An electron is travelling with a velocity v in free space and when...

The de Broglie wavelength of an electron in free space is given by $\lambda_0 = \dfrac{h}{mv}$.

When the electron enters the medium, its velocity is reduced by $20\%$. The new velocity is $v' = v - 0.20v = 0.80v$.

The de Broglie wavelength in the medium is $\lambda' = \dfrac{h}{mv'}$.

Substituting $v' = 0.80v$, we get $\lambda' = \dfrac{h}{m(0.80v)} = \dfrac{1}{0.80} \left(\dfrac{h}{mv}\right).<br><br>Since \dfrac{1}{0.80} = \dfrac{100}{80} = 1.25$, we have $\lambda' = 1.25\lambda_0$.

Comparing this with $\lambda' = \alpha\lambda_0$, we get $\alpha = 1.25$.

Answer: $1.25$