How to simplify the diatomic molecule Hamiltonian using an expansion?

[MathematicalPhysicist]

Homework Statement

I have the diatomic molecule hamiltonian given by:
$$-\hbar^2/(2\mu)d^2/dr^2+\hbar^2\ell(\ell+1)/(2\mu r^2)+(1/4)K(r-d_0)^2$$

Now it's written in my solutions that if we put:
$$K\equiv 2\mu \omega_0^2, \hbar^2\ell(\ell+1)/(2\mu d_0^4)\equiv \gamma_{\ell} K, r-d_0\equiv \rho$$

Expand to second order in $\rho$ and drop terms in $\gamma_{\ell}^2$ since $\gamma_{\ell}\ll 1$, to get:
$$-\hbar^2/(2\mu)d^2/dr^2+(1/2)\mu \omega_0^2[(1+12\gamma_{\ell})(\rho - 4\gamma_{\ell}d_0)^2+4\gamma_{\ell}d_0^2]$$

How to get the last expression explicitly?

Homework Equations

The Attempt at a Solution

I thought of expanding $1/(\rho+d_0)^2 \approx 1/(d_0^2)[1-2\rho/d_0+3\rho^2/d_0^2]$
But I don't see how did they get this expression for the Hamiltonian?
edit: I have edited and corrected the typo.

 

[TSny]

Expand to second order in $\rho$ and drop terms in $\gamma_{\ell} since \gamma_{\ell}\ll 1$, to get:

Due to the typo, we can't tell what order of approximation in $\gamma_{\ell}$ is to be made.

I thought of expanding $1/(\rho+d_0)^2 \approx 1/(d_0^2)[1-2\rho/d_0+3\rho^2/d_0^2]$

Sounds good.

But I don't see how did they get this expression for the Hamiltonian?

The only thing that is required is for their way of writing H to agree with your way of writing H to the assumed orders of approximation in $\rho$ and $\gamma_{\ell}$. They are probably choosing to write H in a particular way in order to simplify further analysis.

 

[MathematicalPhysicist]

Due to the typo, we can't tell what order of approximation in $\gamma_{\ell}$ is to be made.

Sounds good.
The only thing that is required is for their way of writing H to agree with your way of writing H to the assumed orders of approximation in $\rho$ and $\gamma_{\ell}$. They are probably choosing to write H in a particular way in order to simplify further analysis.

I corrected the typo, we should drop terms of order $\gamma_{\ell}^2$ since $\gamma_{\ell}\ll 1$.

 

[TSny]

I corrected the typo, we should drop terms of order $\gamma_{\ell}^2$ since $\gamma_{\ell}\ll 1$.

OK.

You are on the right track to use your expansion $1/(\rho+d_0)^2 \approx 1/(d_0^2)[1-2\rho/d_0+3\rho^2/d_0^2]$ in the expression

$V(r) \doteq \hbar^2\ell(\ell+1)/(2\mu r^2)+(1/4)K(r-d_0)^2$

Then you can manipulate your expression for $V(r)$ into their corresponding expression. (Keep in mind that any terms of order $\gamma_{\ell}^2$ may be neglected.) The motivation for doing all this is that, in their expression, the potential energy part of $H$ is just that of a 1D simple harmonic oscillator.