The ground-state wave function for a particle confined to a one-dimensional box of length l is ψ=(2/l)^1/2 sin(πx/l). suppose the box is 100 nm long. calculate the probability that the particle is (a) between x= 4.95 nm and 5.05 nm (b) between x= 1.95 nm and 2.05 nm (c) between x=9.90 nm and 10.00 nm (d) in the right half of the box (e) in the central third of the box.

(a) 4.98x10⁻⁵

(b) 7.89x10⁻⁶

(c) 1.89x10⁻⁴

(d) 0.5

(e) 2.9x10⁻²  

Step-by-step explanation:  

The probability (P) to find the particle is given by:

    (1)

The solution of the intregral of equation (1) is:

(a) The probability to find the particle between x = 4.95 nm and 5.05 nm is:

(b) The probability to find the particle between x = 1.95 nm and 2.05 nm is:

(c) The probability to find the particle between x = 9.90 nm and 10.00 nm is:

(d) The probability to find the particle in the right half of the box, that is to say, between x = 0 nm and 50 nm is:

(e) The probability to find the particle in the central third of the box, that is to say, between x = 0 nm and 100/6 nm is:

I hope it helps you!