# [QM] Darwin term is not hermitian

1. Jun 24, 2010

### eoghan

Hi there! I'm solving the dirac equation to get the fine structure hamiltonian of the hydrogen atom. In the hamiltonian there is this term:
$$\frac{\hbar ^2 e}{4m^2c^2}\frac{dV}{dr}\frac{\partial}{\partial r}$$

This term gives rise to some difficulty because it is not hermitian. So Darwin proposed to use instead the symmetrical combination:
$$\frac{1}{2}\left[\left(\frac{\hbar ^2 e}{4m^2c^2}\frac{dV}{dr}\frac{\partial}{\partial r}\right)+[\left(\frac{\hbar ^2 e}{4m^2c^2}\frac{dV}{dr}\frac{\partial}{\partial r}\right)^*\right]=\frac{\hbar^2}{8m^2c^2}\nabla^2V$$

but the adjoint of $$\frac{\hbar ^2 e}{4m^2c^2}\frac{dV}{dr}\frac{\partial}{\partial r}$$ is the adjoint of the spatial derivative which is anti-hermitian, so the symmetric combination should be 0... where am I wrong??