# In the quantum hamiltonian, why does kinetic energy turn into an operator while potential doesn't?

When we go from the classical many-body hamiltonian

$$H = \sum_i \frac{\vec{p}_i^2}{2m_e} - \sum_{i,I} \frac{Z_I e^2 }{|\vec{r}_i - \vec{R}_I|} + \frac{1}{2}\sum_{i,j} \frac{ e^2 }{|\vec{r}_i - \vec{r}_j|} + \sum_I \frac{\vec{p}_I^2}{2M_I}+ \frac{1}{2}\sum_{I,J} \frac{Z_IZ_J e^2 }{|\vec{R}_I - \vec{R}_J|}$$

to the quantum many-body hamiltonian

$$H = -\sum_i \frac{\hbar^2}{2m_e}\nabla_i^2 - \sum_{i,I} \frac{Z_I e^2 }{|\vec{r}_i - \vec{R}_I|} + \frac{1}{2}\sum_{i,j} \frac{ e^2 }{|\vec{r}_i - \vec{r}_j|} - \sum_I \frac{\hbar^2}{2M_I} \nabla_I^2+ \frac{1}{2}\sum_{I,J} \frac{Z_IZ_J e^2 }{|\vec{R}_I - \vec{R}_J|}$$

only the kinetic energy parts turn into operators. I mean the other parts are also operators but merely numbers.

Why is this the case? My guess is, it has to be with the representation we are working with, but that's as far as I go, I don't know how it affects.

Can someone give a heuristic explanation also?

• The potential operator is actually: $\hat{V} \equiv V(x)\hat{I}$ with $\hat{I}$ the identity operator for the 1D case here. Often you omit the $\hat{I}$. Mar 11, 2019 at 14:57

Actually the potential is also an operator. It just so happens that, in the position representation, $$\hat x\psi(x)=x\psi(x)$$, so that the potential energy operator $$V(\hat x)$$ acts by multiplication: $$V(\hat x)\psi(x)=V(x)\psi(x)$$.
In the momentum representation $$\hat x$$ acts by differentiation so in this case the potential energy operator becomes a (usually quite complicated) differential operator since one needs to use the formal expansion of potential to convert it to a polynomial.
• Pseudodifferential operators are not complicated, since they are conveniently defined by Fourier transform. And acting on $L^2$, as in quantum mechanics, they can even be defined directly by spectral calculus. No power expansion (formal or not) of the function is needed, only measurability (essentially with respect to the Lebesgue measure). Mar 11, 2019 at 12:30