Suppose I have a system of N identical bosons interacting via pairwise potential [itex]V(\vec{x} - \vec{x}')[/itex].(adsbygoogle = window.adsbygoogle || []).push({});

I want to show that the expectation of the Hamiltonian in the non-interacting ground state is

[itex]\frac{N(N-1)}{2\mathcal{V}}\widetilde{V}(0)[/itex]

where

[itex]\widetilde{V}(q) = \int d^3 \vec{x} e^{i \vec{q} \cdot \vec{x}}(\vec{x})[/itex]

and [itex]\mathcal{V}[/itex] is the volume of the `box'.

My attempt:

First I need to find the ground state in the absence of potential.

The second-quantized Hamiltonian is

[itex]\hat{H} = \int d^{3}\vec{x} \hat{\psi}^\dag (\vec{x})\left( -\frac{\hbar^2}{2m}\nabla_{\vec{x}}^2 \right) \hat{\psi}(\vec{x}) \quad +\quad \int\int d^3\vec{x}d^3\vec{x}'\hat{\psi}^\dag(\vec{x}')\hat{\psi}^\dag(\vec{x})V(\vec{x},\vec{x}')\hat{\psi}(\vec{x})\hat{\psi}(\vec{x}')[/itex]

Set V = 0 and then

[itex]\hat{H} = \int d^{3}\vec{x} \hat{\psi}^\dag (\vec{x})\left( -\frac{\hbar^2}{2m}\nabla_{\vec{x}}^2 \right) \hat{\psi}(\vec{x}) [/itex]

Now use the definition [itex]\hat{\psi}(x) \equiv \sum_{\lambda} \langle \vec{x} | a^{(\lambda)} \rangle \hat{a}_{\lambda} , \quad \hat{\psi}(x) \equiv \sum_{\lambda} \langle a^{(\lambda)} | \vec{x}} \rangle \hat{a}^\dag_{\lambda}[/itex]

where [itex]a_\lambda,a^\dag_\lambda[/itex] are the annihilation and creation operators that subtract or add a particle to the single-particle state [itex]|a^{(\lambda)}\rangle[/itex].

Now I'm going to let the single-particle states be momentum eigenstates so

[itex]\hat{\psi}(\vec{x}) = \frac{1}{\sqrt{\mathcal{V}}}\sum_{\vec{k}} e^{i\vec{k} \cdot\vec{x}} \hat{a}_{\vec{k}}[/itex]

Plugging this in and using the fact that [itex]\int d^3\vec{x} e^{i(\vec{k}' - \vec{k})\cdot\vec{x}} = \delta_{\vec{k},\vec{k}'}[/itex] gives

[itex]\hat{H} = \sum_{\vec{k}} \frac{\hbar^2 \vec{k}^2}{2m}\hat{a}^\dag_{\vec{k}}\hat{a}_\vec{k}[/itex]

so the eigenstates of the non-interacting Hamiltonian are the occupation number states [itex]| n_{\vec{k}_1},n_{\vec{k}_2},\ldots\rangle[/itex]

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# 2nd quantization

**Physics Forums | Science Articles, Homework Help, Discussion**