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Chapter 10: Q61E (page 470)

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

Short Answer

Expert verified

Answer: -

The effective mass of a free particle is m=h2d2Edk2-1.

Step by step solution

01

- Relation and momentum and wavenumber

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

02

- Deriving equation of effective mass

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

E=p22m

Substitute the value of P as hk. Thus,

E=hk22mE=h2k22m

Differentiate the above equation with respect to K.

dEdk=ddkh2k22m=2h2k2m=h2km

Differentiate the above equation again with respect to K.

d2Edk2=ddkh2km=h2m

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

m=h2d2Edk2-1

Hence, The effective mass of a free particle ism=h2d2Edk2-1 .

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