Question

Question: - Verify using equation (10-12) that the effective mass of a free particle is m.

Short answer

Answer: -

The effective mass of a free particle is $m=h^2{\left(\frac{d^{2}E}{d{k}^2}\right)}^{-1}$.

Step-by-step solution

- Relation and momentum and wavenumber

The relation between the momentum p of a particle and a wavenumber is, where is a reduced plank constant.

- Deriving equation of effective mass

The kinetic energy E of the particle is related to the momentum as:

$E=\frac{p^2}{2m}$

Substitute the value of P as h_k. Thus,

$$\begin{gathered} E=\frac{{\left(hk\right)}^2}{2m} \\ E=\frac{h^2{k}^2}{2m} \end{gathered}$$

Differentiate the above equation with respect to K.

$$\begin{gathered} \frac{dE}{dk}=\frac{d}{dk}\left(\frac{h^2{k}^2}{2m}\right) \\ =\frac{2h^{2}k}{2m} \\ =\frac{h^{2}k}{m} \end{gathered}$$

Differentiate the above equation again with respect to K.

$$\begin{gathered} \frac{d^{2}E}{d{k}^2}=\frac{d}{dk}\left(\frac{h^{2}k}{m}\right) \\ =\frac{h^2}{m} \end{gathered}$$

The effective mass is calculated by rearranging the above equation for :

$m=h^2{\left(\frac{d^{2}E}{d{k}^2}\right)}^{-1}$

Hence, The effective mass of a free particle is$m=h^2{\left(\frac{d^{2}E}{d{k}^2}\right)}^{-1}$ .