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# 物理代写|量子力学代写Quantum mechanics代考|PHY350 The Born Approximation

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## 物理代写|量子力学代写Quantum mechanics代考|The Born Approximation

Our task now is to calculate the scattering amplitude $f\left(\mathbf{k}^{\prime}, \mathbf{k}\right)$ for some given potential energy function $V(\mathbf{x})$. This amounts to calculating the matrix element
$$\left\langle\mathbf{k}^{\prime}|V| \psi^{(+)}\right\rangle=\left\langle\mathbf{k}^{\prime}|T| \mathbf{k}\right\rangle .$$
This task is not straightforward, however, since we do not have closed analytic expressions for either $\left\langle\mathbf{x}^{\prime} \mid \psi^{(+)}\right\rangle$or $T$. Consequently, one typically resorts to approximations at this point.

We have already alluded to a useful approximation scheme in (6.32). Again replacing $\hbar \varepsilon$ with $\varepsilon$, this is
$$T=V+V \frac{1}{E-H_0+i \varepsilon} V+V \frac{1}{E-H_0+i \varepsilon} V \frac{1}{E-H_0+i \varepsilon} V+\cdots$$
which is an expansion in powers of $V$. We will shortly examine the conditions under which truncations of this expansion should be valid. First, however, we will make use of this scheme and see where it leads us.

Taking the first term in the expansion, i.e. $T=V$ or, equivalently, $\left|\psi^{(+)}\right\rangle=|\mathbf{k}\rangle$, is called the first-order Born approximation. In this case, the scattering amplitude is denoted by $f^{(1)}$, where
$$f^{(1)}\left(\mathbf{k}^{\prime}, \mathbf{k}\right)=-\frac{m}{2 \pi \hbar^2} \int d^3 x^{\prime} e^{i\left(\mathbf{k}-\mathbf{k}^{\prime}\right) \cdot \mathbf{x}^{\prime}} V\left(\mathbf{x}^{\prime}\right)$$
after inserting a complete set of states $\left|\mathbf{x}^{\prime}\right\rangle$ into (6.58). In other words, apart from an overall factor, the first-order amplitude is just the three-dimensional Fourier transform of the potential $V$ with respect to $\mathbf{q} \equiv \mathbf{k}-\mathbf{k}^{\prime}$.

## 物理代写|量子力学代写Quantum mechanics代考|The Higher-Order Born Approximation

Now, write $T$ to second order in $V$, using (6.71), namely
$$T=V+V \frac{1}{E-H_0+i \varepsilon} V .$$
It is natural to continue our Born approximation approach and write
$$f\left(\mathbf{k}^{\prime}, \mathbf{k}\right) \approx f^{(1)}\left(\mathbf{k}^{\prime}, \mathbf{k}\right)+f^{(2)}\left(\mathbf{k}^{\prime}, \mathbf{k}\right)$$

where $f^{(1)}\left(\mathbf{k}^{\prime}, \mathbf{k}\right)$ is given by $(6.72)$ and
\begin{aligned} f^{(2)}=&-\frac{1}{4 \pi} \frac{2 m}{\hbar^2}(2 \pi)^3 \int d^3 x^{\prime} \int d^3 x^{\prime \prime}\left\langle\mathbf{k}^{\prime} \mid \mathbf{x}^{\prime}\right\rangle V\left(\mathbf{x}^{\prime}\right) \ & \times\left\langle\mathbf{x}^{\prime}\left|\frac{1}{E-H_0+i \varepsilon}\right| \mathbf{x}^{\prime \prime}\right\rangle V\left(\mathbf{x}^{\prime \prime}\right)\left(\mathbf{x}^{\prime \prime} \mid \mathbf{k}\right) \ =&-\frac{1}{4 \pi} \frac{2 m}{\hbar^2} \int d^3 x^{\prime} \int d^3 x^{\prime \prime} e^{-i \mathbf{k}^{\prime} \cdot x^{\prime}} V\left(\mathbf{x}^{\prime}\right) \ & \times\left[\frac{2 m}{\hbar^2} G_{+}\left(\mathbf{x}^{\prime}, \mathbf{x}^{\prime \prime}\right)\right] V\left(\mathbf{x}^{\prime \prime}\right) e^{i \mathbf{k} \cdot \mathbf{x}^{\prime \prime}} \end{aligned}
This scheme can obviously be continued to higher orders
A physical interpretation of (6.86) is given in Figure 6.7, where the incident wave interacts at $\mathbf{x}^{\prime \prime}$, which explains the appearance of $V\left(\mathbf{x}^{\prime \prime}\right)$, and then propagates from $\mathbf{x}^{\prime \prime}$ to $\mathbf{x}^{\prime}$ via Green’s function for the Helmholtz equation (6.48). Subsequently, a second interaction occurs at $\mathbf{x}^{\prime}$, thus the appearance of $V\left(\mathbf{x}^{\prime}\right)$, and, finally, the wave is scattered into the direction $\mathbf{k}^{\prime}$. In other words, $f^{(2)}$ corresponds to scattering viewed as a two-step process. Likewise, $f^{(3)}$ can be viewed as a three-step process, and so on.

