Regarding Drude theory and AC conductivity of metal

[otaKu]

This snapshot is from the book Solid State Physics by Ashcroft,Mermin. They represented the Field and Momentum as the real part of a complex function. As far as what I understand, we represent fields in complex notations for our own convenience over the standard real representation which results in computational complexities. Why then does the imaginary part of the field which we have designated by ourselves need to satisfy the equation 1.24 which the real part of the field(the actual field) satisfies?

 

[DrDu]

You combine two real functions into two complex functions, the second one being the adjoint function of the first.E.g you can combine cos x and sin x into exp ix and exp -ix. Hence both the real and the imaginary part of the complex function are solutions of the equation you want to solve. Whether you recover the cos or sin function taking the real part, you can decide by multiplying the complex function with a complex phase factor.

 

[otaKu]

Thanks for the input! It was helpful.