# Perturbations and harmonic oscillator

## Homework Statement

In a diatomic molecule with atom masses m1 and m2, the atoms are bound by a potential
$$V(r)=V_0\big[ \big(\frac{r_0}r\big)^{12} - 2\big(\frac{r_0}r\big)^{6}\big]$$
where r is the distance between the atom centra, r0 is the equilibrium istance, and V0 is a constant depending on the atoms.

1) Expand V(r) in series close to r0 and keep only the term that corresponds to a harmonic oscillator. Find for this case the ground state wave function and the corresponding energy.

2) Keep another term in the series of V(r) that gives a contribution to the ground state energy calculated with first order perturbation theory. Find the new (approximate) energy for this state.

## Homework Equations

Wave function for harmonic oscillator, ground state:
$$\psi_0(x) = \big(\frac{m \omega}{\pi \hbar}\big)^{1/4} e^{-\frac{m \omega}{2\hbar}x^2}$$

$$E_0^1 = <\psi_0^0 | H' |\psi_0^0>$$
where
$$E_0^1=$$ ground state energy with first order perturbation

$$\psi_0^0=$$ ground state wave function in unperturbed state

$$H'=$$ the perturbation

## The Attempt at a Solution

Serialize V (I'll drop the ordo-terms):
$$V(r)= V(r_0)+V'(r_0)(r-r_0) + \frac 12 V''(r_0) (r-r_0)^2$$
V'(r0)=0 because it's the equilibrium point
V(r0) = constant and can be removed since this doesn't change the force
Conclusion:
$$V(r) = \frac 12 V''(r_0) (r-r_0)^2$$
$$V''(r_0)=\frac{72}{r_0^2}V_0$$
$$V(r) = \frac{36}{r_0^2}V_0 (r-r_0)^2 = \alpha (r-r_0)^2$$
So compared with the harmonic oscillator's
$$V(x) = \frac 12 \mu \omega^2x^2$$
I get that
$$\alpha = \frac {\mu \omega^2}2 \rightarrow \omega = \frac 6r_0 \sqrt{\frac{2V_0}{\mu}} \rightarrow E=\frac 12 \hbar \omega = 3\hbar \sqrt{\frac{2V_0}{r_0^2 \mu}}$$

So far all good...

The wave function should become
$$\psi_0(r) = \big( \frac{6\sqrt{2\mu V_0}}{\pi r_0 \hbar}\big)^{1/4} e^{-\frac{3\sqrt{2\mu V_0}}{r_0 \hbar}r^2}$$
1) the given wave function under 2 is in x, and has been normalized with an integral from -infinity to infinity. As I'm using r, and it's a distance between two atoms, it should be 0 to infinity the get the correct wavefunction.
2) the given wave function is for an oscillator with r0=0. Is it correct to use r in the exponential function or should it be r-r0?

That was for part 1. Second question:
Serialize like before but keep the third order term...
$$V'''(r_0) = -\frac{1512}{r_0^3}V_0$$
$$V(r) = \frac{36}{r_0^2}V_0 (r-r_0)^2 - 252\frac{V_0}{r_0^3} (r-r_0)^3$$
Last term will be H' - the perturbation.
$$E_0^1 = <\psi_0^0 | H' |\psi_0^0> = -252 \frac{V_0}{r_0^3} \big(\frac{6\sqrt{2\mu V_0}}{\pi r_0 \hbar}\big)^{1/4} \int_{-\infty}^\infty e^{- \frac{6\sqrt{2\mu V_0}}{r_0 \hbar}r^2}(r-r_0)^3 dr = 252 V_0 + \frac{63\hbar}{r_0} \sqrt{\frac{V_0}{2\mu}}$$
Once again I'm confused about the integration limits.

And this is definitely not the correct answer. If I would divide the energy from the first question with the perturbed one, I would get a constant term of 21/4 that is way too great to make sense.