Method for calculatingTrHnin solid-state theory

Expressing the quantum mechanics of Bloch electrons in a solid in terms of the Weyl transform instead of quantum operators and the Wigner function instead of state vectors, the method used by Wannier and Upadhyaya for calculating the trace of the n th power of the Hamiltonian operator is generalized as a series expansion in powers of ${\hslash }^2$ of a cosine function of a sum of Poisson-bracket operators. The expression is calculated explicitly to order ${\hslash }^2$. The result is applied to the derivation of the magnetic susceptibility of solids with substitutional impurities, to order ${\hslash }^2$, as well as to other problems where spatial inhomogeneity is present.