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Relation between HO and this hamiltonian

  1. Sep 14, 2015 #1
    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 = hw1a+a+hw2a+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
  2. jcsd
  3. Sep 15, 2015 #2


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    Formally, this new Hamiltonian is only an operator on the same Hilbert space as that used for the harmonic oscillator and expressed in terms of the operators ##a## and ##a^\dagger##, which have a known action on the energy eigenstates of the harmonic oscillator. You can therefore simply treat it as a (hermitian) operator on that Hilbert space and compute its eigenvalues.
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