Question

The transition rate for a particle moving between states in Fermi’s Golden rule is directly proportional to:

• A. The square of the matrix element of the perturbation
• B. The initial state wave function only
• C. The difference in energy between the final and initial states
• D. The potential energy of the system

Hint:

Fermi’s Golden Rule: \( W \propto |\langle f | H' | i \rangle|^2 \). Always focus on how the perturbation connects the initial and final states to determine transition probability.

Answer 1

The Correct Option is A

Step 1: Fermi’s Golden Rule is a fundamental result in time-dependent perturbation theory in quantum mechanics. It provides the transition rate \( W \) from an initial quantum state \( |i\rangle \) to a final state \( |f\rangle \) due to a perturbation.
Step 2: Mathematically, the transition rate is given by: \[ W_{i \to f} = \frac{2\pi}{\hbar} \left| \langle f | H' | i \rangle \right|^2 \rho(E_f) \] where \( H' \) is the perturbation Hamiltonian and \( \rho(E_f) \) is the density of final states at energy \( E_f \).
Step 3: The key factor controlling the transition rate is \( \left| \langle f | H' | i \rangle \right|^2 \), i.e., the square of the matrix element of the perturbation between the initial and final states. This element captures how effectively the perturbation can cause the transition.
Step 4: Other factors such as the initial wave function, energy difference, or potential energy do not alone determine the transition rate unless they are part of this matrix element.
Hence, the transition rate is directly proportional to the square of the matrix element of the perturbation.