In quantum mechanics, the variational method is one way of finding approximations to the lowest energy eigenstate or ground state, and some excited states.
[2][3] The method consists of choosing a "trial wavefunction" depending on one or more parameters, and finding the values of these parameters for which the expectation value of the energy is the lowest possible.
Suppose we are given a Hilbert space and a Hermitian operator over it called the Hamiltonian
Ignoring complications about continuous spectra, we consider the discrete spectrum of
(see spectral theorem for Hermitian operators for the mathematical background):
Once again ignoring complications involved with a continuous spectrum of
If we were to vary over all possible states with norm 1 trying to minimize the expectation value of
Varying over the entire Hilbert space is usually too complicated for physical calculations, and a subspace of the entire Hilbert space is chosen, parametrized by some (real) differentiable parameters αi (i = 1, 2, ..., N).
This, in general, is not an easy task, since we are looking for a global minimum and finding the zeroes of the partial derivatives of ε over all αi is not sufficient.
If ψ(α) is expressed as a linear combination of other functions (αi being the coefficients), as in the Ritz method, there is only one minimum and the problem is straightforward.
This has been demonstrated by calculations using a modified harmonic oscillator as a model system, in which an exactly solvable system is approached using the variational method.
A wavefunction different from the exact one is obtained by use of the method described above.
[citation needed] Although usually limited to calculations of the ground state energy, this method can be applied in certain cases to calculations of excited states as well.
If the ground state wavefunction is known, either by the method of variation or by direct calculation, a subset of the Hilbert space can be chosen which is orthogonal to the ground state wavefunction.
This defect is worsened with each higher excited state.
This holds for any trial φ since, by definition, the ground state wavefunction has the lowest energy, and any trial wavefunction will have energy greater than or equal to it.
Proof: φ can be expanded as a linear combination of the actual eigenfunctions of the Hamiltonian (which we assume to be normalized and orthogonal):
Therefore, if the guessed wave function φ is normalized:
Another facet in variational principles in quantum mechanics is that since
can be varied separately (a fact arising due to the complex nature of the wave function), the quantities can be varied in principle just one at a time.
[4] The helium atom consists of two electrons with mass m and electric charge −e, around an essentially fixed nucleus of mass M ≫ m and charge +2e.
where ħ is the reduced Planck constant, ε0 is the vacuum permittivity, ri (for i = 1, 2) is the distance of the i-th electron from the nucleus, and |r1 − r2| is the distance between the two electrons.
If the term Vee = e2/(4πε0|r1 − r2|), representing the repulsion between the two electrons, were excluded, the Hamiltonian would become the sum of two hydrogen-like atom Hamiltonians with nuclear charge +2e.
where a0 is the Bohr radius and Z = 2, helium's nuclear charge.
The expectation value of the total Hamiltonian H (including the term Vee) in the state described by ψ0 will be an upper bound for its ground state energy.
A tighter upper bound can be found by using a better trial wavefunction with 'tunable' parameters.
Each electron can be thought to see the nuclear charge partially "shielded" by the other electron, so we can use a trial wavefunction equal with an "effective" nuclear charge Z < 2: The expectation value of H in this state is:
This is minimal for Z = 27/16 implying shielding reduces the effective charge to ~1.69.
[5] Even closer estimations of this energy have been found using more complicated trial wave functions with more parameters.
This is done in physical chemistry via variational Monte Carlo.