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  1. Express the operators $a$ and $a^\dagger$ defined by $$ a = {( p -i m\omega x) \over \sqrt{2m\omega \hbar}}, \qquad a^\dagger = {( p +i m\omega x) \over \sqrt{2m\omega \hbar}} $$ in the co-ordinate representation and solve the equation $$ a \psi(x) = 0 $$ to determine the ground state wave function.
  2. Applying $a^\dagger$ on the ground state wave function, find the first two excited state eigen functions for the harmonic oscillator.
  3. Normalize the ground state and the two excited state eigen functions found above.

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