Question

What is the probability of finding the particle in the region a/4 < x < a/2? show your work.

Answer 1

The probability of finding the particle in the region a/4 < x < a/2

The probability of particle in given range is given by ^*dx = ²dx-------------------(1)

     In 1-D Particle    =(2/a)^1/2 sin[nπx/a]     

                                      lower limit is a/4     upper limit is a/2

        

^*dx = ²dx= (2/a)sin²[nπx/a]dx     sin² x= (1-cos2x/2)

                               =2/a[1-cos2(nπx/a)]dx

                             = 2/a{1/2[1 dx] - [cos2(nπx/a) dx]}

                           =    2/a{1/2[x]_^a/2a/4 - a/[2nπ] [sin2(nπx/a)/]_a/4^a/2}

                          = 2/a{1/2[a/4-a/2] - a/[2nπ] [sin(2nπa/4 1/a)/]_^-[sin(2nπa/2 1/a]}

                         = (2/a )1/2{[a/4] - a/[2nπ] [sin(nπ)/]_^-[sin(nπ/2]}

                         =   (2/a)(1/2){(a/4)- [a/2nπ][-(-1)ⁿ] }

                        =    (2/a)(1/2){(a/4)- [a/2nπ][-(-1)ⁿ] }

                        = (2/a)(a/4){[1/2+(1/nπ) (-1)ⁿ]}

                       =1/2{(1/2) +(1/nπ)(-1)ⁿ}

   probability P ={(1/4) +(1/2nπ)(-1)ⁿ}

                     for ground state i.e n=0

                    P= 1/4=0.25

              For first excited state i,e n=1

   P= [(1/4) -(1/2π)]