In various branches of physics,
the Ritz method is a technique to approximately find the lowest solutions to an eigenvalue problem.
Some call it the Rayleigh-Ritz method, the Ritz-Galerkin method,
or simply the variational method.
In the context of variational calculus,
consider the following functional to be optimized:
Where is the unknown function,
and are given.
In addition, is the norm of , which we demand be constant
with respect to a weight function :
To handle this normalization requirement,
we introduce a Lagrange multiplier ,
and define the Lagrangian for the full constrained optimization problem as:
The resulting Euler-Lagrange equation is then calculated in the standard way, yielding:
Which is clearly satisfied if and only if the following equation is fulfilled:
This has the familiar form of a Sturm-Liouville problem (SLP),
with representing an eigenvalue.
SLPs have useful properties, but before we can take advantage of those,
we need to handle an important detail: the boundary conditions (BCs) on .
The above equation is only a valid SLP for certain BCs,
as seen in the derivation of Sturm-Liouville theory.
Let us return to the definition of ,
and integrate it by parts:
The boundary term vanishes for a subset of the BCs that make a valid SLP,
including Dirichlet BCs , Neumann BCs , and periodic BCs.
Therefore, we assume that this term does indeed vanish,
such that we can use Sturm-Liouville theory later:
Where is the self-adjoint Sturm-Liouville operator.
Because the constrained Euler-Lagrange equation is now an SLP,
we know that it has an infinite number of real discrete eigenvalues with a lower bound,
corresponding to mutually orthogonal eigenfunctions .
To understand the significance of this result,
suppose we have solved the SLP,
and now insert one of the eigenfunctions into :
Where is the normalization of .
In other words, when given ,
the functional yields the corresponding eigenvalue :
This powerful result was not at all clear from ’s initial definition.
But what if we do not know the eigenfunctions? Is still useful?
Yes, as we shall see. Suppose we make an educated guess
for the ground state (i.e. lowest-eigenvalue) solution :
Here, we are using the fact that the eigenfunctions of an SLP form a complete set,
so our (known) guess can be expanded in the true (unknown) eigenfunctions .
We are assuming that is already quite close to its target ,
such that the (unknown) expansion coefficients are small;
Let us start from what we know:
This quantity is known as the Rayleigh quotient.
Inserting our ansatz ,
and using that the true have corresponding eigenvalues :
For convenience, we switch to Dirac notation
before evaluating further.
Using orthogonality ,
and the fact that by definition, we find:
It is always possible to choose our normalizations such that for all , leaving:
And finally, after rearranging the numerator, we arrive at the following relation:
Thus, if we improve our guess ,
then approaches the true eigenvalue .
For numerically finding and , this gives us a clear goal: minimize , because:
In the context of quantum mechanics, this is not surprising,
since any superposition of multiple states
is guaranteed to have a higher energy than the ground state.
Note that the convergence to goes as ,
while converges to as by definition,
so even a fairly bad guess will give a decent estimate for .
In the following, we stick to Dirac notation,
since the results hold for both continuous functions and discrete vectors ,
as long as the operator is self-adjoint.
Suppose we express our guess as a linear combination
of known basis vectors with weights :
For numerical tractability, we truncate the sum at terms,
and for generality, we allow to be non-orthogonal,
as described by an overlap matrix with elements :
From the discussion above,
we know that the ground-state eigenvalue is estimated by:
And we also know that our goal is to minimize ,
so we vary to find its extremum:
Clearly, this is only satisfied if the following holds for all :
For illustrative purposes,
we can write this as a matrix equation
Note that this looks like an eigenvalue problem for .
Indeed, demanding that cannot simply be inverted
(i.e. the solution is non-trivial)
yields a characteristic polynomial for :
This gives a set of ,
which are the exact eigenvalues of ,
and the estimated eigenvalues of
(recall that is expressed in a truncated basis).
The eigenvector of the lowest
gives the optimal weights to approximate in the basis .
Likewise, the higher ’s eigenvectors approximate
excited (i.e. non-ground) eigenstates of ,
although in practice the results are less accurate the higher we go.
The overall accuracy is determined by how good our truncated basis is,
i.e. how large a subspace it spans
of the Hilbert space in which the true resides.
Clearly, adding more basis vectors will improve the results,
at the cost of computation.
For example, if represents a helium atom,
a good choice for would be hydrogen orbitals,
since those are qualitatively similar.
You may find this result unsurprising;
it makes some intuitive sense that approximating
in a limited basis would yield a matrix giving rough eigenvalues.
The point of this discussion is to rigorously show
the validity of this approach.
If we only care about the ground state,
then we already know from ,
so all we need to do is solve the above matrix equation for .
Keep in mind that is singular,
and are only defined up to a constant factor.
Nowadays, there exist many other methods to calculate eigenvalues
of complicated operators ,
but an attractive feature of the Ritz method is that it is single-step,
whereas its competitors tend to be iterative.
That said, the Ritz method cannot recover from a poorly chosen basis.
- G.B. Arfken, H.J. Weber,
Mathematical methods for physicists, 6th edition, 2005,
- O. Bang,
Applied mathematics for physicists: lecture notes, 2019,