I Form of potential operator of two interacting particles

Kashmir
Messages
466
Reaction score
74
Considering two interacting particles in 3d, the corresponding Hilbert space ##H## is the tensor product of the two individual Hilbert spaces of the two particles.
If the particle interaction is given by a potential ##V(\mathbf r_1 -\mathbf r_2)## ,what is the corresponding potential operator for it?
 
Physics news on Phys.org
It's most simple in the position representation
$$V(\hat{\vec{r}}_1-\hat{\vec{r}_2}) \psi(\vec{r}_1,\vec{r}_2)=V(\vec{r}_1-\vec{r}_2) \psi(\vec{r}_1,\vec{r}_2).$$
 
vanhees71 said:
It's most simple in the position representation
$$V(\hat{\vec{r}}_1-\hat{\vec{r}_2}) \psi(\vec{r}_1,\vec{r}_2)=V(\vec{r}_1-\vec{r}_2) \psi(\vec{r}_1,\vec{r}_2).$$
Thank you. I was looking for the abstract expression involving the tensor product of individual spaces.
 
If you write it in this way, it just reads
$$V(\hat{\vec{r}}_1 \otimes \hat{1} - \hat{1} \otimes \hat{\vec{r}}_2) |\vec{r}_1 \rangle \otimes |\vec{r}_2 \rangle = V(\vec{r}_1-\vec{r}_2) |\vec{r}_1 \rangle \otimes |\vec{r}_2 \rangle.$$
Now you use it in the definition of the two-particle wave function,
$$\psi(\vec{r}_1,\vec{r}_2) = \langle \vec{r}_1| \otimes \langle \vec{r}_2|\psi \rangle.$$
 
Not an expert in QM. AFAIK, Schrödinger's equation is quite different from the classical wave equation. The former is an equation for the dynamics of the state of a (quantum?) system, the latter is an equation for the dynamics of a (classical) degree of freedom. As a matter of fact, Schrödinger's equation is first order in time derivatives, while the classical wave equation is second order. But, AFAIK, Schrödinger's equation is a wave equation; only its interpretation makes it non-classical...
I am not sure if this falls under classical physics or quantum physics or somewhere else (so feel free to put it in the right section), but is there any micro state of the universe one can think of which if evolved under the current laws of nature, inevitably results in outcomes such as a table levitating? That example is just a random one I decided to choose but I'm really asking about any event that would seem like a "miracle" to the ordinary person (i.e. any event that doesn't seem to...
Back
Top