Born Approximation and Its Validity

Hello! Welcome back to our module on Quantum Scattering Theory.

In our last lesson, we successfully reformulated the Schrödinger equation into the Lippmann-Schwinger equation, an integral equation that elegantly incorporates the boundary conditions for a scattering experiment. We arrived at this exact, but implicit, equation for the scattering state $|{\psi }^{(+)}⟩$:

$$|{\psi }^{(+)}⟩=|\mathbf{k}⟩+\frac{1}{E-H_0+iϵ}V|{\psi }^{(+)}⟩$$

While exact, this form isn't an explicit solution because $|{\psi }^{(+)}⟩$ appears on both sides. Its real power lies in paving the way for approximate solutions. Today, we will develop the most fundamental of these approximations.

This lesson addresses the learning outcome: Analyze scattering in the Born approximation and determine its validity regime. We will derive an iterative solution to the Lippmann-Schwinger equation, known as the Born series, and focus on its first and most important term. This will give us a practical tool to calculate scattering cross-sections and connect directly to experimental observables. For your work in condensed matter physics, this approximation is the cornerstone for understanding how probes like X-rays, neutrons, and even electrons themselves scatter within a material, revealing its structure and properties.

1. The Born Series: An Iterative Approach

The Lippmann-Schwinger equation is of the form $\psi =\varphi +K\psi$, where $\varphi$ is the known incident wave and $K$ is an integral operator. The key idea of the Born series is to solve this by iteration.

The physical reasoning is as follows: if the potential $V$ is weak, the total wavefunction $\psi$ inside the integral (i.e., within the scattering region) should not be drastically different from the incident plane wave $\varphi$.

So, as a zeroth-order approximation, we can say:

$$|{\psi }^{(0)}⟩=|\mathbf{k}⟩$$

This is simply the incident plane wave, ignoring the potential entirely. To get a better approximation, we can substitute this guess into the right-hand side of the Lippmann-Schwinger equation:

$$|{\psi }^{(1)}⟩=|\mathbf{k}⟩+G_0^{+}V|{\psi }^{(0)}⟩=|\mathbf{k}⟩+G_0^{+}V|\mathbf{k}⟩$$

This is the first Born approximation for the wavefunction. It includes the incident wave plus a term representing a single scattering event, where the incident wave $|\mathbf{k}⟩$ scatters off the potential $V$.

We can continue this process. Substituting $|{\psi }^{(1)}⟩$ back into the Lippmann-Schwinger equation gives the second-order approximation:

$$\begin{array}{rl}|{\psi }^{(2)}⟩& =|\mathbf{k}⟩+G_0^{+}V|{\psi }^{(1)}⟩\\ & =|\mathbf{k}⟩+G_0^{+}V\left(|\mathbf{k}⟩+G_0^{+}V|\mathbf{k}⟩\right)\\ & =|\mathbf{k}⟩+G_0^{+}V|\mathbf{k}⟩+G_0^{+}V{G}_0^{+}V|\mathbf{k}⟩\end{array}$$

The new term, $G_0^{+}V{G}_0^{+}V|\mathbf{k}⟩$, has a clear physical interpretation: the incident particle scatters, propagates freely (described by $G_0^{+}$), and then scatters a second time. Repeating this process generates the full Born series:

$$|{\psi }^{(+)}⟩=|\mathbf{k}⟩+G_0^{+}V|\mathbf{k}⟩+G_0^{+}V{G}_0^{+}V|\mathbf{k}⟩+G_0^{+}V{G}_0^{+}V{G}_0^{+}V|\mathbf{k}⟩+\dots$$

This is a perturbation series in powers of the potential $V$. The following video provides a clear, step-by-step construction of this series.

2. The First Born Approximation for the Scattering Amplitude

For the rest of this lesson, we will focus on the first Born approximation, which is often sufficient and widely used. Our goal is to find the scattering amplitude $f({\mathbf{k}}^{\prime },\mathbf{k})$, which is defined by the asymptotic behavior of the wavefunction far from the scatterer:

$${\psi }_{\mathbf{k}}^{(+)}(\mathbf{r})\stackrel{r\to \infty }{\to }{e}^{i\mathbf{k}\cdot \mathbf{r}}+f({\mathbf{k}}^{\prime },\mathbf{k})\frac{e^{ikr}}{r}$$

where ${\mathbf{k}}^{\prime }=k\hat{\mathbf{r}}$ is the wavevector in the direction of observation.

