Low dimensional systems

In this type of experiment the role of the impurity can also be played by a laser beam with small enough size and intensity. The Fourier components of the perturbed wave function $\delta\psi_{{\bf k}}$ are given by the formula (7.5), which is derived in an arbitrary number of dimensions. The only difference is in the substitution of $d^3k/(2\pi)^3$ with $d^Dk/(2\pi)^D$ in the integrals:

$$F_V = \frac{2i{g_{i}}^2\phi_0^2}{V} \int \frac{{\bf k V} \frac{\h... ...2 k^2}{2m} \left(\frac{\hbar^2 k^2}{2m}+2mc^2\right)} \,\frac{\bf dk}{(2\pi)^D}$$ (7.29)

In the expression for the energy (7.13), the term quadratic in velocity is of a great interest, as the coefficient in front of $V^2/2$ has physical meaning of an effective mass. We develop further this term:

$$\Delta E = {g_{i}}^2\phi_0^2\int\frac{(\hbar{\bf kV})^2}{\frac{\h... ...(2m)^3V^2}{D\hbar^4} \int \frac{1}{(k^2+(2mc/\hbar)^2)^2}\frac{d^Dk}{(2\pi)^D},$$ (7.30)

where we used symmetry properties $\int f(k) ({\bf kV})^2 d^Dk = \frac{1}{D}\int f(k) (kV)^2 d^Dk$.