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## 物理代写|统计力学代写Statistical mechanics代考|Approximate localization for multi-particle states

In the preceding subsection a loop expansion for the symmetrization factor was given that in the case of single-particle energy states allowed an infinite resummation of the partition function that gave the grand potential as a simple series of loop potentials. Here is briefly discussed an approximate treatment of multi-particle energy states that allows a similar factorization and resummation. Here we focus on a dimer permutation, which is sufficient to illustrate the idea.

For $N$ particles in the multi-particle energy state $\mathbf{n}$, the transposition of particles $j$ and $k$ gives a dimer symmetrization or overlap factor
\begin{aligned} \chi_{\mathbf{n} ; j k}^{\pm,(2)} &=\pm\left\langle\zeta_{\mathbf{n}}\left(\hat{P}{j k} \mathbf{r}\right) \mid \zeta{\mathbf{n}}(\mathbf{r})\right\rangle \ &=\pm \int \mathrm{d} \mathbf{r} \zeta_{\mathrm{n}}\left(\ldots \mathbf{r}{k} \ldots \mathbf{r}{j \ldots} \ldots\right)^{} \zeta_{\mathrm{n}}\left(\ldots \mathbf{r}{\left.j \ldots \mathbf{r}{k} \ldots\right)} .\right. \end{aligned}
The question addressed in the present subsection is to what extent the states $\mathbf{n}$ that give a non-zero overlap factor correspond to a localization of the permuted particles’ states. For example, ideally the inner product of the non-permuted particles would cancel these particles out leaving just a two-particle state,
$$\chi_{\mathbf{n}, j k}^{\pm,(2)} \approx \pm \int \mathrm{d} \mathbf{r}{j} \mathrm{~d} \mathbf{r}{k} \zeta_{\mathbf{n}{j k}}\left(\mathbf{r}{k}, \mathbf{r}{j}\right)^{} \zeta{\mathbf{n}{j k}}\left(\mathbf{r}{j}, \mathbf{r}_{k}\right) .$$
Obviously this would be exact if the eigenfunction were the product of one-particle eigenfunctions, as for the non-interacting subsystem analyzed above in sections $3.4 .1 .2$ and $3.4 .3$.

## 物理代写|统计力学代写Statistical mechanics代考|Single-particle states

For ideal particles, the potential energy is zero, $U(\mathbf{r})=0$, and the energy operator is just the kinetic energy operator,
$$\hat{\mathcal{H}}^{\mathrm{id}}(\mathbf{r})=\hat{\mathcal{K}}(\mathbf{r})=\frac{-\hbar^{2}}{2 m} \nabla^{2}=\frac{-\hbar^{2}}{2 m} \sum_{j=1}^{N} \nabla_{j}^{2} .$$
The energy eigenfunction equation is
$$\hat{\mathcal{H}}^{\mathrm{id}}(\mathbf{r}) \zeta_{\mathrm{n}}(\mathbf{r})=\mathcal{H}{\mathrm{n}} \zeta{\mathrm{n}}(\mathbf{r})$$
For the ideal gas, the energy eigenfunctions are just momentum eigenfunctions, which are the product of plane waves,$$\zeta_{\mathbf{n}}(\mathbf{r})=\frac{1}{V^{N / 2}} \prod_{j=1}^{N} e^{\mathrm{i}{j} \mathbf{r}{j} / h}, \quad \mathbf{p}{j}=\frac{2 \pi \hbar}{L} \mathbf{n}{j}, \quad n_{j \alpha}=0, \pm 1, \pm 2, \ldots$$
Here $\mathbf{p}{j}$ is the momentum eigenvalue of of particle $j, \mathbf{n}$ is a $3 \mathrm{~N}$-dimensional integer vector, $\alpha \in{x, y, z}$, and $V=L^{3}$ is the volume of the subsystem. The energy eigenvalues are $$\mathcal{H}{\mathbf{n}}=\frac{1}{2 m} \sum_{j=1}^{N} p_{j}^{2} .$$
The symmetrization or overlap factor for an l-loop, equation (3.49), is
\begin{aligned} \chi_{\mathbf{n}}^{\pm,(l)} &=(\pm 1)^{l-1}\left\langle\zeta_{\mathbf{n}}\left(\mathbf{r}{2}, \mathbf{r}{3}, \ldots, \mathbf{r}{l}, \mathbf{r}{\mathbf{1}}\right) \mid \zeta_{\mathbf{n}}\left(\mathbf{r}{1}, \mathbf{r}{2}, \ldots, \mathbf{r}{l}\right)\right\rangle \ &=(\pm 1)^{l-1} \frac{1}{V^{l}} \int \mathrm{d} \mathbf{r} \prod{j=1}^{l} e^{\mathbf{i p}{j}\left[\mathbf{r}{j}-\mathbf{r}_{j-1}\right] / \hbar}, \end{aligned}