The first Born approximation for the wavefunction in coordinate space is:

$${\psi }_{\mathbf{k}}^{(1)}(\mathbf{r})=e^{i\mathbf{k}\cdot \mathbf{r}}+\int d^3{\mathbf{r}}^{\prime }{G}_0^{+}(\mathbf{r}-{\mathbf{r}}^{\prime })V({\mathbf{r}}^{\prime })e^{i\mathbf{k}\cdot {\mathbf{r}}^{\prime }}$$

The second term is the scattered wave. Let's analyze its asymptotic form using what we know about the Green's function from the last lesson:

$$G_0^{+}(\mathbf{r}-{\mathbf{r}}^{\prime })=-\frac{2m}{4\pi {\hslash }^2}\frac{e^{ik|\mathbf{r}-{\mathbf{r}}^{\prime }|}}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}$$

For large $r$ (i.e., $r\gg r^{\prime }$), we can approximate $|\mathbf{r}-{\mathbf{r}}^{\prime }|\approx r-\hat{\mathbf{r}}\cdot {\mathbf{r}}^{\prime }$. The scattered wave becomes:

$${\psi }_{\text{scatt}}(\mathbf{r})\approx -\frac{m}{2\pi {\hslash }^2}\int d^3{\mathbf{r}}^{\prime }\frac{e^{ik(r-\hat{\mathbf{r}}\cdot {\mathbf{r}}^{\prime })}}{r}V({\mathbf{r}}^{\prime })e^{i\mathbf{k}\cdot {\mathbf{r}}^{\prime }}$$

Factoring out the terms that don't depend on the integration variable ${\mathbf{r}}^{\prime }$, we get:

$${\psi }_{\text{scatt}}(\mathbf{r})\approx \frac{e^{ikr}}{r}\left[-\frac{m}{2\pi {\hslash }^2}\int d^3{\mathbf{r}}^{\prime }{e}^{-ik\hat{\mathbf{r}}\cdot {\mathbf{r}}^{\prime }}V({\mathbf{r}}^{\prime })e^{i\mathbf{k}\cdot {\mathbf{r}}^{\prime }}\right]$$

Comparing this with the definition of the scattering amplitude, we arrive at our main result.

The scattering amplitude in the first Born approximation is:

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

Let's define the momentum transfer vector $\mathbf{Q}={\mathbf{k}}^{\prime }-\mathbf{k}$.

CAPTION: The momentum transfer vector $\mathbf{Q}={\mathbf{k}}^{\prime }-\mathbf{k}$. For elastic scattering, $|\mathbf{k}|=|{\mathbf{k}}^{\prime }|=k$, forming an isosceles triangle. The magnitude of $\mathbf{Q}$ is $Q=2k\sin (\theta /2)$, where $\theta$ is the scattering angle.

Using this definition, the scattering amplitude takes a beautifully simple form:

$$f^{(1)}(\mathbf{Q})=-\frac{m}{2\pi {\hslash }^2}\tilde{V}(\mathbf{Q})$$

where $\tilde{V}(\mathbf{Q})=\int d^3\mathbf{r}V(\mathbf{r})e^{-i\mathbf{Q}\cdot \mathbf{r}}$ is the Fourier transform of the potential.

This is a profound result: in the first Born approximation, the scattering amplitude is simply proportional to the Fourier transform of the potential evaluated at the momentum transfer vector.

The following video walks through this derivation and highlights the significance of this Fourier transform relationship.

3. Differential Cross Section and an Example: Yukawa Potential

The quantity measured in experiments is the differential cross section, $d\sigma /d\Omega$, which is the number of particles scattered into a solid angle $d\Omega$ per unit time, per unit incident flux. In quantum mechanics, it's given by:

$$\frac{d\sigma }{d\Omega }=|f({\mathbf{k}}^{\prime },\mathbf{k}){|}^2$$

In the Born approximation, this becomes:

$${\left(\frac{d\sigma }{d\Omega }\right)}^{(1)}={\left|-\frac{m}{2\pi {\hslash }^2}\tilde{V}(\mathbf{Q})\right|}^2={\left(\frac{m}{2\pi {\hslash }^2}\right)}^2|\tilde{V}(\mathbf{Q}){|}^2$$

This formula is the workhorse of scattering analysis in many fields. In condensed matter physics, scattering of X-rays or neutrons from a crystal lattice involves a periodic potential $V(\mathbf{r})$. The Fourier transform $\tilde{V}(\mathbf{Q})$ is non-zero only for specific values of $\mathbf{Q}$ corresponding to the reciprocal lattice vectors, leading to the sharp Bragg peaks seen in diffraction patterns.

