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PROBLEM/QM-10006

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  1. Express the following operators in terms of $a$ and $a^\dagger$. $$ (i) \ \hat{x}\qquad (ii)\ \hat{p}\qquad (iii)\ \hat{x}^2 \qquad (iv)\ \hat{p}^2 $$
  2. Using the properties of operators $a$ and $a^\dagger$ compute the $m n$ matrix elements of the four operators given in part (a) in the harmonic oscillator basis.
  3. What answer do you expect for the matrix elements of the Hamiltonian operator $$ \hat{H} = { \hat{p}^2\over 2m} +{1\over2}m \omega^2 \hat{x}^2 $$ Using the answers obtained in part (b) check if your guess is correct.

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