Quantum Mechanics Course

Total: 450+ pages covering mathematical foundations through advanced quantum mechanics

Part VIII, Chapter 2 | Page 2 of 3

Worked Examples

Applying the Born approximation to Yukawa, Coulomb, square well, and hard sphere potentials

The Yukawa Potential

The Yukawa (or screened Coulomb) potential models a short-range interaction with an exponential decay:

$$V(r) = \beta\,\frac{e^{-\mu r}}{r}$$

Here $\beta$ characterizes the strength of the potential and $\mu$is the inverse screening length ($1/\mu$ is the range). This potential was originally proposed by Yukawa to model nuclear forces mediated by meson exchange, where$\mu = m_\pi c/\hbar$.

Applying the Born formula for a spherically symmetric potential:

$$f(\theta) = -\frac{2m}{\hbar^2 q}\int_0^\infty \beta\,\frac{e^{-\mu r'}}{r'}\cdot r'\sin(qr')\,dr' = -\frac{2m\beta}{\hbar^2 q}\int_0^\infty e^{-\mu r'}\sin(qr')\,dr'$$

The integral evaluates to $q/(\mu^2 + q^2)$, giving the remarkably clean result:

Yukawa Scattering Amplitude
$$\boxed{f(\theta) = -\frac{2m\beta}{\hbar^2(q^2 + \mu^2)}}$$

The differential cross section follows immediately:

$$\frac{d\sigma}{d\Omega} = \frac{4m^2\beta^2}{\hbar^4(q^2 + \mu^2)^2} = \frac{4m^2\beta^2}{\hbar^4\!\left(4k^2\sin^2\!\frac{\theta}{2} + \mu^2\right)^2}$$

The total cross section is obtained by integrating over solid angle. Using$d\Omega = 2\pi\sin\theta\,d\theta$ and the substitution$q^2 = 4k^2\sin^2(\theta/2)$:

$$\sigma_{\text{tot}} = \frac{16\pi m^2\beta^2}{\hbar^4\,\mu^2(4k^2 + \mu^2)}$$

The Coulomb Limit and Rutherford Formula

The pure Coulomb potential is recovered from the Yukawa potential in the limit $\mu \to 0$with $\beta = Z_1 Z_2 e^2/(4\pi\epsilon_0)$. Taking this limit in the differential cross section:

$$\frac{d\sigma}{d\Omega}\bigg|_{\mu\to 0} = \frac{4m^2\beta^2}{\hbar^4 \cdot 16k^4\sin^4\!\frac{\theta}{2}} = \left(\frac{Z_1 Z_2 e^2}{16\pi\epsilon_0 E}\right)^2 \frac{1}{\sin^4(\theta/2)}$$

This is precisely the Rutherford scattering formula. It is a remarkable coincidence that the first Born approximation yields the exact result for the Coulomb potential—a consequence of the special $1/r$ form. Note, however, that the total cross section diverges ($\mu\to 0$ gives $\sigma\to\infty$), reflecting the infinite range of the Coulomb force.

Square Well Potential

Consider a spherical square well:

$$V(r) = \begin{cases} -V_0 & r < a \\ 0 & r > a \end{cases}$$

The Born integral becomes:

$$f(\theta) = \frac{2mV_0}{\hbar^2 q}\int_0^a r'\sin(qr')\,dr' = \frac{2mV_0}{\hbar^2 q^3}\Big[\sin(qa) - qa\cos(qa)\Big]$$

The differential cross section exhibits a diffraction-like pattern:

$$\frac{d\sigma}{d\Omega} = \frac{4m^2V_0^2}{\hbar^4 q^6}\Big[\sin(qa) - qa\cos(qa)\Big]^2$$

Features of the Diffraction Pattern

  • Oscillations in $\theta$ resembling optical diffraction from a sphere
  • First minimum at $qa \approx 4.49$ (first zero of $\sin(qa) - qa\cos(qa)$)
  • Angular width of central maximum: $\Delta\theta \sim \lambda/a$
  • Sharper minima at higher $ka$ (shorter wavelength resolves the well boundary)

