Categories: Mathematics, Numerical methods, Perturbation, Physics, Quantum mechanics.

Ritz method

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.

Background

In the context of variational calculus, consider the following functional to be optimized:

R[u]=1Sabp(x)ux(x)2q(x)u(x)2dx\begin{aligned} R[u] = \frac{1}{S} \int_a^b p(x) \big|u_x(x)\big|^2 - q(x) \big|u(x)\big|^2 \dd{x} \end{aligned}

Where u(x)Cu(x) \in \mathbb{C} is the unknown function, and p(x),q(x)Rp(x), q(x) \in \mathbb{R} are given. In addition, SS is the norm of uu, which we demand be constant with respect to a weight function w(x)Rw(x) \in \mathbb{R}:

S=abw(x)u(x)2dx\begin{aligned} S = \int_a^b w(x) \big|u(x)\big|^2 \dd{x} \end{aligned}

To handle this normalization requirement, we introduce a Lagrange multiplier λ\lambda, and define the Lagrangian Λ\Lambda for the full constrained optimization problem as:

Λ1S((pux2qu2)λ(wu2))\begin{aligned} \Lambda \equiv \frac{1}{S} \bigg( \big( p |u_x|^2 - q |u|^2 \big) - \lambda \big( w |u|^2 \big) \bigg) \end{aligned}

The resulting Euler-Lagrange equation is then calculated in the standard way, yielding:

0=Λuddx(Λux)=1S(qu+λwu+ddx(pux))\begin{aligned} 0 &= \pdv{\Lambda}{u^*} - \dv{}{x}\Big( \pdv{\Lambda}{u_x^*} \Big) \\ &= - \frac{1}{S} \bigg( q u + \lambda w u + \dv{}{x}\big( p u_x \big) \bigg) \end{aligned}

Which is clearly satisfied if and only if the following equation is fulfilled:

ddx(pux)+qu=λwu\begin{aligned} \dv{}{x}\big( p u_x \big) + q u = - \lambda w u \end{aligned}

This has the familiar form of a Sturm-Liouville problem (SLP), with λ\lambda 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 uu. 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 R[u]R[u], and integrate it by parts:

R[u]=1Sabpuxuxquudx=1S[puxu]ab1Sabddx(pux)u+quudx\begin{aligned} R[u] &= \frac{1}{S} \int_a^b p u_x u_x^* - q u u^* \dd{x} \\ &= \frac{1}{S} \Big[ p u_x u^* \Big]_a^b - \frac{1}{S} \int_a^b \dv{}{x}\Big(p u_x\Big) u^* + q u u^* \dd{x} \end{aligned}

The boundary term vanishes for a subset of the BCs that make a valid SLP, including Dirichlet BCs u(a)=u(b)=0u(a) = u(b) = 0, Neumann BCs ux(a)=ux(b)=0u_x(a) = u_x(b) = 0, and periodic BCs. Therefore, we assume that this term does indeed vanish, such that we can use Sturm-Liouville theory later:

R[u]=1Sab(ddx(pux)+qu)udx1SabuH^udx\begin{aligned} R[u] &= - \frac{1}{S} \int_a^b \bigg( \dv{}{x}\Big(p u_x\Big) + q u \bigg) u^* \dd{x} \equiv - \frac{1}{S} \int_a^b u^* \hat{H} u \dd{x} \end{aligned}

Where H^\hat{H} 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 λn\lambda_n with a lower bound, corresponding to mutually orthogonal eigenfunctions un(x)u_n(x).

To understand the significance of this result, suppose we have solved the SLP, and now insert one of the eigenfunctions unu_n into RR:

R[un]=1SnabunH^undx=1Snabunλnwundx=1Snλnabwun2dx=SnSnλn\begin{aligned} R[u_n] &= - \frac{1}{S_n} \int_a^b u_n^* \hat{H} u_n \dd{x} = \frac{1}{S_n} \int_a^b u_n^* \lambda_n w u_n \dd{x} \\ &= \frac{1}{S_n} \lambda_n \int_a^b w |u_n|^2 \dd{x} = \frac{S_n}{S_n} \lambda_n \end{aligned}

Where SnS_n is the normalization of unu_n. In other words, when given unu_n, the functional RR yields the corresponding eigenvalue λn\lambda_n:

R[un]=λn\begin{aligned} \boxed{ R[u_n] = \lambda_n } \end{aligned}

This powerful result was not at all clear from RR’s initial definition.

