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i have studied the annihilation and creation operators and number operator N in relation with the simple harmonic oscillator that is governed by:

H = hw(N+1/2)

i don't understand the relation between the harmonic oscillator and for example, this hamiltonian:

H = hw_{1}a^{+}a+hw_{2}a^{+}a^{+}aa

that i have found it in an example in the lecture notes of a course. they calculate the energies of this system. They use the annihilation operator that is defined from the simple HO to solve that system.

what is physically this system? why i can use the SHO to calculate the energies?

i feel that i am confused with the a operator . i thought that it was defined from the hamiltonian of the simple harmonic oscillator ,,,, isn't it ?

thanks in advance

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# Relation between HO and this hamiltonian

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