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## 物理代写|费曼图代写Feynman diagram代考|Properly symmetrized products as a basis set

Although the products $\phi_{v_1}(1) \phi_{v_2}(2) \ldots \phi_{v_i}(N)$ of single-particle states may serve as a basis for the expansion of the $N$-particle wave function, they are not useful as such. This is because $\Psi(1 \ldots \ldots i \ldots, \ldots . . N)$ must be symmetric (antisymmetric) under the exchange of $i$ and $j$ if the $N$ identical particles are bosons (fermions). The product $\phi_{1_1}(1) \phi_{1,2}(2) \ldots \phi_{1, N}(N)$ lacks this property, and the symmetry (antisymmetry) property must be buried in the constants $C_{Y_1, v_2 \ldots r_Y}$. It is far more convenient to incorporate the appropriate symmetry into the product of the functions, so that $C_{\text {yt }}$ vy…1″ will be completely symmetric upon the exchange of any two indices. For bosons, we can achieve this by summing the product over the $N$ ! permutations of $1,2, \ldots, N$; the basis states are thus given by
$$\Phi_{v_1 v_2 \ldots 1_N}^B(1.2, \ldots, N)=\frac{1}{\prod_\mu \sqrt{n_{k} !}} \frac{1}{\sqrt{N !}} \sum_P \phi_{i_1}[P(1)] \phi_{v_2}[P(2)] \ldots \phi_{v_N}[P(N)]$$
Here $P(1) . P(2), \ldots, P(N)$ is a permutation of $\left[, 2 \ldots, N\right.$, and $n_\mu$ is the number of times the index $\mu$ appears in the product. The factor before the summation ensures that $\Phi^B$ is normalized.

For fermions, a similar expression for the basis states is used, except for the following two modifications. First, $n_\mu$ is either 0 or I (Pauli exclusion principle), so that $n_{\mu} !=1(0 !=1$ and $1 !=1)$. Second, we must insert a minus sign whenever $P(1), P(2) \ldots, P(N)$ is an odd permutation of $1,2, \ldots, N$. The fermionic basis functions are given by
$$\Phi_{v_1 v_2 \ldots v_N}^{\Gamma}(1,2, \ldots . N)=\frac{1}{\sqrt{N !}} \sum_\rho(-1)^P \phi_{v_1}[P(1)] \phi_{v_j}[P(2)] \ldots \phi_{v_M}[P(N)]$$
Equivalently, we may permute the indices instead of the coordinates
$$\Phi_{v_1 v_2 \ldots w_N}^F(1,2, \ldots . N)=\frac{1}{\sqrt{N !}} \sum_P(-1)^P \phi_{P\left(v_1\right)}(1) \phi_{P\left(v_2\right)}(2), \ldots \phi_{P\left(v_N\right)}(N)$$

## 物理代写|费曼图代写Feynman diagram代考|One-body operators

The Hamiltonian for a system of $N$ identical, interacting particles is generally the sum of a one-body operator $\sum_{i=1}^N h(i)$ and a two-body operator $(1 / 2) \sum_{i \neq j} v(i, j)$. For now, we will focus on the one-body operator and give its expression in terms of creation and annihilation operators.

Let $H_0=\sum_{i=1}^N h(i)$, where $h(i)$ is an operator that depends on the coordinates of particle $i$. For example. $h(i)$ could be the kinetic cnergy $-\left(\hbar^2 / 2 m\right) \nabla_i^2$ of particle $i$, or it could be the sum of the kinetic energy and the potential energy $v(i)$ produced by some external field. In general. $h$ may depend on both spatial and spin coordinates.
Suppose that $\left.\left|\phi_1\right\rangle, \mid \phi_2\right}, \ldots$ constitute a complete, orthonormal set of singleparticle states. For example, if for a system of electrons $|\phi\rangle=|\mathbf{k} \sigma\rangle$, the complete set of single-particle states will be $\left|\mathbf{k}_1 \uparrow\right\rangle,\left|\mathbf{k}_1 \downarrow\right\rangle,\left|\mathbf{k}_2 \uparrow\right\rangle, \ldots$

We can express the operator $H_0$ in terms of the creation and annihilation operators $c_v^{\dagger}$ and $c_y$. The derivation of such an expression is somewhat lengthy; it is given in Appendix A. Here, we merely state the result:
$$H_0=\sum_{w{ }^{\prime}}\left\langle\phi_{v^{\prime}}|h| \phi_{1 ;}\right\rangle c_{v^{\prime}}^{\dagger} c_1 .$$
This is the second quantized form of $H_0$, and it holds true for both fermions and bosons. The expression is plausible: a one-body operator is the sum of singleparticle operators $h(1), h(2), \ldots, h(N)$. The effect of a single-particle operator is to scatter a particle from a state $\left|\phi_v\right\rangle$ into a state $\left|\phi_u\right\rangle$. The scattering process can be viewed as the annihilation of a particle in state $\left|\phi_y\right\rangle$, followed by the creation of a particle in state $\left|\phi_{v^{\prime}}\right\rangle$. The amplitude for this process is the matrix element $\left\langle\phi_{v^{\prime}}|h| h \mid \phi_1\right\rangle .$

# 费曼图代考

## 物理代写|费曼图代写Feynman diagram代考|Properly symmetrized products as a basis set

$$\Phi_{v_1 v_2 \ldots v_N}^{\Gamma}(1,2, \ldots N)=\frac{1}{\sqrt{N !}} \sum_\rho(-1)^P \phi_{v_1}[P(1)] \phi_{v_j}[P(2)] \ldots \phi_{v_M}[P(N)]$$

$$\Phi_{v_1 v_2 \ldots w_N}^F(1,2, \ldots N)=\frac{1}{\sqrt{N !}} \sum_P(-1)^P \phi_{P\left(v_1\right)}(1) \phi_{P\left(v_2\right)}(2), \ldots \phi_{P\left(v_N\right)}(N)$$

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