Justification

But what if we do not know the eigenfunctions? Is RR still useful? Yes, as we shall see. Suppose we make an educated guess u(x)u(x) for the ground state (i.e. lowest-eigenvalue) solution u0(x)u_0(x):

u(x)=u0(x)+n=1cnun(x)\begin{aligned} u(x) = u_0(x) + \sum_{n = 1}^\infty c_n u_n(x) \end{aligned}

Here, we are using the fact that the eigenfunctions of an SLP form a complete set, so our (known) guess uu can be expanded in the true (unknown) eigenfunctions unu_n. We are assuming that uu is already quite close to its target u0u_0, such that the (unknown) expansion coefficients cnc_n are small; specifically cn21|c_n|^2 \ll 1. Let us start from what we know:

R[u]=uH^udxuwudx\begin{aligned} \boxed{ R[u] = - \frac{\displaystyle\int u^* \hat{H} u \dd{x}}{\displaystyle\int u^* w u \dd{x}} } \end{aligned}

This quantity is known as the Rayleigh quotient. Inserting our ansatz uu, and using that the true unu_n have corresponding eigenvalues λn\lambda_n:

R[u]=(u0+ncnun)H^{u0+ncnun}dxw(u0+ncnun)(u0+ncnun)dx=(u0+ncnun)( ⁣ ⁣λ0wu0ncnλnwun)dxw(u0+ncnun)(u0+ncnun)dx\begin{aligned} R[u] &= - \frac{\displaystyle\int \Big( u_0^* + \sum_n c_n^* u_n^* \Big) \: \hat{H} \Big\{ u_0 + \sum_n c_n u_n \Big\} \dd{x}} {\displaystyle\int w \Big( u_0 + \sum_n c_n u_n \Big) \Big( u_0^* + \sum_n c_n^* u_n^* \Big) \dd{x}} \\ &= - \frac{\displaystyle\int \Big( u_0^* + \sum_n c_n^* u_n^* \Big) \Big( \!-\! \lambda_0 w u_0 - \sum_n c_n \lambda_n w u_n \Big) \dd{x}} {\displaystyle\int w \Big( u_0^* + \sum_n c_n^* u_n^* \Big) \Big( u_0 + \sum_n c_n u_n \Big) \dd{x}} \end{aligned}

For convenience, we switch to Dirac notation before evaluating further.

R=(u0+ncnun)(λ0wu0+ncnλnwun)(u0+ncnun)(wu0+ncnwun)=λ0u0|wu0+λ0n=1cnun|wu0+n=1cnλnu0|wun+mncncmλnum|wunu0|wu0+n=1cnun|wu0+n=1cnu0|wun+mncncmum|wun\begin{aligned} R &= \frac{\displaystyle \Big( \Bra{u_0} + \sum_n c_n^* \Bra{u_n} \Big) \cdot \Big( \lambda_0 \Ket{w u_0} + \sum_n c_n \lambda_n \Ket{w u_n} \Big)} {\displaystyle \Big( \Bra{u_0} + \sum_n c_n^* \Bra{u_n} \Big) \cdot \Big( \Ket{w u_0} + \sum_n c_n \Ket{w u_n} \Big)} \\ &= \frac{\displaystyle \lambda_0 \Inprod{u_0}{w u_0} + \lambda_0 \sum_{n = 1}^\infty c_n^* \Inprod{u_n}{w u_0} + \sum_{n = 1}^\infty c_n \lambda_n \Inprod{u_0}{w u_n} + \sum_{m n} c_n c_m^* \lambda_n \Inprod{u_m}{w u_n}} {\displaystyle \Inprod{u_0}{w u_0} + \sum_{n = 1}^\infty c_n^* \Inprod{u_n}{w u_0} + \sum_{n = 1}^\infty c_n \Inprod{u_0}{w u_n} + \sum_{m n} c_n c_m^* \Inprod{u_m}{w u_n}} \end{aligned}

Using orthogonality um|wun=Snδmn\Inprod{u_m}{w u_n} = S_n \delta_{mn}, and the fact that n0n \neq 0 by definition, we find:

