
$$
{\vec F(x)} = - k \thinspace {\vec x}
$$


$$
{U(x)} = {1 \over 2} \thinspace k \thinspace x^2
$$


$$
{{-\hbar^2}\over{2 m}} {{d^2 \Psi}\over{dx^2}} + U \Psi = E \Psi 
$$

$$
{{-\hbar^2}\over{2 m}} {{d^2 \Psi}\over{dx^2}}  + {{1}\over{2}}k x^2  \Psi = E \Psi 
$$

$$
{{d^2 \Psi}\over{dx^2}}  + {-{{2 m} \over {\hbar^2}}} {{1}\over{2}} k x^2  \Psi 
          = - {{2 m}\over{\hbar^2}} E \Psi 
$$

$$
{{d^2 \Psi}\over{dx^2}}  =  {{{m} \over {\hbar^2}}} k x^2  \Psi 
            - {{2 m}\over{\hbar^2}} E \Psi 
$$

$$
{{d^2 \Psi}\over{dx^2}}  =  ({{{k m} \over {\hbar^2}}}) x^2  \Psi 
            - ({{2 m E}\over{\hbar^2}}) \Psi 
$$


$$
\Psi(x)  =  A e^{-\alpha x^2}
$$

$$
{{d \Psi(x)}\over{dx}}  =  
$$

$$
{{d \Psi(x)}\over{dx}}  =  - 2 \alpha x A e^{-\alpha x^2}  =  - 2 x \alpha \Psi
$$

$$
{{d^2 \Psi}\over{dx^2}}  =  - 2 \alpha A e^{-\alpha x^2} +
                            - 2 \alpha x A (- 2 \alpha x) e^{-\alpha x^2}
$$

$$
{{d^2 \Psi}\over{dx^2}}  =  - 2 \alpha A e^{-\alpha x^2} 
                            + 4 \alpha^2 x^2 A e^{-\alpha x^2}
$$

$$
{{d^2 \Psi}\over{dx^2}}  =  - 2 \alpha \Psi + 4 \alpha^2 x^2 \Psi
$$

$$
    x(t)  =  A \thinspace cos(\omega t)
$$

$$
    E  =  {{1}\over{2}} k A^2
$$

$$
    \omega  =  \sqrt{{k}\over{m}}
$$

$$
   {{d \Psi}\over{dx}} =  ( ... )  x \thinspace \Psi
$$

$$
   \alpha  =  {{\sqrt{k m}}\over{2 \hbar}}
$$

$$
    E      =  {{1}\over{2}} \hbar \sqrt{{k}\over{m}}
$$

$$
    L      \sim  {{1}\over{\sqrt{\alpha}}}
$$

$$
    E      =  {{1}\over{2}} \hbar \thinspace \omega
$$

$$
    E      =  h \thinspace \nu
$$

$$
{\Psi_0}(x)  =  A e^{-\alpha x^2}
$$

$$
{\Psi_1}(x)  =  B x e^{-\alpha x^2}
$$

$$
    E_0    =  {{1}\over{2}} \hbar \thinspace \omega
$$

$$
    E_1    =  {{3}\over{2}} \hbar \thinspace \omega
$$

$$
    E_n    =  {{(2n + 1)}\over{2}} \thinspace \hbar \thinspace \omega
$$
   

\bye
