While variation of the energy functional yields the Schrödinger equation in the usual, linear case, no such statement can be formulated in the general nonlinear situation when the Hamiltonian depends on its eigenvector. In this latter case, as we illustrate by sample numerical calculations, the points of the energy expectation value hypersurface where the eigenvalue equation is satisfied separate from those where the energy is stationary. We show that the variation of the energy at the eigensolution is determined by a generalized Hellmann–Feynman theorem. Functionals, other than the energy, can, however be constructed, that result the nonlinear Schrödinger equation upon setting their variation zero. The second centralized moment of the Hamiltonian is one example.

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关于非线性薛定谔方程的变分原理

虽然能量函数的变化在通常的线性情况下产生薛定谔方程，但当哈密顿量取决于其特征向量时，在一般非线性情况下无法表述这样的陈述。在后一种情况下，正如我们通过示例数值计算说明的那样，满足特征值方程的能量期望值超曲面的点与能量静止的点分开。我们表明，本征解处能量的变化是由广义 Hellmann-Feynman 定理决定的。然而，可以构造除能量之外的泛函，在将它们的变化设为零时产生非线性薛定谔方程。哈密​​顿量的第二中心矩就是一个例子。