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You can use the annihilation operator to find the groundstate wavefunction, and then use the hamiltonian in terms of annhiliation and creation operators to find the energy eigen value of that state. (or you could put the wavefunction into the Schrodinger equation)

By acting the creation operatior on the groundstate wavefunction you can find the next excited state wavefunction.

My question is if a question asks you to find the eigenstates and eigenvalues for a particular system is it best to use these annihilation / creation operators or would it be better to use the Schrodinger equation in its 'original' form?

I think the benefit of the annihilation / creation operator method is that it allows you to find the quantised amount that each new energy level would be raised by, but you could also do this using the Schrodinger equation if you find the ground state energy and the next excited state energy.

I'm trying to get an idea of the methods used more generally at an advanced level.

Thanks in advance for any input!