(a) Explanation Write the expression for wave function. ψ n ( x ) = A sin ( n π x L ) (I) Here, A is normalization constant, n is energy state, x is position of box, L is the length of box and ψ n ( x ) is the wave function. Substitute x − L 2 for x in equation (I). ψ n ( x ) = A sin [ ( n π L ) ( x − L 2 ) ] = A sin [ ( n π x L ) − n π 2 ] Expand the above expression. ψ n ( x ) = A [ sin ( n π x L ) cos ( n π 2 ) − cos ( n π x L ) sin ( n π 2 ) ] (II) Write the expression for probability density. P n ( x ) = | ψ n ( x ) | 2 (III) Conclusion: Substitute ( 2 L ) 1 / 2 for A and 1 for n in equation (II) (b) To determine The sketch of wave function and probability density for given states.