Starting from the time-dependent Schrödinger equation with

PHYSICS · Subjective · Fermi University Physics · Quantum Mechanics · Particle Physics & Quantum Fields · Particles, interactions and experiment

Symmetry and Flavour, Ch From amplitude to rate: Lorentz-invariant phase space, widths and cross sections

Problem

H = H_0 + V

, where the perturbation

V

is switched on at

t=0

and held constant thereafter, derive Fermi's golden rule

\Gamma = 2\pi|V_{fi}|^2\rho(E_f)

for the transition rate into a continuum of final states of the same energy as the initial state. Then use the non-relativistic box density of states

\rho(E) = \frac{Vp^2}{2\pi^2}\frac{\mathrm{d}p}{\mathrm{d}E}

to compute

\rho(E)

, in states per MeV, for a proton of kinetic energy

2.0\,\mathrm{MeV}

confined to a box of volume

8.0\,\mathrm{fm^3}

. Take

m_pc^2 = 938.3\,\mathrm{MeV}

and ignore relativistic corrections to the proton's kinetic energy.