Feynman–Hellmann theorem

Feynman–Hellmann theorem. In quantum mechanics and quantum chemistry, the theorem relates the derivative of the total energy of a system with respect to a parameter to the expectation value of the derivative of the Hamiltonian with respect to that same parameter. It provides a powerful tool for calculating various physical quantities without solving the full Schrödinger equation. The result is attributed independently to Hans Hellmann in the context of molecular physics and Richard Feynman in his seminal work on quantum electrodynamics.

Statement of the theorem

Consider a Hamiltonian \(\hat{H}(\lambda)\) that depends on a real parameter \(\lambda\). Let \(E_n(\lambda)\) and \(|\psi_n(\lambda)\rangle\) be the \(n\)-th eigenvalue and corresponding normalized eigenstate, respectively, satisfying the time-independent Schrödinger equation. The theorem states that the derivative of the energy level with respect to the parameter is equal to the expectation value of the derivative of the Hamiltonian: \[ \frac{\partial E_n}{\partial \lambda} = \left\langle \psi_n \left| \frac{\partial \hat{H}}{\partial \lambda} \right| \psi_n \right\rangle. \] This holds provided the eigenstates are non-degenerate and vary smoothly with \(\lambda\). The theorem is foundational in perturbation theory and is extensively used in calculations involving molecular forces, polarizability, and magnetic susceptibility.

Derivation

The derivation begins with the Schrödinger equation \(\hat{H} |\psi_n\rangle = E_n |\psi_n\rangle\). Taking the derivative with respect to \(\lambda\) and then projecting onto the bra vector \(\langle \psi_n |\) yields: \[ \langle \psi_n | \frac{\partial \hat{H}}{\partial \lambda} | \psi_n \rangle + \langle \psi_n | \hat{H} | \frac{\partial \psi_n}{\partial \lambda} \rangle = \frac{\partial E_n}{\partial \lambda} \langle \psi_n | \psi_n \rangle + E_n \langle \frac{\partial \psi_n}{\partial \lambda} | \psi_n \rangle. \] Using the hermiticity of \(\hat{H}\) and the normalization condition \(\langle \psi_n | \psi_n \rangle = 1\), the terms involving derivatives of the state vectors cancel. This elegant cancellation, a hallmark of variational methods, leaves the simple result stated above. The proof assumes the eigenstates are differentiable and relies on properties of inner product spaces.

Example applications

A classic application is calculating the force on a nucleus in a molecule by taking the parameter \(\lambda\) as a nuclear coordinate. The Hellmann–Feynman force theorem directly gives the force as the expectation value of the derivative of the Hamiltonian. In atomic physics, choosing \(\lambda\) as the fine-structure constant yields insights into relativistic corrections. For a hydrogen atom in a uniform electric field (Stark effect), the theorem simplifies the computation of the polarizability. In solid-state physics, it is used to derive sum rules for optical conductivity and to compute pressure in electron gas models.

Relation to other theorems

The theorem is intimately connected to the Ehrenfest theorem, which relates time derivatives of expectation values to expectation values of commutators with the Hamiltonian. Both results emerge from the foundational principles of quantum mechanics. It is also a special case of the more general hypervirial theorem developed by Joseph O. Hirschfelder. Furthermore, the approach shares a deep conceptual link with the virial theorem in classical mechanics and its quantum counterpart, as all relate derivatives of energies to expectation values of specific operators.

Generalizations

Generalizations extend the theorem to handle degenerate states and time-dependent Hamiltonians. The Hellmann–Feynman theorem for density-functional theory is a cornerstone in computational chemistry, enabling efficient force calculations in methods like DFT. A broader mathematical framework exists within the context of Rayleigh–Ritz variational principles and stationary perturbation theory. Extensions also appear in quantum field theory, particularly in the diagrammatic techniques pioneered by Richard Feynman himself, and in statistical mechanics for derivatives of the partition function. Category:Quantum mechanics Category:Physics theorems Category:Theoretical chemistry