If a particle is accelerating, how does this affect its de Broglie wavelength?

The de Broglie wavelength \(\lambda\) of a particle is defined as: \(\lambda = \frac{h}{p}\) where \(h\) is the Planck's constant and \(p\) is the momentum of the particle.

The momentum \(p\) is defined as the product of mass \(m\) and velocity \(v\): \(p = mv\)

Substitute the momentum equation into the de Broglie wavelength equation: \(\lambda = \frac{h}{mv}\)

If a particle is accelerating (either speeding up or slowing down), its velocity changes with time. Let's denote the acceleration as \(a\). If the initial velocity is \(v_0\), after a certain time \(t\), the final velocity \(v\) can be expressed as: \(v = v_0 + at\)

Substitute the new velocity equation \(v = v_0 + at\) into the de Broglie wavelength equation: \(\lambda = \frac{h}{m(v_0 + at)}\)

Given this equation, we can see that as the particle accelerates (i.e., as \(a\) and/or \(t\) increase), its velocity and consequently its momentum will change. This results in a change in the de Broglie wavelength \(\lambda\). If the particle accelerates in the direction of its motion (positive \(a\)), its velocity increases, and its wavelength decreases. However, if the particle decelerates, or accelerates against the direction of its motion (negative \(a\)), its velocity decreases, and its wavelength increases. In conclusion, the de Broglie wavelength of an accelerating particle changes as its momentum changes due to the acceleration. The wavelength decreases if the particle accelerates in the direction of its motion, or increases if it decelerates or accelerates against the direction of its motion.