In the forward direction ($q \to 0$), L'Hopital's rule gives:

$$f(0) = -\frac{2mV_0 a^3}{3\hbar^2}$$

Hard Sphere at Low Energy

For a hard sphere ($V = \infty$ for $r < a$), the Born approximation is not directly applicable since the potential is not weak. However, we can study the low-energy limit of the exact result (from partial wave analysis) to compare. The exact s-wave scattering at low energy ($ka \ll 1$) gives:

$$f \approx -a, \qquad \sigma_{\text{tot}} = 4\pi a^2$$

This is four times the classical geometric cross section $\pi a^2$—a purely quantum effect arising from diffraction around the sphere. For the finite square well at low energy, the Born approximation gives:

$$\sigma_{\text{tot}} = \frac{16\pi m^2 V_0^2 a^6}{9\hbar^4} \qquad (ka \ll 1)$$

Total Cross Section Calculations

The total cross section is obtained by integrating $d\sigma/d\Omega$ over all solid angles. For the Born amplitude $f(q)$ with $q = 2k\sin(\theta/2)$, we change variables from $\theta$ to $q$:

$$\sigma_{\text{tot}} = \int |f(\theta)|^2\,d\Omega = \frac{2\pi}{k^2}\int_0^{2k}|f(q)|^2\,q\,dq$$

This change of variable is frequently more convenient for evaluating the integral analytically.

Validity of the Born Approximation

The Born approximation requires that the scattered wave be small compared to the incident wave. For a potential of depth $V_0$ and range $a$, this translates to:

Born Validity Criterion
$$\boxed{V_0 \ll \frac{\hbar^2}{2ma^2}}$$

This is the condition at low energy ($ka \lesssim 1$). At high energy ($ka \gg 1$), the condition relaxes to:

$$V_0 \ll \frac{\hbar^2 k}{2ma} = \frac{E}{ka} \qquad (ka \gg 1)$$

Thus the Born approximation always improves at higher energies—the fast particle spends less time in the potential region and is less perturbed.

When Born Fails

  • Strong potentials: nuclear forces, deep attractive wells
  • Low energies: bound states and resonances cannot be captured
  • Long-range potentials: Coulomb has logarithmic phase corrections missed by Born
  • Near resonances: cross sections much larger than Born predicts

Gaussian Potential

As another instructive example, consider the Gaussian potential:

$$V(r) = -V_0\,e^{-r^2/a^2}$$

The Born amplitude for a spherically symmetric potential requires the integral:

$$f(\theta) = \frac{2mV_0}{\hbar^2 q}\int_0^\infty r\,e^{-r^2/a^2}\sin(qr)\,dr = \frac{mV_0 a^3\sqrt{\pi}}{2\hbar^2}\,e^{-q^2 a^2/4}$$

The differential cross section is a Gaussian in momentum transfer:

$$\frac{d\sigma}{d\Omega} = \frac{\pi m^2 V_0^2 a^6}{4\hbar^4}\,e^{-q^2 a^2/2} = \frac{\pi m^2 V_0^2 a^6}{4\hbar^4}\,\exp\!\left(-2k^2 a^2\sin^2\!\frac{\theta}{2}\right)$$

The Gaussian potential produces a smooth, monotonically decreasing cross section without diffraction minima, in contrast to the square well. The angular width is$\Delta\theta \sim 1/(ka)$, and the total cross section is:

$$\sigma_{\text{tot}} = \frac{\pi^2 m^2 V_0^2 a^4}{2\hbar^4 k^2}\left(1 - e^{-2k^2 a^2}\right)$$

Comparing Potentials: Key Patterns

Born Cross Section Characteristics

  • Sharp-edged potentials (square well): produce diffraction minima and oscillatory patterns, analogous to optical diffraction from a hard-edged aperture
  • Smooth potentials (Gaussian): produce smooth, monotonic angular distributions without sharp features
  • Long-range potentials (Yukawa, Coulomb): strongly forward-peaked cross sections with possible divergence at $\theta = 0$
  • Universal feature: the angular width of the diffraction peak scales as$\Delta\theta \sim 1/(ka)$, where $a$ is the range of the potential
  • High energy limit: forward scattering dominates for all potentials as$k\to\infty$; only very small-angle scattering survives
← Page 1: The Born ApproximationPage 3: Higher Born & Form Factors →