# 统计力学代考

## 物理代写|统计力学代写Statistical mechanics代考|Approximate localization for multi-particle states

$$\chi_{\mathbf{n} ; j k}^{\pm,(2)}=\pm\left\langle\zeta_{\mathbf{n}}(\hat{P} j k \mathbf{r}) \mid \zeta \mathbf{n}(\mathbf{r})\right\rangle \quad=\pm \int \mathrm{d} \mathbf{r} \zeta_{\mathbf{n}}(\ldots \mathbf{r} k \ldots \mathbf{r} j \ldots \ldots) \zeta_{\mathbf{n}}(\ldots \mathbf{r} j \ldots \mathbf{r} k \ldots)$$

$$\chi_{\mathbf{n}, j k}^{\pm,(2)} \approx \pm \int \mathrm{d} \mathbf{r} j \mathrm{~d} \mathbf{r} k \zeta_{\mathbf{n} j k}(\mathbf{r} k, \mathbf{r} j) \zeta \mathbf{n} j k\left(\mathbf{r} j, \mathbf{r}_{k}\right)$$

## 物理代写|统计力学代写Statistical mechanics代考|Single-particle states

$$\hat{\mathcal{H}}^{\mathrm{id}}(\mathbf{r})=\hat{\mathcal{K}}(\mathbf{r})=\frac{-\hbar^{2}}{2 m} \nabla^{2}=\frac{-\hbar^{2}}{2 m} \sum_{j=1}^{N} \nabla_{j}^{2} .$$

$$\hat{\mathcal{H}}^{\mathrm{id}}(\mathbf{r}) \zeta_{\mathrm{n}}(\mathbf{r})=\mathcal{H} \mathrm{n} \zeta \mathrm{n}(\mathbf{r})$$

$$\zeta_{\mathbf{n}}(\mathbf{r})=\frac{1}{V^{N / 2}} \prod_{j=1}^{N} e^{\mathrm{i} j \mathbf{r} j / h}, \quad \mathbf{p} j=\frac{2 \pi \hbar}{L} \mathbf{n} j, \quad n_{j \alpha}=0, \pm 1, \pm 2, \ldots$$

$$\mathcal{H} \mathbf{n}=\frac{1}{2 m} \sum_{j=1}^{N} p_{j}^{2}$$

$$\chi_{\mathbf{n}}^{\pm,(l)}=(\pm 1)^{l-1}\left\langle\zeta_{\mathbf{n}}(\mathbf{r} 2, \mathbf{r} 3, \ldots, \mathbf{r} l, \mathbf{r} 1) \mid \zeta_{\mathbf{n}}(\mathbf{r} 1, \mathbf{r} 2, \ldots, \mathbf{r} l)\right\rangle \quad=(\pm 1)^{l-1} \frac{1}{V^{l}} \int \mathrm{d} \mathbf{r} \prod j=1^{l} e^{\mathbf{i} \mathbf{p} j[\mathbf{r} j-\mathbf{r} j 11] / \hbar},$$

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