1. Use the coordinate representation and write the differential equation which will give energy eigenvalues of a harmonic oscillator using  coordinate representation.
2. How will the equation for energy eigenvalues of a harmonic oscillator look in the momentum representation?
3. Verify that the position and momentum operators satisfy the  canonical commutation relation using

              $$ \text{ Hint:~~~~Apply }  (\hat{x} \, \hat{p} -\hat{p}\hat{x})\,$$   on an arbitrary function.

4.    Do this using (i) coordinate and (ii) momentum representation.

What vector space will be needed for describing only spin degrees of freedom of a spin 1/2 particle? for a particle with general spin $s$? How do we get spin operators $\vec{S}_x, \vec{S}_y, \vec{S}_z$ ? You need to know some properties of spin before you can answer this
question.

Remarks

1. We have bypassed a lot of mathematical details and have  given important results only.
2. It will be sufficient for restricted purpose of class room activities in Quantum Mechanics and Quantum Information courses.
3. However, for a complete understanding  and understanding at a little deeper or more technical level, the mathematical portions bypassed will be critical.