To which energy level does this estimate apply?
9.4 Consider a particle in a one-dimensional potential well such that
4
2
V(x) ˆ (b" =ma )x(x ÿ a), 0 < x < a
ˆ1, x , 0, x . a
where b is a dimensionless parameter. Using the particle in a box with V(x) ˆ 0
for 0 < x < a, V(x) ˆ1 for x , 0, x . a as the unperturbed system, calculate
the ®rst-order perturbation correction to the energy levels. (See Appendix A for
the evaluation of the resulting integrals.)
9.5 Consider a particle in a one-dimensional potential well such that
2
3
V(x) ˆ (b" =ma )x, 0 < x < a
ˆ1, x , 0, x . a
where b is a dimensionless parameter. Using the particle in a box with V(x) ˆ 0
for 0 < x < a, V(x) ˆ1 for x , 0, x . a as the unperturbed system, calculate
the ®rst-order perturbation correction to the energy levels. (See Appendix A for
the evaluation of the resulting integral.)
9.6 Calculate the second-order perturbation correction to the ground-state energy for
the system in problem 9.5. (Use integration by parts and see Appendix A for the
evaluation of the resulting integral.)
9.7 Apply the linear variation function
ö ˆ c 1 (2=a) 1=2 sin(ðx=a) ‡ c 2 (2=a) 1=2 sin(2ðx=a)
for 0 < x < a to the system in problem 9.5. Set the parameter b in the potential
2
equal to ð =8. Solve the secular equation to obtain estimates for the energies E 1
and E 2 of the ground state and ®rst-excited state. Compare this estimate for E 1
with the ground-state energies obtained by ®rst-order and second-order perturba-
tion theory. Then determine the variation functions ö 1 and ö 2 that correspond to
E 1 and E 2 .
2
9.8 Consider a particle in a one-dimensional champagne bottle for which
2 2
2
V(x) ˆ (ð " =8ma ) sin(ðx=a), 0 < x < a
ˆ1, x , 0, x . a

2 G. R. Miller (1979) J. Chem. Educ. 56, 709.