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# 物理代写|凝聚态物理代写Condensed Matter Physics代考|PH4101 The Born–Oppenheimer adiabatic approximation

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## 物理代写|凝聚态物理代写Condensed Matter Physics代考|The Born–Oppenheimer adiabatic approximation

The cores are much heavier than the electrons; for example, $M \sim 5 \times 10^{4} \mathrm{~m}$ for aluminum. At densities corresponding to typical solids, the cores behave like classical particles, but the electrons form a degenerate electron gas, leading to a difference of orders of magnitude in the kinetic energy of the two species. This difference in mass and kinetic energy implies that the cores are significantly more sluggish than the electrons; therefore the electrons can react almost instantaneously to any motion of the cores.

For most systems, it can be assumed that the electrons feel the instantaneous potential produced by the cores in their “frozen” or “fixed” positions. Therefore, we define an electronic Hamiltonian for fixed cores,
$$H_{e}=\sum_{i}\left[\frac{\mathbf{p}{i}^{2}}{2 m}+\sum{n} V_{n}\left(\mathbf{r}{i}-\mathbf{R}{n}\right)\right]+\frac{1}{2} \sum_{i j}^{\prime} \frac{e^{2}}{\left|\mathbf{r}{i}-\mathbf{r}{j}\right|}+H_{R},$$
and determine its energy spectrum $E_{e}^{l}\left(\left{\mathbf{R}{n}\right}\right)$, where $l$ labels the ith excited state which depends implicitly on the core positions $\left{\mathbf{R}{n}\right}$. In particular, with the electronic system in the ground state, the energy $E_{e}^{0}({\mathbf{R}})$ defines the potential $V_{\mathrm{ec}}({\mathbf{R}})$ within which the cores move. Remaining contributions of Eq. (2.1) can be assigned to the core part of the Hamiltonian,
$$H_{c}=\sum_{n} \frac{\mathbf{p}{n}^{2}}{2 M{n}}+\frac{1}{2} \sum_{n n^{\prime}}^{\prime} \frac{Z_{n} Z_{n^{\prime}} e^{2}}{\left|\mathbf{R}{n}-\mathbf{R}{n^{\prime}}\right|}+V_{\mathrm{ec}}\left(\left{\mathbf{R}{n}\right}\right) .$$ The term $V{\mathrm{ec}}\left(\left{\mathbf{R}{n}\right}\right)$, which is $E{e}^{0}\left(\left{\mathbf{R}_{n}\right}\right)$, represents an electron-core term which can be evaluated once Eq. (2.2) is solved and then used in Eq. (2.3) to determine the energy specturm for the cores.

The procedure described above is the Born-Oppenheimer ${ }^{1}$ adiabatic approximation. It provides the important step of separating the electronic and core degrees of freedom. The electronic part (Eq. (2.2)) leads primarily to the determination of the properties of the electrons, holes, excitons, plasmons, and magnons. The core part (Eq. (2.3)) is used to describe the core motions and phonons. When the phonons are coupled to the electrons, including terms in Eq. (2.1) that go beyond Eqs. (2.2) and (2.3), one can examine polarons, superconductivity, resistivity, and other properties of solids. At this point, we restrict ourselves to the electronic Hamiltonian (Eq. (2.2)).

## 物理代写|凝聚态物理代写Condensed Matter Physics代考|The mean-field approximation

Even though the Hamiltonian (Eq. (2.2)) does not include the cores as dynamical variables, it still contains a very large number of particles, namely, all the valence electrons in the solid. Another approximation is needed. The simplest commonly employed one is the Hartree $^{2}$ mean-field approximation, which assumes that each electron moves in the average or mean field created by the cores together with all the other electrons. In the Hartree approach, one assumes that the electronic wavefunction of all the valence electrons is approximated by a product of one-electron wavefunctions, and each one is characterized by some one-electron spatial and spin quantum numbers. The effects of the Pauli exclusion principle are taken into account by requiring that no pairs of one-electron wavefunctions (orbitals) in the product have an identical set of quantum numbers. As discussed in standard quantum textbooks, demanding that the ground state has the lowest energy for the electronic system results in a set of self-consistent Euler-Lagrange equations (the Hartree equations) for the one-electron orbitals and energies with potential $V\left(\mathbf{r},\left{\mathbf{R}_{n}\right}\right)$.

The Hartree mean-field approximation accomplishes the important task of separating the Hamiltonian (Eq. (2.2)) into a sum of one-electron Hamiltonians
$$H_{e}=\sum_{i} H\left(\mathbf{r}{i},\left{\mathbf{R}{n}\right}\right),$$
where
$$H\left(\mathbf{r},\left{\mathbf{R}{n}\right}\right)=\frac{p^{2}}{2 m}+V\left(\mathbf{r},\left{\mathbf{R}{n}\right}\right) .$$
Another approach involves the Hartree-Fock approximations, which is also a meanfield approach for the electronic ground state. The Hartree-Fock method approximates the wavefunction by a determinant of one-electron orbitals, and thus automatically satisfies all the symmetry requirements of the Pauli principle. However, although more accurate, this approach has several complicating features which often make it inconvenient for many applications. Further discussion of this point appears in Chapter $6 .$

## 物理代写儗聚态物理代写Condensed Matter Physics代考|The Born-Oppenheimer adiabatic approximation

$$H_{e}=\sum_{i}\left[\frac{\mathbf{p} i^{2}}{2 m}+\sum n V_{n}(\mathbf{r} i-\mathbf{R} n)\right]+\frac{1}{2} \sum_{i j}^{\prime} \frac{e^{2}}{|\mathbf{r} i-\mathbf{r} j|}+H_{R},$$

〈left 的分隔符缺失或无法识别 . 特别是，当电子系统处于基态时，能量 $E_{e}^{0}(\mathbf{R})$ 定义湝力 $V_{e c}(\mathbf{R})$ 核心在其中移 动。方程式的剩余贡献。(2.1) 可以分配给哈密顿量的核心部分，
Yleft 的分隔符缺失或无法识别

## 物理代写儗聚态物理代写Condensed Matter Physics代考|The mean-field approximation

Hartree 方法中，假设所有价电子的电子波函数近似为单电子波函数的乘积，并且每个波函数都由一些单电子空间数和自旋量子数 来表征。通过要求乘积中没有一对单电子波函数（轨道）具有相同的一组量子数，考虑了泡利不相容原理的影响。正如标准量子教 科书中所讨论的，要求基态具有电子系统的最低能量会导致一组自洽的欧拉-拉格朗日方程 (哈特里方程) 用于单电子轨道和具有 势能的能荲 left 的分隔符缺失或无法识别
Hartree 平均场近似完成了将哈密顿量 (Eq. (2.2)) 分蓠为单电子哈密顿量之和的重要任务
\left 的分隔符缺失或无法识别

、left 的分隔符缺失或无法识别

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