费曼-海尔曼定理

title: Example applications.
title: Molecular forces.
content: The most common application of the Hellmann–Feynman theorem is to the calculation of intramolecular forces in molecules. This allows for the calculation of equilibrium geometries – the nuclear coordinates where the forces acting upon the nuclei, due to the electrons and other nuclei, vanish. The parameter λ corresponds to the coordinates of the nuclei. For a molecule with 1 ≤ "i" ≤ "N" electrons with coordinates {r}, and 1 ≤ α ≤ "M" nuclei, each located at a specified point {R={"X","Y","Z")} and with nuclear charge "Z", the clamped nucleus Hamiltonian is  The force acting on the x-component of a given nucleus is equal to the negative of the derivative of the total energy with respect to that coordinate. Employing the Hellmann–Feynman theorem this is equal to  Only two components of the Hamiltonian contribute to the required derivative – the electron-nucleus and nucleus-nucleus terms. Differentiating the Hamiltonian yields  Insertion of this in to the Hellmann–Feynman theorem returns the force on the x-component of the given nucleus in terms of the electronic density ("ρ"(r)) and the atomic coordinates and nuclear charges:
title: Expectation values.
content: An alternative approach for applying the Hellmann–Feynman theorem is to promote a fixed or discrete parameter which appears in a Hamiltonian to be a continuous variable solely for the mathematical purpose of taking a derivative. Possible parameters are physical constants or discrete quantum numbers. As an example, the radial Schrödinger equation for a hydrogen atom is  which depends upon the discrete azimuthal quantum number "l". Promoting "l" to be a continuous parameter allows for the derivative of the Hamiltonian to be taken:  The Hellmann–Feynman theorem then allows for the determination of the expectation value of formula_8 for hydrogen-like atoms:
title: Van der Waals forces.
content: In the end of Feynman's paper, he states that, "Van der Waals's forces can also be interpreted as arising from charge distributions with higher concentration between the nuclei. The Schrödinger perturbation theory for two interacting atoms at a separation "R", large compared to the radii of the atoms, leads to the result that the charge distribution of each is distorted from central symmetry, a dipole moment of order 1/"R" being induced in each atom. The negative charge distribution of each atom has its center of gravity moved slightly toward the other. It is not the interaction of these dipoles which leads to van der Waals's force, but rather the attraction of each nucleus for the distorted charge distribution of its "own" electrons that gives the attractive 1/"R" force".
title: 随时间变化的波函数的费曼–海尔曼定理.
content: 因为一个一般的随时间变化的波函数满足含时薛定谔方程,所以费曼–海尔曼定理不再适用。  但是,该波函数满足以下关系:  其中ψ满足:
title: 证明.
content: 以下证明只依赖于薛定谔方程,以及对于λ和t求偏导时,可以互换顺序的假设。

Summary: 量子力学中,费曼–海尔曼定理描述的是一个体系的能量对某个参量的导数与哈密顿量算符对同一参量的导数的期望值之间的关系。根据这一定理,通过求解薛定谔方程得到电子密度的空间分布后,体系中的所有力都能通过经典静电学求出。

Hellmann–Feynman theorem: In quantum mechanics, the Hellmann–Feynman theorem relates the derivative of the total energy with respect to a parameter, to the expectation value of the derivative of the Hamiltonian with respect to that same parameter. According to the theorem, once the spatial distribution of the electrons has been determined by solving the Schrödinger equation, all the forces in the system can be calculated using classical electrostatics.