Evaluate 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)

1. Ground State (n=1):

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_1(x)|^2 dx.

|ψ_1(x)|^2 = 2/L * sin^2(πx/L)

Integrating this from 0.45L to 0.55L gives the probability for the ground state.

2. First Excited State (n=2):

Similarly, the probability for the first excited state is given by the integral from 0.45L to 0.55L of |ψ_2(x)|^2 dx.

|ψ_2(x)|^2 = 2/L * sin^2(2πx/L)

Integrating this from 0.45L to 0.55L gives the probability for the first excited state.

These integrals can be solved using standard techniques of integration. The exact values will depend on the specific values of L, but the procedure outlined above gives the general method for solving this type of problem.