[Physics] The virial theorem and a delta function potential

classical-mechanicsdirac-delta-distributionsquantum mechanicsstatistical mechanicsvirial-theorem

So the virial theorem tells us that:

$2\langle T\rangle = \langle \textbf{r}\cdot\nabla V\rangle$.

Now I was wondering what would happen if V has te form:

$V(\textbf{r}-\textbf{r}') = V_0\delta^{(D)}(\textbf{r}-\textbf{r}')$, where $\delta^{(D)}(\textbf{r}-\textbf{r}')$ is the delta-function in D dimensions. I'm not sure why, but I think that I should get that:

$\langle \textbf{r}\cdot\nabla V\rangle = \frac{1}{D}\langle V_0\rangle$ since the delta written out as a product of different components is:

$\delta^{(D)}(\textbf{r}-\textbf{r}') = \frac{1}{\sqrt{det(G)}}\prod\limits_{i=1}^D\delta(x_i-x_i')$, with $x_i$ the different components of the vector $\textbf{r}$, given in the base with metric G, where $\sqrt{det(G)}$ gives the D-dimensional volume-element in the basis $\textbf{e}_i$.

I don't know wether there is a more rigorous reasoning for this? Or wether this is even correct ?

Addendum: a different perspective:

Another way to look at it, is that if I rescale my vector $\mathbb{r}$ bij a factor $\lambda$, I get:

$\delta^{(D)}(\lambda\textbf{r}-\lambda\textbf{r}') = \frac{1}{\lambda^D}\delta^{(D)}(\textbf{r}-\textbf{r}')$. This makes me also think that i should get the above relation for the virial theorem. But still I'm not sure of my reasoning !

Extra demand on potential (necessary for finite system)

Next to my delta-potential, I also have an extra confining potential to keep the particles together. For simplicity I'll take an harmonic trap $V(r) = \frac{1}{2}m\omega^2r^2$ which keeps the particles together! So this is the other term of the potential, but this one I didn't consider in my question because that one posed no problem to my calculations!

Best Answer

Comments to the question (v8):

  1. The Virial theorem usually apply to periodic or bounded systems, but pairwise attractive delta function potentials would not constitute a bounded system unless the system is additionally confined in a box. (OP has in an update (v9) of the question introduced an additional potential to confine the particles.)

  2. If we focus on one pairwise interaction (out of the many pairwise interactions), then the attractive delta function potential $$\tag{A} V(r)~:=~ -A\delta^d(\vec{r}),\qquad A~>~0$$ is classically ill-defined, and needs to be regularized. One could hope that quantum mechanical smearing of the wave function renders the potential (A) well-defined. However, this is not possible for $d>2$: The potential (A) is quantum mechanically unbounded from below for $d>2$ (See e.g. this Phys.Se post. The limiting dimension $d=2$ case is only bounded from below for sufficiently weak attractive delta function potentials (A).)

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