= Effective-mass tensor
{title2=$(m^{*-1})_{ij}=\hbar^{-2}\partial_{k_i}\partial_{k_j}\varepsilon$}
For a semiclassical <wave packet> in an isolated <energy band>, $v_i=\hbar^{-1}\partial_{k_i}\varepsilon$ and $\hbar\dot k_j=F_j$ imply $\dot v_i=\sum_j(m^{*-1})_{ij}F_j$. The inverse <effective-mass tensor> is the <Hessian matrix> of the dispersion divided by $\hbar^2$. Along a unit vector $\mathbf e$, $1/m^*_{\mathbf e}=\sum_{ij}e_i(m^{*-1})_{ij}e_j$. The <Laplacian> is its trace: $\hbar^{-2}\nabla_k^2\varepsilon=\operatorname{tr}(m^{*-1})$. Thus in $d$ isotropic dimensions the scalar mass is $d\hbar^2/(\nabla_k^2\varepsilon)$, rather than $\hbar^2/(\nabla_k^2\varepsilon)$.
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