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Hello everyone! For my quantum mechanics class I have to study the problem of two quantum oscillator coupled to each other and in particular to find the eigenstates and eigenergies for a subspace of the Fock space.
I know that, in general, to solve this kind of problem I have to diagonalize the hamiltonian of the system that in this case is the following one:
$$
H=\hbar\omega_0 (a^+a+b^+b)+\hbar J(a^+b+b^+a)
$$
with a and b bosonic creation and annhilation operator for the two harmonic oscillator.
What I do not understand is how to write the matrix in a given subspace. For example in the case of one quanta of energy present in the oscillators my two eigenstate would be: |00> a superposition of |01> and |10> (one quanta of energy in the first oscillator and 0 in the second one and viceversa). Is that correct?
I don't know how to set the problem,
thank to everyone for the help.
I know that, in general, to solve this kind of problem I have to diagonalize the hamiltonian of the system that in this case is the following one:
$$
H=\hbar\omega_0 (a^+a+b^+b)+\hbar J(a^+b+b^+a)
$$
with a and b bosonic creation and annhilation operator for the two harmonic oscillator.
What I do not understand is how to write the matrix in a given subspace. For example in the case of one quanta of energy present in the oscillators my two eigenstate would be: |00> a superposition of |01> and |10> (one quanta of energy in the first oscillator and 0 in the second one and viceversa). Is that correct?
I don't know how to set the problem,
thank to everyone for the help.