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MathematicalPhysicist
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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.
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