The Ehrenfest theorem, named after Paul Ehrenfest, the Austrian physicist and mathematician, relates the time derivative of the expectation value for a quantum mechanicaloperator to the commutator of that operator with the Hamiltonian of the system. It is
where A is some QM operator and is its expectation value. Ehrenfest’s theorem is obvious in the Heisenberg picture of quantum mechanics, where it is just the expectation value of the Heisenberg equation of motion.
Ehrenfest’s theorem is closely related to Liouville’s theorem from Hamiltonian mechanics, which involves the Poisson bracket instead of a commutator. In fact, it is a rule of thumb that a theorem in quantum mechanics which contains a commutator can be turned into a theorem in classical mechanics by changing the commutator into a Poisson bracket and multiplying by .
Jacobian matrix and determinant
In vector calculus, the Jacobian matrix is the matrix of all first-order partial derivatives of a vector– or scalar-valued function with respect to another vector. Suppose F :Rn → Rm is a function from Euclidean n-space to Euclidean m-space. Such a function is given by m real-valued component functions, y1(x1,…,xn), …, ym(x1,…,xn). The partial derivatives of all these functions (if they exist) can be organized in an m-by-n matrix, the Jacobian matrix J of F, as follows:
This matrix is also denoted by and . If (x1,…,xn) are the usual orthogonal Cartesian coordinates, the i th row (i = 1, …, m) of this matrix corresponds to the gradient of the ith component function yi: . Note that some books define the Jacobian as the transpose of the matrix given above.
The Jacobian determinant (often simply called the Jacobian) is the determinant of the Jacobian matrix.
- For the mechanical engineering and architecture usage, see isometric projection. For isometry in differential geometry, seeisometry (Riemannian geometry).