All Problems Wave Properties of particles. Schrodinger Equation.
Problem 6.55
Derive the expression for a de Broglie wavelength \(\lambda\) of a relativistic particle moving with kinetic energy \(T .\) At what values of \(T\) does the error in determining \(\lambda\) using the non-relativistic formula not exceed \(1 \%\) for an electron and a proton?
\[ \text { 6.55. } \lambda=2 \pi \hbar / \sqrt{2 m T\left(1+T / 2 m c^{2}\right)}, T \leqslant 4 m c^{2} \Delta \lambda / \lambda=20.4 \mathrm{keV} \text { (for } \] an electron) and \(37.5 \mathrm{MeV}\) (for a proton).