# How small for perturbation theory to be valid?

1. Aug 5, 2010

### czaroffishies

1. The problem statement, all variables and given/known data

A particle of mass m is in the ground state in the harmonic oscillator potential

V(x) = $$\frac{1}{2}Kx^{2}$$

A small perturbation $$\beta x^{6}$$ is added to this potential.

How small must $$\beta$$ be in order for perturbation theory to be valid?

2. Relevant equations

All here:
http://en.wikipedia.org/wiki/Pertur...chanics)#Time-independent_perturbation_theory

3. The attempt at a solution

Well, this is kind of a conceptual question, and I'm not sure how to start. It feels like I am guessing.

All I know is that:

$$\beta x^{6} << \frac{1}{2}Kx^{2}$$

I would really appreciate a pointer in the right direction... thanks =)

2. Aug 6, 2010

### nrqed

Well, these are operators so it is not well-defined to write an inequality between operators.
What you must do is impose that the first order correction to the energy due to the perturbation must much smaller than the unperturbed energies.