Saved Bookmarks
| 1. |
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). Findthe probability of the particle with the lowesr energy to be located with in a region 0 lt x lt a//3 |
|
Answer» Solution :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%` |
|