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$\def\rr{{\bf r}}
\def\ii{{\rm i}}$
There is indeed a continuity equation for the particle density $\rho(\rr)=\Psi^\dagger(\rr)\Psi(\rr),$ where the field operator $\Psi^\dagger(\rr)$ creates a particle at position $\rr$. To derive it, you need only the canonical commutation relations for the fie...
In 1-particle non-relativistic QM we have the continuity equation for the 1-electron probability current density for an electron in the magnetic field in an eigenstate of the full Hamiltonian (=static problem) and ignoring spin
$$ \nabla\cdot{\bf j}=0$$
with the definition of
$$
{\bf j}=-\fra...
In mechanics, the virial theorem provides a general equation that relates the average over time of the total kinetic energy, ⟨T⟩, of a stable system consisting of N particles, bound by potential forces, with that of the total potential energy, ⟨VTOT⟩, where angle brackets represent the average over time of the enclosed quantity. Mathematically, the theorem states
⟨
T
⟩
=
−
1
2
∑
k
=...
In mathematics, the indefinite orthogonal group, O(p, q) is the Lie group of all linear transformations of an n-dimensional real vector space that leave invariant a nondegenerate, symmetric bilinear form of signature (p, q), where n = p + q. The dimension of the group is n(n − 1)/2.
The indefinite special orthogonal group, SO(p, q) is the subgroup of O(p, q) consisting of all elements with determinant 1. Unlike in the definite case, SO(p, q) is not connected – it has 2 components – and there are two additional finite index subgroups, namely the connected SO+(p, q) and O+(p, q), which has 2 components...
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