\magnification=\magstep3

$$
   { {\rm Prob}(x_0 < x < x_0 + dx)} = { {|\Psi(x_0)|}^2 dx }
$$

$$\eqalign{
   { {\rm Prob}(r_0 < r < r_0 + dr)} &= { {|\Psi(r_0)|}^2 dV } \cr
                                     &\phantom{=} \cr
                                     &= { {|\Psi(r_0)|}^2 r^2 dr } \cr
}
$$

$$
   { {\rm Prob}(R_1 < r < R_2)} = { {\int_{R_1}^{R_2}} {{|\Psi(r)|}^2 r^2 dr}  } 
$$


$$
  E  =  - {{\hbar^2} \over{2 m a^2}}  =  
         - {{ {m_e}^2 e^4} \over{32 \pi^2 {\epsilon_0}^2 \hbar^2 }} 
$$

$$
  {\Psi(r) } \thinspace =  \thinspace { Q e^{-r/a}  }
$$

$$
  {{\rm Prob\ }(r_1 < r < r_2) } \thinspace 
            =  
            \thinspace {\int_{r_1}^{r_2} { {\Psi(r)}^2 \thinspace dr  } }
            =  
            \thinspace {\int_{r_1}^{r_2} { {Q^2 e^{-2r/a}} \thinspace dr  } }
$$

$$
  dV  \thinspace = \thinspace dx
$$

$$
  dV  \thinspace = \thinspace dx \thinspace dy
$$


$$
  dV  \thinspace = \thinspace dr \thinspace r d\theta
$$

$$
  dV  \thinspace = \thinspace dx \thinspace dy \thinspace dz
$$

$$\eqalign{
  dV  \thinspace &= \thinspace (dr) \thinspace (r d\theta) \thinspace (r \sin\theta \thinspace d\phi) \cr
                 &= \thinspace r^2 dr \thinspace \sin\theta d\theta \thinspace\thinspace d\phi \cr
}
$$


$$\eqalign{
  { {\rm Prob }(r, \theta, \phi) } \thinspace
        &= \thinspace
        {\int \left({\Psi(r, \theta, \phi)}\right)^2  \thinspace dV } \cr
        &= \thinspace
        {\int \left({\Psi(r, \theta, \phi)}\right)^2  \thinspace r^2 dr \sin\theta d\theta \thinspace d\phi } \cr
}
$$

$$\eqalign{
  { {\rm Prob }(r, \theta, \phi) } \thinspace
        &= \thinspace
        {\int \left({ R(r) \Theta(\theta) \Phi(\phi) }\right)^2  \thinspace r^2 dr \sin\theta d\theta \thinspace d\phi } \cr
        &= \thinspace
        {\int { \left( R^2(r) r^2 dr \right)  \thinspace  
                \left( \Theta^2(\theta) \sin\theta d\theta  \right) \thinspace 
                \left( \Phi^2(\phi) d\phi \right) }  } \cr
        &= \thinspace
        {  { \int R^2(r) r^2 dr  }  \thinspace  
           { \int \Theta^2(\theta) \sin\theta d\theta   } \thinspace 
           { \int \Phi^2(\phi) d\phi }  } \cr
}
$$


$$
  \int { {1} \thinspace \sin\theta d\theta   }
$$


$$
  \int { {1} \thinspace d\phi   }
$$

$$
  {R(r) } \thinspace =  \thinspace { Q e^{-r/a}  }
$$

$$\eqalign{
  { {\rm Prob }(r_1 < r < r_2) } \thinspace
        &= \thinspace
        { 4 \pi  { \int_{r_1}^{r_2} R^2(r) r^2 dr  }    }  \cr
}
$$

$$\eqalign{
  { {\rm Prob }(r_1 < r < r_2) } \thinspace
        &= \thinspace
        { 4 \pi  \thinspace (Q^2 {{a^3}\over{8}}) \thinspace
              { \int_{2{r_1}/a}^{2{r_2}/a} u^2 e^{-u} du  }    }  \cr
}
$$


$$\eqalign{
  { {\rm Prob }(r_1 < r < r_2) } \thinspace
        &= \thinspace
        { 4 \pi  \thinspace (Q^2 {{a^3}\over{8}}) \thinspace
              { \left[ -e^{-u} ( u^2 + 2u + 2 ) \right]_{2r_1/a}^{2r_2/a}   }  }  \cr
}
$$

$$
  <r> \quad = \quad {  { \int { \Psi^2 \thinspace r \thinspace dV } } \over
           { \int { \Psi^2 \thinspace dV } }  }
$$


\bye
