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I'd like to find the normalized ground state wavefunction for the anharmonic oscillator (Duffing) whose potential for which there is no analytic solution; an oscillator with a quartic potential, in addition to the quadratic potential. Any comments would be highly appreciated

$$V(x_{-}) = \frac{x^2}{2}+\lambda \frac{x^4}{4}$$

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What have you tried so far?

We already know the wave functions and energy levels when $\lambda=0$, they're Hermite functions and integers + a half, respectively. If you know that the parameter $\lambda$ determining the anharmonicity is small, you could use perturbation theory to expand the eigenfunctions and energy levels in a power series in $\lambda$, e.g.

$\psi = \pi^{-1/4}e^{-x^2/2}+\lambda\psi_1+\lambda^2\psi_2+\ldots$,

$E = \frac{1}{2}+\lambda E_1+\lambda_2 E_2+\ldots$

I'm happy to elaborate, but the wiki article on quantum mechanical perturbation theory has a better explanation than I could give. It also uses the anharmonic oscillator as an example, although just to find the ground-state energy. The Landau & Lifshitz book has a pretty good chapter on perturbation theory, as does Baym's book.

Alternatively, it's probably just a hypergeometric function, because experience has demonstrated that everything is some kind of hypergeometric function.

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  • $\begingroup$ @ Daniel Shapero, I tryed using matrix methods. A common approach is to take a finite basis set and diagonalize it numerically. The ground state of this reduced basis set will not be the exact ground state, but by increasing the size of the basis we can improve the accuracy and check if the energy converges as we increase the basis size. $\endgroup$
    – David H
    Dec 19 '13 at 9:36
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I solved using the time independent perturbation theory. The corrected wavefunction in nondegenerate first order perturbation theory is given by

$|\psi{_n}\rangle\approx|\psi_{n}^{(0)}\rangle+\sum_{k\neq n}\frac{\langle \psi_{k}^{(0)}|\hat H^{(1)}|\psi_{n}^{(0)}\rangle}{E_{n}^{(0)}-E_{k}^{(0)}}|\psi_{k}\rangle$

Here, $|\psi_{n}^{(0)}\rangle$ is the original wavefunction and $|\psi_{k}^{(0)}\rangle$ is the wavefunction of the $k$ level. This means that the perturbation could lead the original state to have contributions from all other levels. The denominator shows that contributions will decrease as the energy difference increases.

Now there are two integrals to evaluate:

$\langle \psi_{1}^{(0)}|\hat H^{(1)}|\psi_{0}^{(0)}\rangle=\int_{-\infty}^{\infty}\psi_{1}^{(0)}(x)\hat H^{(1)}\psi_{0}^{(0)}(x)dx=0$=odd function

$\langle \psi_{2}^{(0)}|\hat H^{(1)}|\psi_{2}^{(0)}\rangle=\int_{-\infty}^{\infty}\psi_{1}^{(0)}(x)\hat H^{(1)}\psi_{0}^{(0)}(x)dx$

Putting all together, the ground state wavefunction interms of the harmonic oscillator's ground state gives $|\psi_{0}\rangle\approx|\psi_{0}^{(0)}\rangle-\frac{3}{4\sqrt{2}}\frac{\lambda} {\alpha^2(2\hbar \omega+{\frac{39}{16}}{\frac{\lambda}{\alpha^2})}}|\psi_{2}^{(0)}\rangle$

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  • $\begingroup$ Cool! How did it compare with the matrix method you were using before? $\endgroup$ Jan 9 '14 at 17:59
  • $\begingroup$ @ Daniel Shapero,The ground state of this reduced basis set will not be the exact ground state, but by increasing the size of the basis we can improve the accuracy and check if the energy converges as we increase the basis size using the matrix method where as the perturbation method is independent of the number of basis. Actually, my aim is to find the normalized ground state wave function to study the nonlinearity parameters (to study how "quantum mechanical" the system is behaving) $\endgroup$
    – David H
    Jan 23 '14 at 10:11

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