Show that the number of states g(E) d E d^2 Ωwith energy in (E, E+d E) and momentum in the soli

step 1 In this problem, we have some boundary conditions that we have to observe. First one being, acco

Show that the number of states g(E) d E d^2 Ωwith energy in...

Show that the number of states $g(E) \mathrm{d} E \mathrm{~d}^{2} \Omega$ with energy in $(E, E+\mathrm{d} E)$ and momentum in the solid angle $\mathrm{d}^{2} \Omega$ around $\mathbf{p}=\hbar \mathbf{k}$ of a particle of mass $m$ that moves freely subject to periodic boundary conditions on the walls of a cubical box of side length $L$ is $$ g(E) \mathrm{d} E \mathrm{~d}^{2} \Omega=\left(\frac{L}{2 \pi}\right)^{3} \frac{m^{3 / 2}}{\hbar^{3}} \sqrt{2 E} \mathrm{~d} E \mathrm{~d} \Omega^{2} $$ Hence show from Fermi's golden rule that the cross-section for elastic scattering of such particles by a weak potential $V(\mathbf{x})$ from momentum $\hbar \mathbf{k}$ into the solid angle $\mathrm{d}^{2} \Omega$ around momentum $\hbar \mathbf{k}^{\prime}$ is $$ \mathrm{d} \sigma=\frac{m^{2}}{(2 \pi)^{2} \hbar^{4}} \mathrm{~d}^{2} \Omega\left|\int \mathrm{d}^{3} \mathbf{x} \mathrm{e}^{\mathrm{i}\left(\mathbf{k}-\mathbf{k}^{\prime}\right) \cdot \mathbf{x}} V(\mathbf{x})\right|^{2} $$ Explain in what sense the potential has to be 'weak' for this Born approximation to the scattering cross-section to be valid.

Show that the number of states $g(E) \mathrm{d} E \mathrm{~d}^{2} \Omega$ with energy in $(E, E+\mathrm{d} E)$ and momentum in the solid angle $\mathrm{d}^{2} \Omega$ around $\mathbf{p}=\hbar \mathbf{k}$ of a particle of mass $m$ that moves freely subject to periodic boundary conditions on the walls of a cubical box of side length $L$ is
$$
g(E) \mathrm{d} E \mathrm{~d}^{2} \Omega=\left(\frac{L}{2 \pi}\right)^{3} \frac{m^{3 / 2}}{\hbar^{3}} \sqrt{2 E} \mathrm{~d} E \mathrm{~d} \Omega^{2}
$$
Hence show from Fermi's golden rule that the cross-section for elastic scattering of such particles by a weak potential $V(\mathbf{x})$ from momentum $\hbar \mathbf{k}$ into the solid angle $\mathrm{d}^{2} \Omega$ around momentum $\hbar \mathbf{k}^{\prime}$ is
$$
\mathrm{d} \sigma=\frac{m^{2}}{(2 \pi)^{2} \hbar^{4}} \mathrm{~d}^{2} \Omega\left|\int \mathrm{d}^{3} \mathbf{x} \mathrm{e}^{\mathrm{i}\left(\mathbf{k}-\mathbf{k}^{\prime}\right) \cdot \mathbf{x}} V(\mathbf{x})\right|^{2}
$$
Explain in what sense the potential has to be 'weak' for this Born approximation to the scattering cross-section to be valid.

Key Concepts

Density of States

This concept refers to the number of quantum states available to a system per unit energy (and per unit solid angle in momentum space). In the context of a particle in a box with periodic boundary conditions, the discrete allowed momentum states become quasi-continuous in the large volume limit, leading to an expression for the density of states that depends on factors like the system size, mass, and the energy of the particle. It is a critical element for counting the number of available final states in processes such as scattering.

Periodic Boundary Conditions

Periodic boundary conditions are used to simplify problems in quantum mechanics by assuming that the wave function repeats itself at the boundaries of the system, effectively modeling an infinite or translationally invariant system. This condition quantizes the allowed momentum states of a particle in a box, leading to the discrete sum over states that, in the thermodynamic limit, can be converted into an integral using the density of states.

Fermi's Golden Rule

Fermi's golden rule provides a formula for the transition rate from one quantum state to another due to a perturbation, such as a scattering potential. It relates the probability per unit time of a transition to the square of the matrix element of the perturbing potential between the initial and final states, multiplied by the density of final states. This rule is fundamental in deriving expressions for scattering cross-sections in quantum mechanics.

Elastic Scattering

Elastic scattering is a process where a particle collides with a potential and changes its direction of motion without any change in its kinetic energy. In quantum scattering theory, elastic scattering implies that the energy of the incoming and outgoing states remain the same, which simplifies the analysis because conservation of energy and momentum can be applied directly to the scattering process.

Born Approximation and Weak Potential

The Born approximation is a perturbative approach used in scattering theory, valid when the scattering potential is 'weak', meaning that its effect on the incident wave can be treated as a small perturbation to the free particle state. 'Weak' in this context implies that the potential energy is much smaller than the kinetic energy of the particle, thereby justifying the use of first-order perturbation theory and the neglect of higher-order terms in the scattering amplitude. This approximation allows the scattering amplitude to be directly related to the Fourier transform of the potential, leading to the given expression for the cross-section.

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