Quantum harmonic oscillator with harmonic perturbation

[Evgn]

Homework Statement

Let's consider a harmonic oscillator with a harmonic perturbation:
$$H = \frac{p^2}{2} + \frac{x^2}{2} + a \frac{x^2}{2}.$$
Exact solution is known, but we want to derive it using perturbation theory. More specifically, suppose we want to obtain a series for the ground state energy $E(a)$. This series is known to be a convegent one (when $|a| < 1$) with sum equal to the exact answer ($1/2 \sqrt{1+a}$). We want to comfirm this fact by a direct calculation.

Homework Equations

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The Attempt at a Solution

Well-knows recurrence relations of Rayleigh-Schrödinger theory are applicable, and I've manually checked first five orders. The calculations quickly become very cumbersome, and there seems to be no easy way to obtain a general expression for n-th order (known in advance from the exact answer).

What is the most straightforward way to obtain this series? Will the Kato's theory be of any help in this case?

Thank you in advance.

 

[dextercioby]

Are you using a ladder operator approach ? The calculations should be the simplest.

 

[Evgn]

Sure, I use them to calculate matrix elements of perturbation. This part of calculation is indeed simple. What seems to be not simple is the calculation of the n-th order of perturbation series and not just 1st and 2nd like most textbooks do.

1st order correction is $A |2\rangle$, 2nd order - $B |2\rangle + C |4\rangle$, 3rd order - $D |2\rangle + F |4\rangle + G |6\rangle$ and so on. The number of terms grows with n.

There should be an elegant way to do the calculation even in the framework of an "old" perturbation theory. I deliberately avoid using diagramms here.