## 物理代写|量子力学代写Quantum mechanics代考|The Born Approximation

$$\left\langle\mathbf{k}^{\prime}|V| \psi^{(+)}\right\rangle=\left\langle\mathbf{k}^{\prime}|T| \mathbf{k}\right\rangle .$$

$$T=V+V \frac{1}{E-H_0+i \varepsilon} V+V \frac{1}{E-H_0+i \varepsilon} V \frac{1}{E-H_0+i \varepsilon} V+\cdots$$

$$f^{(1)}\left(\mathbf{k}^{\prime}, \mathbf{k}\right)=-\frac{m}{2 \pi \hbar^2} \int d^3 x^{\prime} e^{i\left(\mathbf{k}-\mathbf{k}^{\prime}\right) \cdot \mathbf{x}^{\prime}} V\left(\mathbf{x}^{\prime}\right)$$

## 物理代写量子力学代写Quantum mechanics代考|The Higher-Order Born

Approximation

$$T=V+V \frac{1}{E-H_0+i \varepsilon} V .$$

$$f\left(\mathbf{k}^{\prime}, \mathbf{k}\right) \approx f^{(1)}\left(\mathbf{k}^{\prime}, \mathbf{k}\right)+f^{(2)}\left(\mathbf{k}^{\prime}, \mathbf{k}\right)$$

$$f^{(2)}=-\frac{1}{4 \pi} \frac{2 m}{\hbar^2}(2 \pi)^3 \int d^3 x^{\prime} \int d^3 x^{\prime \prime}\left\langle\mathbf{k}^{\prime} \mid \mathbf{x}^{\prime}\right\rangle V\left(\mathbf{x}^{\prime}\right) \quad \times\left\langle\mathbf{x}^{\prime}\left|\frac{1}{E-H_0+i \varepsilon}\right| \mathbf{x}^{\prime \prime}\right\rangle V\left(\mathbf{x}^{\prime \prime}\right)\left(\mathbf{x}^{\prime \prime} \mid \mathbf{k}\right)=-\frac{1}{4 \pi} \frac{2 m}{\hbar^2} \int d^3 x^{\prime} \int d^3 x^{\prime \prime} e^{-i \mathbf{k}^{\prime} \cdot x^{\prime}} V\left(\mathbf{x}^{\prime}\right)$$

• (6.86) 的物理解释在图 $6.7$ 中给出，其中入射波在 $\mathbf{x}^{\prime \prime}$, 这解释了 $V\left(\mathbf{x}^{\prime \prime}\right)$, 然后从 $\mathbf{x}^{\prime \prime}$ 至 $\mathbf{x}^{\prime}$ 通过亥姆霍兹方程 (6.48) 的格林函
数。随后，第二次交互发生在 $\mathbf{x}^{\prime}$ ，因此出现 $V\left(\mathbf{x}^{\prime}\right)$ ，最后，波被散射到方向 $\mathbf{k}^{\prime}$. 换句话脱， $f^{(2)}$ 对应于被视为两步过程的散射。
同样地, $f^{(3)}$ 可以看作是 个三步过程，依此类推。

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