I was wondering if there was a way to construct the ladder operators for a nonlinear oscillator given by the Hamiltonian $$H=x^2+p^2+\lambda x^4$$ If we were to just calculate scattering amplitudes, then it makes sense to just use the harmonic oscillator ladder operators and use perturbation theory. But I am looking to re-express this Hamiltonian in terms of ladder operators. So, is there a way the generalise their definitions to different Hamiltonians?
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1$\begingroup$ One could go second quantized representation and write the operators that annihilate state $n$ and create state $n\pm1$: $X_{n\pm 1} = a^\dagger_{n\pm1}a_n$. However having the ladder operators nicely expressed in terms of the first derivative and $x$ is impossible (otherwise, this Hamiltonian would be exactly solvable in terms fo some special functions - I doubt that this is the case). I remember myself struggling with it for a few days when I began reading Haken's book... I should have instead continued reading to learn about the second quantization. $\endgroup$– Roger V.Commented Jun 5, 2020 at 12:57
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$\begingroup$ Non harmonic Hamiltonians are no problem for second quantization. For example see the Gross–Pitaevskii equation that was extensively studied and applied to interpretation of experimental data. $\endgroup$– AlexanderCommented Jun 8, 2020 at 3:56
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It depends a bit on what you have in mind but, in this specific problem, it is possible to express the Hamiltonian in terms of the $\mathfrak{su}(1,1)\sim\mathfrak{sp}(2,\mathbb{R})$ algebra, with (complexified) generators \begin{align} \hat K_+=\textstyle\frac{1}{2}\hat a^\dagger \hat a^\dagger\, ,\qquad \hat K_-=\frac{1}{2}\hat a\, \hat a\, ,\qquad \hat K_0=\frac{1}{4}\left(\hat a^\dagger \hat a+\hat a\hat a^\dagger\right)\, . \tag{1} \end{align} It turns out that $p^2+x^2\sim \hat K_0$ (up to some constant) and $x^4$ is also expressible in terms of the operators in (1).
A very accessible reference on this is
Novaes, Marcel. "Some basics of su (1, 1)." Revista Brasileira de Ensino de Fisica 26.4 (2004): 351-357
but Google also throws up a number of canonical papers.
See also this question for a slightly more general setting.
answered Jun 8, 2020 at 3:23