Quantum Resonant Harmonic Oscillator

[rbwang1225]

The Hamiltonian is $H=\hbar \omega (a^\dagger a+b^\dagger b)+\hbar\kappa(a^\dagger b+ab^\dagger)$ with commutation relations $[a,a^\dagger]=1 \hspace{1 mm} and \hspace{1 mm}[b,b^\dagger]=1$.
I want to calculate the Heisenberg equations of motion for a and b.
Beginning with $\dot a=\frac{i}{\hbar}[H,a]=-i\omega a-i\kappa b$ and
$\dot b=\frac{i}{\hbar}[H,b]=-i\omega b-i\kappa a$,
I got $\ddot a=-(\omega^2+\kappa^2)a-2\omega\kappa b$ and
$\ddot b=-(\omega^2+\kappa^2)b-2\omega\kappa a$.
The solution is $a+b=[a(0)+b(0)]e^{-i(\omega+\kappa)t}$ and from this I got
$b=[a(0)+b(0)]e^{-i(\omega+\kappa)t}-a$ and then
$\dot a=-i(\omega+\kappa)a-i\kappa[a(0)+b(0)]e^{-i(\omega+\kappa)t}$.
The solution of $a$ is $a=-i\kappa[a(0)+b(0)]te^{-i(\omega+\kappa)t}+a(0)e^{-i(\omega+\kappa)t}$ and therefore
$b=i\kappa[a(0)+b(0)]te^{-i(\omega+\kappa)t}+b(0)e^{-i(\omega+\kappa)t}$.
However, my result did not preserve the commutator, i.e., $[a,a^\dagger]=2\kappa^2t^2+1$.
I don't know which step is wrong in my derivation.

The solution in the book of Carmichael is $a=e^{-i\omega t}[a(0)\cos\kappa t-ib(0)\sin\kappa t]$ and
$b=e^{-i\omega t}[b(0)\cos\kappa t-ia(0)\sin\kappa t]$, which preserves the commutators.

 

[Bill_K]

a¨=−(ω²+κ²)a−2ωκb and b¨=−(ω²+κ²)b−2ωκa.

From this, (a + b)¨ = - (ω + κ)² (a + b) and (a - b)¨ = - (ω - κ)² (a - b).

Thus each of a and b are superpositions of exponentials, e^{i(ω + κ)t} and e^{i(ω - κ)t}