Find De Broglie Wavelength Scaling Factor When Electron Speed Decreases

The de Broglie wavelength $\lambda$ of a particle of mass $m$ moving with velocity $v$ is given by the formula: $\lambda = \frac{h}{m v}$

where $h$ is Planck's constant.

Step 1: Express the de Broglie wavelength in free space In free space, the electron travels with a velocity $v$. Therefore, its initial de Broglie wavelength $\lambda_0$ is: $\lambda_0 = \frac{h}{m v}$

Step 2: Express the velocity of the electron in the medium When the electron enters the medium, its velocity is reduced by $20\%$. Thus, its new velocity $v'$ is: $v' = v - 0.20 v = 0.80 v = \frac{4}{5} v$

Step 3: Calculate the new de Broglie wavelength The de Broglie wavelength of the electron in the medium, $\lambda$, is given by: $\lambda = \frac{h}{m v'} = \frac{h}{m (0.80 v)}$

Substituting $\lambda_0 = \frac{h}{m v}$ into the equation: $\lambda = \frac{\lambda_0}{0.80} = \frac{1}{0.80} \lambda_0 = 1.25 \lambda_0$

Step 4: Find the value of $\alpha$ Given that $\lambda = \alpha \lambda_0$, comparing this with our derived equation gives: $\alpha = 1.25$

Thus, the correct option is C.