The document provides steps to calculate an upper bound and lower bound for the ground state energy of a particle in a box. It first uses a trial quadratic wavefunction to find an upper bound E* for the ground state energy Egs. It then uses the uncertainty principle and assuming Δx=L/2 to find a lower bound for Egs. The exact ground state energy E0 is compared to the upper bound E* calculated, and the lower bound inequality for Egs is stated.

1. Consider a particle of mass m in a box of size L so the wave
function vanishes at x=±L/2.
Find an upper bound E∗ on the ground state energy Egs using a
simple trial wavefunction: a quadratic function of x which
vanishes at ±L/2.
Useful integral:(limit 0 to 1) ∫(1−u2)^2du=8/5.
Egs (less than or equal) E∗ = ??
Compare the bound E∗ with the exact ground state energy E0:
E*/E0 =???
Consider a particle of mass m in a box of size L so the wave
function vanishes at x=±L/2.
Use the uncertainty principle to calculate a lower bound on the
ground state energy (Hint: assume the maximum possible value
for Δx is Δx=L/2.)
Egs≥ ??