Question

For a particle trapped in a one-dimensional infinite square well potential of length ?, find the probability that the particle is in its ground state is in

a) The left third of the box: 0 ≤ ? ≤ ?/3

b) The middle third of the box: ?/3 ≤ ? ≤ 2?/3

c) The right third of the box: 2?/3 ≤ ? ≤ L

After doing parts a), b), and c):

d) Calculate the sum of the probabilities you got for parts a), b), and c). Do you get what you expect? Why or why not?

Answer 1

The wavefunction of the particle in a 1-D infinite box can be given as

Here given that hence

The required probability can be calculated as

(a) For the left third of the box in the ground state( n=1)

(b) For the middle third of the box in the ground state( n=1)

(c) For the right third of the box in the ground state( n=1)

By adding all the probability we get

As the particle is inside the box then the sum of all the probability should be the unity which is true in this case. It shows that particle will be definitely somewhere inside the box.