An electron (mass m ) with an initial velocity v = v₀î(v₀ > 0) is in an electric field E = -E₀î(E₀ = constant > 0).  It’s de Broglie wavelength at time t is given by

Concept:

According to wave-particle duality, the de-Broglie wavelength in Quantum Mechanics determines the probability density of finding the object in a space

\(\lambda = {h \over mv}\)

Solution:

Initial Wavelength

\(λ_0={h\over mv_0}\)

At time t,

\(λ ={h \over mv}\)

where v = v₀ + \(e E_0t \over m\)

The ratio between these two wavelengths is given by

\({λ \over λ_0} = {v_0 \over v}\)

= \(v_0 \over v_0 + {eE_0t\over m}\)

= \(v_0 \over v_0 (1+ {eE_0t\over mv_0})\)

⇒ λ = \({\lambda_0 \over 1+ {eE_0t\over m v_0}}\)

The correct answer is option (1).