Show that for a relativistic particle of rest energy E0, the...

Show that for a relativistic particle of rest energy $E_{0}$, the de Broglie wavelength in $\AA$ is given by
$$
\lambda=\frac{1.24 \times 10^{-2}}{E_{0}(\mathrm{MeV})} \frac{\left(1-\beta^{2}\right)^{1 / 2}}{\beta} \quad \text { where } \beta=v / c
$$

Key Concepts

Rest Energy

Rest energy, expressed as E? = mc², represents the intrinsic energy of a particle when it is at rest. In relativistic derivations, it provides a fundamental energy scale against which kinetic energies and relativistic modifications are compared, and it is pivotal in linking mass and energy through quantum mechanical relations.

Lorentz Factor

The Lorentz factor, defined as ? = 1/(1 ? ?²)^(1/2) where ? = v/c, quantifies the relativistic corrections necessary when particles approach the speed of light. It appears not only in the expression for momentum but also in energy relations, playing a crucial role in deriving formulas that bridge classical and relativistic physics.

Dimensional Analysis and Unit Conversion

Accurate derivations in physics require careful dimensional analysis and appropriate unit conversions. When expressing quantities like the de Broglie wavelength in specific units (such as angstroms), constants like Planck’s constant and the speed of light must be incorporated with the correct conversion factors to ensure that the final expression is both dimensionally consistent and practically usable.

de Broglie Hypothesis

The de Broglie hypothesis states that any particle exhibits wave-like behavior, with its wavelength being inversely proportional to its momentum. This concept is fundamental in quantum mechanics, serving as the basis for understanding the wave-particle duality of matter.

Relativistic Momentum

In the context of special relativity, momentum is modified to account for the effects of high speeds. Instead of the classical expression, the momentum of a particle is given by p = ?mv, where ? (the Lorentz factor) increases as the particle’s velocity approaches the speed of light. This ensures that computed quantities like the de Broglie wavelength remain valid even at relativistic velocities.

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