$V(r) = \begin{cases} V_0 & \text{for } r < a \\ 0 & \text{for } r \ge a \end{cases}$
where $V_0$ and $a$ are positive constants. In the low energy limit, the total scattering cross-section is $\sigma = 4\pi a^2 \left(\frac{1}{ka}\tanh ka - 1\right)^2$, where $k^2 = \frac{2m}{\hbar^2}(V_0 - E) > 0$. In the limit $V_0 \to \infty$ the ratio of $\sigma$ to the classical scattering cross-section off a sphere of radius $a$ is
Scattering Cross Section Ratio Analysis
This problem requires calculating the ratio of the quantum mechanical scattering cross-section ($\sigma$) to the classical scattering cross-section ($\sigma_{classical}$) for a repulsive spherical potential. The calculation is performed in the specific limit where the potential height $V_0$ approaches infinity ($V_0 \to \infty$).
Classical Scattering Cross Section
For a repulsive potential modeled as a hard sphere of radius $a$, the classical scattering cross-section represents the effective area that deflects incoming particles. This area is geometrically determined by the radius of the sphere:
$ \sigma_{classical} = \pi a^2 $
Quantum Mechanical Cross Section in Limit
The quantum mechanical total scattering cross-section is provided by the formula:
$ \sigma = 4\pi a^2 \left(\frac{1}{ka}\tanh ka - 1\right)^2 $
The parameter $k$ is defined as $k^2 = \frac{2m}{\hbar^2}(V_0 - E)$, where $m$ is the particle mass, $\hbar$ is the reduced Planck constant, $V_0$ is the potential height, and $E$ is the particle energy. We are interested in the limit as $V_0 \to \infty$. Since $E$, $m$, and $\hbar$ are constants, this implies $k \to \infty$.
To evaluate the expression, let $x = ka$. As $k \to \infty$, it follows that $x \to \infty$. We need to find the limit of the term within the parenthesis:
$ \lim_{x \to \infty} \left(\frac{1}{x}\tanh x - 1\right) $
The hyperbolic tangent function, $\tanh x$, approaches $1$ as $x$ tends to infinity:
$ \lim_{x \to \infty} \tanh x = 1 $
Substituting this into the limit expression:
$ \lim_{x \to \infty} \left(\frac{1}{x}(1) - 1\right) = 0 - 1 = -1 $
Now, we substitute this limiting value back into the formula for $\sigma$:
$ \sigma = 4\pi a^2 (-1)^2 $
$ \sigma = 4\pi a^2 (1) = 4\pi a^2 $
Ratio Calculation
The final step is to compute the ratio of the quantum mechanical cross-section ($\sigma$) to the classical cross-section ($\sigma_{classical}$):
$ \frac{\sigma}{\sigma_{classical}} = \frac{4\pi a^2}{\pi a^2} $
$ \frac{\sigma}{\sigma_{classical}} = 4 $
Important Questions from Scattering Theory
- When a neutron of $1 \text{ keV}$ kinetic energy impinges on a $^{12}\text{C}$ target, the total scattering cross section is $1000 \text{ barns}$. The approximate value of the phase shift $\delta_0$ is
A particle of mass $m$ and energy $E > 0$, in one dimension is scattered by the potential shown below.
If the particle was moving from $x = -\infty$ to $x = \infty$, which of the following graphs gives the best qualitative representation of the wavefunction of this particle?
- In the partial wave expansion, the differential scattering cross-section is given by
$$ \frac{d\sigma}{d(\cos\theta)} = \left| \sum_l (2l+1)e^{i\delta_l} \sin\delta_l P_l(\cos\theta) \right|^2 $$
where $\theta$ is the scattering angle. For a certain neutron-nucleus scattering, it is found that the two lowest phase shifts $\delta_0$ and $\delta_1$ corresponding to $s$-wave and $p$-wave, respectively, satisfy $\delta_1 \approx \delta_0/2$. Assuming that the other phase shifts are negligibly small, the differential cross-section reaches its minimum for $\cos\theta$ equal to - Consider the potential
$$V(\vec{r}) = \sum_i V_0 a^3 \delta^{(3)}(\vec{r} - \vec{r}_i)$$
where $\vec{r}_i$ are the position vectors of the vertices of a cube of length $a$ centered at the origin and $V_0$ is a constant. If $V_0 a^2 \ll \frac{\hbar^2}{m}$, the total scattering cross-section, in the low-energy limit, is - A sphere G of radius $b$ is fixed mid-air and several spheres identical to the first one are shot at it with their velocities parallel to each other. If the shot spheres fall within an imaginary cylinder of radius $a \ (b \ll a)$ then the fraction of spheres that will hit G is
