Find the probability that a particle trapped in a box L wide can be found between 0.45L and0.55L forcthe ground and first excited states?

For a particle in a one-dimensional box of width ( L ), the probability density for the ground state (n=1) and the first excited state (n=2) can be calculated using the wave functions. The probability ( P ) of finding the particle between ( 0.45L ) and ( 0.55L ) can be obtained by integrating the square of the wave function over that interval. For the ground state, this probability is approximately 0.1, while for the first excited state, it is slightly higher due to the increased oscillation, generally around 0.2. The exact values can be calculated using the specific wave functions for each state.