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# 物理代写|量子力学代写Quantum mechanics代考|The eikonal approximation

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## 物理代写|量子力学代写Quantum mechanics代考|The eikonal approximation

When $V(r)$ is a slowly varying function in $r$, and the incoming energy is sufficiently large so as to satisfy $\hbar^2 k^2 / 2 m \gg V(r)$, we can find an approximation based on semi-classical methods. This is the basis of the eikonal approximation discussd in this section. The wavefunction may be written in the form
$$\Psi_{\mathbf{k}}(\mathbf{r})=\frac{1}{(2 \pi)^{3 / 2}} e^{i S(\mathbf{k}, \mathbf{r}) / \hbar}$$
for a complex $S(\mathbf{k}, \mathbf{r})$. Then the Schrödinger equation reduces to an equation for $S(\mathbf{k}, \mathbf{r})$
$$(\nabla S)^2-i \hbar \nabla^2 S=2 m(E-V(r))$$
Since $V(r)$ is slowly varying with $r$, one expects that $S(\mathbf{k}, \mathbf{r})$ is also a slowly varying function in $r$, and neglect second order derivatives of $S(\mathbf{k}, \mathbf{r})$. This can also be justified on the basis of large momenta which is represented by the first derivative term $(\nabla S)^2$. Under this assumption the equation reduces to the classical Hamilton-Jacobi equation
$$(\nabla S)^2=2 m(E-V(r))$$

This is like the WKB approximation, but now it is argued on the basis of the large incoming energy, rather than a small $\hbar$. The solution to this equation will be taken as the approximation for $\Psi_{\mathbf{k}}(\mathbf{r})$ for the purpose of computing the scattering amplitude
\begin{aligned} f\left(\mathbf{k}, \mathbf{k}^{\prime}\right) & =-\frac{1}{4 \pi} \int d^3 \mathbf{r} e^{-i \mathbf{k}^{\prime} \cdot \mathbf{r}} v(r)(2 \pi)^{3 / 2} \Psi_{\mathbf{k}}(\mathbf{r}) \ & =-\frac{1}{4 \pi} \int d^3 \mathbf{r} e^{-i \mathbf{k}^{\prime} \cdot \mathbf{r}} v(r) e^{i S(\mathbf{k}, \mathbf{r}) / \hbar} \end{aligned}

## 物理代写|量子力学代写Quantum mechanics代考|Partial waves

Let us consider scattering from a rotationaly invariant potential $V=V(r)$. In our previous discussion the beam of incoming particles was represented by the momentum ket $|\mathbf{k}\rangle$. Since the Hamiltonian $H_0=\mathbf{p}^2 / 2 m$ commutes with $\mathbf{L}^2$ and $L_z$, the free particle can also be represented in the angular momentum and energy basis as $|k l m\rangle$. One basis has an axpansion in terms of the other
$$|\mathbf{k}\rangle=\sum_{l m}|k l m\rangle\langle l m \mid \hat{\mathbf{k}}\rangle=\sum_{l m}|k l m\rangle Y_{l m}(\hat{\mathbf{k}})$$
where $E=\hbar^2 k^2 / 2 m$. The total Hamiltonian, including the potential is also invariant under rotations. By the same arguments, the exact eigenstate $|\mathbf{k}\rangle^{+}$ can also be expressed in the angular momentum basis $|k l m\rangle^{+}$, and have the expansion
$$|\mathbf{k}\rangle^{+}=\sum_{l m}|k l m\rangle^{+}\langle l m \mid \hat{\mathbf{k}}\rangle=\sum_{l m}|k l m\rangle^{+} Y_{l m}(\hat{\mathbf{k}})$$
Since $\mathbf{L}$ commutes with the total Hamiltonian, angular momentum is conseerved during the scattering process.

Let us see what is the advantage of using the angular momentum basis. Consider the following sketch of a beam of particles hitting the target of a size determined by a radius $a$. Semi-classically, the radius $a$ represents the range of the interaction potential.

## 物理代写|量子力学代写Quantum mechanics代考|The eikonal approximation

$$\Psi_{\mathbf{k}}(\mathbf{r})=\frac{1}{(2 \pi)^{3 / 2}} e^{i S(\mathbf{k}, \mathbf{r}) / \hbar}$$

$$(\nabla S)^2-i \hbar \nabla^2 S=2 m(E-V(r))$$

$$(\nabla S)^2=2 m(E-V(r))$$

$$f\left(\mathbf{k}, \mathbf{k}^{\prime}\right)=-\frac{1}{4 \pi} \int d^3 \mathbf{r} e^{-i \mathbf{k}^{\prime} \cdot \mathbf{r}} v(r)(2 \pi)^{3 / 2} \Psi_{\mathbf{k}}(\mathbf{r}) \quad=-\frac{1}{4 \pi} \int d^3 \mathbf{r} e^{-i \mathbf{k}^{\prime} \cdot \mathbf{r}} v(r) e^{i S(\mathbf{k}, \mathbf{r}) / \hbar}$$

## 物理代写|量子力学代写Quantum mechanics代考|Partial waves

$$|\mathbf{k}\rangle=\sum_{l m}|k l m\rangle\langle l m \mid \hat{\mathbf{k}}\rangle=\sum_{l m}|k l m\rangle Y_{l m}(\hat{\mathbf{k}})$$

$$|\mathbf{k}\rangle^{+}=\sum_{l m}|k l m\rangle^{+}\langle l m \mid \hat{\mathbf{k}}\rangle=\sum_{l m}|k l m\rangle^{+} Y_{l m}(\hat{\mathbf{k}})$$

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