(Solved): \( 10 . \) Consider a particle in a \( 1 \mathrm{D} \) infinite square well potential within \( 0 ...

\( 10 . \) Consider a particle in a \( 1 \mathrm{D} \) infinite square well potential within \( 0 \leq x \leq a \). (i) Find the wavefunction of the particle over the entire 1D space by considering its potential \( V(x) \) and solving the Time Independent Schrödinger Equation (TISE); (ii) By considering the boundary conditions, find allowed eigenenergies \( E_{n} \); (iii) Find the normalization constant of the wavefunction and write it explicitly; (iv) Write the probability density \( P_{n} \) and draw the graphs of the first three eigenenergies, eignefunctions and corresponding probability densities; (v) Show the Bohr Correspondence principle for the potential well for the probability in Quantum Mechanics (noting that \( P_{n} \) has peaks at the certain values given by \( x_{j}=a(2 j+1) /(2 n) \) with \( j=0 \), \( 1,2, \ldots, n-1) \) and the Classical probability of finding a classical particle bouncing between the walls in the interval from \( x \) to \( x+d x \) at any time.