$V(r) = \begin{cases} V_0 & \text{for } r < a \\ 0 & \text{for } r \ge a \end{cases}$
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
- A phase shift of $30^\circ$ is observed when a beam of particles of energy $0.1$ MeV is scattered by a target. When the beam energy is changed, the observed phase shift is $60^\circ$. Assuming that only s-wave scattering is relevant and that the cross-section does not change with energy, the beam energy is
- 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 - 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 - The range of the inter-atomic potential in gaseous hydrogen is approximately $5\text{ \AA}$. In thermal equilibrium, the maximum temperature for which the atom-atom scattering is dominantly $s$-wave, 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?