R=λ0S0+λ0ncnSnδn0+ncnλnSnδn0+mncncmλnSnδmnS0+ncnSnδn0+ncnSnδn0+mncncmSnδmn=λ0S0+0+0+ncncnλnSnS0+0+0+ncncnSn=λ0S0+ncn2λnSnS0+ncn2Sn\begin{aligned} R &= \frac{\displaystyle \lambda_0 S_0 + \lambda_0 \sum_n c_n^* S_n \delta_{n0} + \sum_n c_n \lambda_n S_n \delta_{n0} + \sum_{m n} c_n c_m^* \lambda_n S_n \delta_{mn}} {\displaystyle S_0 + \sum_n c_n^* S_n \delta_{n0} + \sum_n c_n S_n \delta_{n0} + \sum_{m n} c_n c_m^* S_n \delta_{mn}} \\ &= \frac{\displaystyle \lambda_0 S_0 + 0 + 0 + \sum_{n} c_n c_n^* \lambda_n S_n} {\displaystyle S_0 + 0 + 0 + \sum_{n} c_n c_n^* S_n} = \frac{\displaystyle \lambda_0 S_0 + \sum_{n} |c_n|^2 \lambda_n S_n} {\displaystyle S_0 + \sum_{n} |c_n|^2 S_n} \end{aligned}

It is always possible to choose our normalizations such that Sn=SS_n = S for all unu_n, leaving:

R=λ0S+ncn2λnSS+ncn2S=λ0+ncn2λn1+ncn2\begin{aligned} R &= \frac{\displaystyle \lambda_0 S + \sum_{n} |c_n|^2 \lambda_n S} {\displaystyle S + \sum_{n} |c_n|^2 S} = \frac{\displaystyle \lambda_0 + \sum_{n} |c_n|^2 \lambda_n} {\displaystyle 1 + \sum_{n} |c_n|^2} \end{aligned}

And finally, after rearranging the numerator, we arrive at the following relation:

R=λ0+ncn2λ0+ncn2(λnλ0)1+ncn2=λ0+ncn2(λnλ0)1+ncn2\begin{aligned} R &= \frac{\displaystyle \lambda_0 + \sum_{n} |c_n|^2 \lambda_0 + \sum_{n} |c_n|^2 (\lambda_n - \lambda_0)} {\displaystyle 1 + \sum_{n} |c_n|^2} = \lambda_0 + \frac{\displaystyle \sum_{n} |c_n|^2 (\lambda_n - \lambda_0)} {\displaystyle 1 + \sum_{n} |c_n|^2} \end{aligned}

Thus, if we improve our guess uu, then R[u]R[u] approaches the true eigenvalue λ0\lambda_0. For numerically finding u0u_0 and λ0\lambda_0, this gives us a clear goal: minimize RR, because:

R[u]=λ0+n=1cn2(λnλ0)1+n=1cn2λ0\begin{aligned} \boxed{ R[u] = \lambda_0 + \frac{\displaystyle \sum_{n = 1}^\infty |c_n|^2 (\lambda_n - \lambda_0)} {\displaystyle 1 + \sum_{n = 1}^\infty |c_n|^2} \ge \lambda_0 } \end{aligned}

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 λ0\lambda_0 goes as cn2|c_n|^2, while uu converges to u0u_0 as cn|c_n| by definition, so even a fairly bad guess uu will give a decent estimate for λ0\lambda_0.

The method

In the following, we stick to Dirac notation, since the results hold for both continuous functions u(x)u(x) and discrete vectors u\vb{u}, as long as the operator H^\hat{H} is self-adjoint. Suppose we express our guess u\Ket{u} as a linear combination of known basis vectors fn\Ket{f_n} with weights anCa_n \in \mathbb{C}:

u=n=0cnun=n=0anfnn=0N1anfn\begin{aligned} \Ket{u} = \sum_{n = 0}^\infty c_n \Ket{u_n} = \sum_{n = 0}^\infty a_n \Ket{f_n} \approx \sum_{n = 0}^{N - 1} a_n \Ket{f_n} \end{aligned}

For numerical tractability, we truncate the sum at NN terms, and for generality, we allow fn\Ket{f_n} to be non-orthogonal, as described by an overlap matrix with elements SmnS_{mn}:

fm|wfn=Smn\begin{aligned} \Inprod{f_m}{w f_n} = S_{m n} \end{aligned}

From the discussion above, we know that the ground-state eigenvalue λ0\lambda_0 is estimated by:

λ0λ=R[u]=uH^uu|wu=mnamanfmH^fnmnamanfm|wfnmnamanHmnmnamanSmn\begin{aligned} \lambda_0 \approx \lambda = R[u] = \frac{\inprod{u}{\hat{H} u}}{\Inprod{u}{w u}} = \frac{\displaystyle \sum_{m n} a_m^* a_n \inprod{f_m}{\hat{H} f_n}}{\displaystyle \sum_{m n} a_m^* a_n \Inprod{f_m}{w f_n}} \equiv \frac{\displaystyle \sum_{m n} a_m^* a_n H_{m n}}{\displaystyle \sum_{m n} a_m^* a_n S_{mn}} \end{aligned}