Let's apply this to a concrete example: the Yukawa potential, which models a screened Coulomb interaction.
$V(r)=V_0\frac{e^{-ar}}{r}$
Here, $1/a$ is the screening length. To find the scattering amplitude, we need its Fourier transform. Since the potential is spherically symmetric, the integral is easiest in spherical coordinates, with the z-axis aligned along $\mathbf{Q}$.

$$\begin{array}{rl}\tilde{V}(\mathbf{Q})& ={\int }_0^{2\pi }d\varphi {\int }_0^{\pi }d\theta \sin \theta {\int }_0^{\infty }{r}^{2}dr\left(V_0\frac{e^{-ar}}{r}\right)e^{-iQr\cos \theta }\\ & =2\pi V_0{\int }_0^{\infty }r{e}^{-ar}dr{\int }_{-1}^{1}d(\cos \theta )e^{-iQr\cos \theta }\\ & =2\pi V_0{\int }_0^{\infty }r{e}^{-ar}dr{\left[\frac{e^{-iQr\cos \theta }}{-iQr}\right]}_{-1}^1\\ & =\frac{4\pi V_0}{Q}{\int }_0^{\infty }r{e}^{-ar}\sin (Qr)dr\end{array}$$

This is a standard integral, which evaluates to $\frac{2Q}{(a^2+Q^2{)}^2}$. Wait, that's not right. The integral ${\int }_0^{\infty }r\sin (Qr)e^{-ar}dr=\frac{2aQ}{(a^2+Q^2{)}^2}$. Let's use a simpler method. The integral ${\int }_0^{\infty }\sin (bx)e^{-ax}dx=b/(a^2+b^2)$. This is not quite it.

Let's re-evaluate the integral over $\cos \theta$.

$${\int }_{-1}^1{e}^{-iQru}du=\frac{e^{-iQr}-e^{iQr}}{-iQr}=\frac{2\sin (Qr)}{Qr}$$

So the Fourier transform is:

$$\tilde{V}(Q)=2\pi V_0{\int }_0^{\infty }r{e}^{-ar}\left(\frac{2\sin (Qr)}{Q}\right)dr=\frac{4\pi V_0}{Q}{\int }_0^{\infty }r\sin (Qr)e^{-ar}dr$$

The integral can be found by taking $-\frac{\partial }{\partial a}$ of ${\int }_0^{\infty }\cos (Qr)e^{-ar}dr$. Let's just use the known result for the Fourier transform of $e^{-ar}/r$, which is $4\pi /(a^2+Q^2)$.
So, $\tilde{V}(Q)=\frac{4\pi V_0}{a^2+Q^2}$.

The differential cross section is then:

$$\frac{d\sigma }{d\Omega }={\left(\frac{m}{2\pi {\hslash }^2}\right)}^2{\left|\frac{4\pi V_0}{a^2+Q^2}\right|}^2={\left(\frac{2m{V}_0}{{\hslash }^2(a^2+Q^2)}\right)}^2$$

Since $Q^2=(2k\sin (\theta /2){)}^2=4k^2{\sin }^2(\theta /2)$, we have:

$$\frac{d\sigma }{d\Omega }={\left(\frac{2m{V}_0}{{\hslash }^2(a^2+4k^2{\sin }^2(\theta /2))}\right)}^2$$

This is the result for Yukawa scattering. In the limit of an unscreened Coulomb potential ($a\to 0$) with $V_0=Z_1{Z}_2{e}^2/4\pi {ϵ}_0$, we recover the famous Rutherford scattering formula.

Test your understanding!

Analyze the angular dependence of the Yukawa scattering cross section.

1. In the low-energy limit ($k\to 0$), how does the cross section depend on the scattering angle $\theta$?
2. In the high-energy limit ($k\gg a$), where is the scattering most intense? What does this imply about high-energy scattering events?

Show answer

1. Low-energy limit ($k\to 0$): In this case, the term $4k^2{\sin }^2(\theta /2)$ in the denominator is negligible compared to $a^2$. The denominator becomes approximately constant (${\hslash }^2{a}^2$). Therefore, the differential cross section $d\sigma /d\Omega$ becomes largely independent of the scattering angle $\theta$. The scattering is nearly isotropic (uniform in all directions).

