Question

A particle is located in a two-dimensional square potential well with absolutely impenetrable walls `(0 lt x lt a, 0 lt y lt b)`. Find the probability of the particle with the lowesr energy to be located with in a region `0 lt x lt a//3`

Answer 1

The wave function for the ground state is
`Psi_(11)(x,y)=A "sin"(pix)/(a) "sin"(piy)/(b)`
we find `A` normalization
`1=A^(2)int_(0)^(a)dxint_(0)^(d)dy"sin"^(2)(pix)/(a)"sin"^(2)(piy)/(b)=A^(2)(ab)/(4)` Thus `A= (2)/(sqrt(ab))`.
Then the requisite probability is
`P=int_(0)^(a//3)dxint_(0)^(b)dy(4)/(ab)"sin"^(2)(pix)/(a)"sin"^(2)(piy)/(b)`
`=(2)/(a)dx "sin"^(2)(pix)/(a)` on doing the `y` intergral
`=(1)/(a)int_(0)^(a//3)d(1-"cos"(2pix)/(a))=(1)/(a)((a)/(3)-("sin"(2pi)/(3))/(2pi//a))`
`(1)/(3)-sqrt(3)/(4pi)=0.196= 19.6%`