Find the probability that a particle trapped in a box L wide can be found between 0.45L and 0.55L for the ground and first excited states.

The probability of finding a particle in a certain region in a box is given by the square of the wave function integrated over that region.

For a particle in a box, the wave functions for the ground state (n=1) and first excited state (n=2) are given by:

ψ_n(x) = sqrt(2/L) * sin(nπx/L)

The probability of finding the particle between 0.45L and 0.55L is given by the integral from 0.45L to 0.55L of |ψ_n(x)|^2 dx.

For the ground state (n=1):

P(0.45L < x < 0.55L) = ∫ from 0.45L to 0.55L of |ψ_1(x)|^2 dx = ∫ from 0.45L to 0.55L of (2/L) * sin^2(πx/L) dx

For the first excited state (n=2):

P(0.45L < x < 0.55L) = ∫ from 0.45L to 0.55L of |ψ_2(x)|^2 dx = ∫ from 0.45L to 0.55L of (2/L) * sin^2(2πx/L) dx

These integrals can be solved using standard techniques of integration. The exact values will depend on the specific values of L, but in general, the probability will be higher for the ground state than for the first excited state, because the ground state wave function has a higher amplitude in the center of the box.