# A What are local and non-local operators in QM?

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1. Nov 29, 2017

### cristianbahena

In Hartree-Fock method, I saw the Fock operator has two integrals: Coulomb integral and exchange integral. One can define two operator. "The exchange operator is no local operator" why? Whats de diference: local and no local operator?

And why do the operators have singularities?

thanks

2. Nov 30, 2017

### DrDu

A general one-particle operator $A$ can be defined via it's action on a wavefunction in position representation
$\{A\psi\}(r)=\int d^3r' \alpha(r,r') \psi(r')$.
The function $\alpha(r,r')$ is called the kernel of the operator.
When $\alpha(r,r')=f(r)\delta^3(r-r')$, with Dirac's delta function, we say that the operator is local.
Obviously, the position operator is local with f(r)=r, while for example, the momentum operator is not local as
$\alpha(r,r') =-i\hbar \partial_r \delta(r-r')$.

3. Nov 30, 2017

### Tio Barnabe

Sorry my ignorance, but what should result application of the position operator into the wavefunction as given above?

4. Nov 30, 2017

### DrDu

The wavefunction gets multiplied at each point with the respective value of r.

5. Nov 30, 2017

### cristianbahena

When F(r)= r
One get:

$${A \phi}(r)= r \phi{r}(r)$$ its a eigenvalue equation
when $$f(r)= -i \hbar \partial_r$$

One get:

$${A \phi}(r)= -i \hbar \partial_r \phi{r}(r)$$ it has no sense.
is the why is local or no local
am i right?

Note: i´m using $$A\phi (r)= \int dr´\phi(r) \alpha(r-r´) \phi(r´)$$

6. Nov 30, 2017