2. High-energy limit ($k\gg a$): The denominator is dominated by the $4k^2{\sin }^2(\theta /2)$ term. The cross section is proportional to $1/{\sin }^4(\theta /2)$. This function is sharply peaked at $\theta =0$ (forward scattering) and falls off very rapidly for larger angles. This implies that high-energy particles are typically only deflected by small angles. The scattering is strongly peaked in the forward direction.

4. Conditions of Validity

When can we trust the Born approximation? The core assumption was that the scattered wave is small compared to the incident wave within the scattering region. Let's make this quantitative.

The first-order wavefunction is ${\psi }^{(1)}=\varphi +{\psi }_{\text{scatt}}$. The approximation is valid if $|{\psi }_{\text{scatt}}(\mathbf{r})|\ll |\varphi (\mathbf{r})|$ for $\mathbf{r}$ inside the potential. Since $|\varphi (\mathbf{r})|=|e^{i\mathbf{k}\cdot \mathbf{r}}|=1$, this condition becomes:

$$\left|\int d^3{\mathbf{r}}^{\prime }{G}_0^{+}(\mathbf{r}-{\mathbf{r}}^{\prime })V({\mathbf{r}}^{\prime })e^{i\mathbf{k}\cdot {\mathbf{r}}^{\prime }}\right|\ll 1(\text{for }\mathbf{r}\text{ in potential region})$$

Evaluating this integral generally is complicated, but it leads to two distinct criteria for two different energy regimes, which can be illustrated using a simple potential like a square well of depth $V_0$ and range $R_0$.

The following resource provides a thorough and clear analysis of these conditions.

Let's summarize the key results from the reading for a potential of characteristic strength $|V_0|$ and range $R_0$:

1. Low Energy Limit ($k{R}_0\ll 1$):
   The condition for validity is
   $\frac{m|V_0|R_0^2}{{\hslash }^2}\ll 1$
   This is a condition on the intrinsic "strength" of the potential. Intriguingly, the quantity on the left is of the order of unity for a potential that is just strong enough to support its first bound state. Therefore, the Born approximation is generally invalid at low energies if the potential is strong enough to have a bound state.

2. High Energy Limit ($k{R}_0\gg 1$):
   The condition becomes
   $\frac{|V_0|R_0}{\hslash v}\ll 1\text{or equivalently}\frac{m|V_0|R_0}{{\hslash }^{2}k}\ll 1$
   where $v=\hslash k/m$ is the particle's velocity. This can be interpreted physically: the time spent in the potential is roughly $R_0/v$, and the change in potential energy is $|V_0|$. The condition states that the "impulse" $(|V_0|\times R_0/v)$ should be much smaller than the incident momentum $\hslash k$. For any potential, no matter how strong, this condition can always be satisfied by going to a sufficiently high energy (large $k$).

In short, the Born approximation is a weak-coupling or high-energy approximation.

Conclusion

In this lesson, we have moved from the formal Lippmann-Schwinger equation to a powerful and practical calculational tool. The first Born approximation provides a direct link between the interaction potential and the experimentally measurable scattering cross-section.

Key Takeaways:

• The Born series is an iterative solution to the Lippmann-Schwinger equation, representing a sequence of multiple scattering events.
• The first Born approximation truncates this series after the single-scattering term. It is valid for weak potentials or high incident energies.
• In this approximation, the scattering amplitude $f(\mathbf{Q})$ is proportional to the Fourier transform of the potential $\tilde{V}(\mathbf{Q})$, where $\mathbf{Q}$ is the momentum transfer.
• The differential cross section is $d\sigma /d\Omega =|f(\mathbf{Q}){|}^2$.
• The approximation fails at low energies for potentials strong enough to support bound states, but becomes valid for any potential at sufficiently high energy.

Preview of the Next Lesson:
The Born approximation is excellent for weak potentials, but what about strong potentials at low energies, where it fails? This regime is common in nuclear physics and in some condensed matter contexts (e.g., scattering from a strong impurity). To handle this, we need a different, non-perturbative method. In the next lesson, we will develop the method of partial wave analysis, which decomposes the scattering problem by angular momentum and is particularly powerful for low-energy scattering from spherically symmetric potentials.