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Quantum Resonant Harmonic Oscillator

  1. Jun 22, 2013 #1
    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.
     
  2. jcsd
  3. Jun 22, 2013 #2

    Bill_K

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    Science Advisor

    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, ei(ω + κ)t and ei(ω - κ)t
     
    Last edited: Jun 22, 2013
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