A particle in a one-dimensional box (potential zero between to a and infinite outside) has the ground state energy $E_{0}=\frac{0.125 h^{2}}{m a^{2}}$. The expectation value of the above Hamiltonian with yields an energy $E_{1}$. Using a linear combination of two even functions $\mathrm{x}(\mathrm{x}-\mathrm{a})$ and $\mathrm{x}^{2}(\mathrm{x}-\mathrm{a})^{2}$ we obtain variational maximum to the ground state energy as E 2 . Which of the following relations holds for $\mathrm{E}_{0}, \mathrm{E}_{1}$ and $\mathrm{E}_{2}$ ?
(a)$\mathrm{E}_{0}<\mathrm{E}_{1}<\mathrm{E}_{2}$
(b)$\mathrm{E}_{0}<\mathrm{E}_{2}<\mathrm{E}_{1}$
(c)$\mathrm{E}_{1}<\mathrm{E}_{0}<\mathrm{E}_{2}$
(d)$\mathrm{E}_{2}<\mathrm{E}_{0}<\mathrm{E}_{1}$
Answer
Answer: B ✓ checked by 4AB · confidence high
The source book printed no answer; this one was worked out and checked — see the explanation.
Explanation
Both variational estimates lie above the exact E₀; the two-function trial space contains the single function, so its optimum is at least as good: E₀ < E₂ < E₁.