And we also know that our goal is to minimize R[u]R[u], so we vary aka_k^* to find its extremum:

0=Rak=(nanHkn)(mnanamSmn)(nanSkn)(mnanamHmn)(mnanamSmn)2=(nanHkn)R[u](nanSkn)u|wu=nan(HknλSkn)u|wu\begin{aligned} 0 = \pdv{R}{a_k^*} &= \frac{\displaystyle \Big( \sum_{n} a_n H_{k n} \Big) \Big( \sum_{m n} a_n a_m^* S_{mn} \Big) - \Big( \sum_{n} a_n S_{k n} \Big) \Big( \sum_{m n} a_n a_m^* H_{mn} \Big)} {\Big( \displaystyle \sum_{m n} a_n a_m^* S_{mn} \Big)^2} \\ &= \frac{\displaystyle \Big( \sum_{n} a_n H_{k n} \Big) - R[u] \Big( \sum_{n} a_n S_{k n}\Big)}{\Inprod{u}{w u}} = \frac{\displaystyle \sum_{n} a_n \big(H_{k n} - \lambda S_{k n}\big)}{\Inprod{u}{w u}} \end{aligned}

Clearly, this is only satisfied if the following holds for all k=0,1,...,N ⁣ ⁣1k = 0, 1, ..., N\!-\!1:

0=n=0N1an(HknλSkn)\begin{aligned} 0 = \sum_{n = 0}^{N - 1} a_n \big(H_{k n} - \lambda S_{k n}\big) \end{aligned}

For illustrative purposes, we can write this as a matrix equation with MknHknλSknM_{k n} \equiv H_{k n} - \lambda S_{k n}:

[M0,0M0,1M0,N1M1,0MN1,0MN1,N1][a0a1aN1]=[000]\begin{aligned} \begin{bmatrix} M_{0,0} & M_{0,1} & \cdots & M_{0,N-1} \\ M_{1,0} & \ddots & & \vdots \\ \vdots & & \ddots & \vdots \\ M_{N-1,0} & \cdots & \cdots & M_{N-1,N-1} \end{bmatrix} \cdot \begin{bmatrix} a_0 \\ a_1 \\ \vdots \\ a_{N-1} \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 0 \end{bmatrix} \end{aligned}

Note that this looks like an eigenvalue problem for λ\lambda. Indeed, demanding that M\overline{M} cannot simply be inverted (i.e. the solution is non-trivial) yields a characteristic polynomial for λ\lambda:

0=det ⁣[M]=det ⁣[HλS]\begin{aligned} 0 = \det\!\Big[ \overline{M} \Big] = \det\!\Big[ \overline{H} - \lambda \overline{S} \Big] \end{aligned}

This gives a set of λ\lambda, which are the exact eigenvalues of H\overline{H}, and the estimated eigenvalues of H^\hat{H} (recall that H\overline{H} is H^\hat{H} expressed in a truncated basis). The eigenvector [a0,a1,...,aN1]\big[ a_0, a_1, ..., a_{N-1} \big] of the lowest λ\lambda gives the optimal weights to approximate u0\Ket{u_0} in the basis {fn}\{\Ket{f_n}\}. Likewise, the higher λ\lambda’s eigenvectors approximate excited (i.e. non-ground) eigenstates of H^\hat{H}, 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 u0\Ket{u_0} resides. Clearly, adding more basis vectors will improve the results, at the cost of computation. For example, if H^\hat{H} represents a helium atom, a good choice for {fn}\{\Ket{f_n}\} would be hydrogen orbitals, since those are qualitatively similar.

You may find this result unsurprising; it makes some intuitive sense that approximating H^\hat{H} in a limited basis would yield a matrix H\overline{H} 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 λ\lambda from R[u]R[u], so all we need to do is solve the above matrix equation for ana_n. Keep in mind that M\overline{M} is singular, and ana_n are only defined up to a constant factor.

Nowadays, there exist many other methods to calculate eigenvalues of complicated operators H^\hat{H}, 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.

References

  1. G.B. Arfken, H.J. Weber, Mathematical methods for physicists, 6th edition, 2005, Elsevier.
  2. O. Bang, Applied mathematics for physicists: lecture notes, 2019, unpublished.

© 2023 Marcus R.A. Newman, CC BY-NC-SA 4.0.
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