This question is about an appendix to chapter 7 of Aspects of Symmetry Erice lectures by Sidney Coleman. We have a SE for a 1-dimensional simple harmonic oscillator with $\omega = 1$, describing the bottom of a well of a symmetric double potential well, with vanishing minima centered at $\pm a$

$$-\frac{1}{2} \hbar \, \partial_x^2 \psi \, +\frac{1}{2}(x-a)^2\psi = E\psi $$

for the ground state, we know $E=\frac{\hbar}{2}$ and the even solution $\psi_1 = e^{-(x-a)^2/2\hbar}$. Because of linearity and reflection symmetry, we know there is another solution, $\phi_1$, odd and increasing for which we don't know an analytic form, but for small $\hbar$ its asymptotic approximation is $\phi_1 = e^{(x-a)^2/2\hbar}$. Here he notes that with the normalizations chosen for $\psi_1$ and $\phi_1$, the Wronskian of the two solutions is $$\phi_1 \partial_x \psi_1 - \psi_1 \partial_x \phi_1 = 2/\hbar$$

Now, we define a small dimensionless quantity $\epsilon$ to represent the first order correction to the ground state energy level, by $E=\hbar \, \big(\frac{1}{2} + \epsilon\big)$. Starting from $\psi_1$ and $\phi_1$, solutions to the SE for $\epsilon=0$, we want to find an approximate solution $\psi$ for small $\epsilon$, neglecting $\epsilon^2$ terms. Here Coleman says that this can be achieved by a standard method, namely turning the above SE into an integral equation and iterating once. He immediately writes this solution to be $$\psi = \psi_1 - \epsilon \int_{x}^{\infty} \mathrm{d}x' \, \psi_1(x')\big[\psi_1(x')\phi_1(x)-\phi_1(x')\psi_1(x)\big]$$

How did he derive this? What is this "standard method of integrating and iterating once" he's talking about? I substituted this solution $\psi$ back into the SE and after computing the derivatives, using the Wronskian written above and the fact that we know $\psi_1$,$\phi_1$ to be solutions I was able to verify by inspection that it is indeed the case that this $\psi$ solves the SE, except for terms of order $\epsilon^2$ which we neglect. But I would not be able to derive this $\psi$ by scratch. How did he do it?


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