20  Weak formulation of the Schrödinger equation

We have now amassed a set of tools that allow us to formulate the Schrödinger equation using Sobolev spaces. This formulation is often called the weak formulation, since we do not require solutions to be classical in the sense that any derivative that occurs is of the weak type.

For simplicity, we consider a single spinless particle in \(\mathbb{R}^3\), with Hamiltonian \[\mathcal{H} = - \frac{1}{2}\nabla^2 + V(\vec{r}),\] where \(V(\vec{r})\) is a multiplicative potential operator, assumed for the moment to be measurable and in \(L^1_\text{loc}(\mathbb{R}^3)\). For the moment we let the potential be otherwise unknown. The calligraphic \(\mathcal{H}\) is to distinguish it from an actual Hilbert-space operator. It is formal, a term mathematicians use about physics formulas that are not rigorous mathematics. Sometimes, but not always, it’s just because they don’t understand it. (It is also a term that physicists use about mathematics they don’t understand, so mathematicians and physicists are really not that different.)

Physicists now want to solve the eigenvalue problem of \(\mathcal{H}\), i.e., find nonzero \(\psi : \mathbb{R}^3 \to \mathbb{C}\) and \(E\in \mathbb{R}\) such that

\[ \mathcal{H}\psi = E\psi. \tag{20.1}\]

This equation is also formal, because we don’t know what kinds of functions \(\mathcal{H}\) can operate on.

In order to connect with spectral theory of self-adjoint operators on Hilbert space, we must find a self-adjoint \(\hat{H} : D(\hat{H}) \to L^2\) with domain \(D(\hat{H})\) that somehow represents \(\mathcal{H}\). If we don’t do this properly, we may end up “missing” eigenfunctions, or even create complex eigenvalues, which is nonsense.

So let \(\phi : \mathbb{R}^3 \to \mathbb{C}\) be a \(C^\infty_0(\mathbb{R}^3)\) function. This is a dense set of \(L^2\), and it is typical in functional analysis to study such well-behaved sets first, then take limits or closures afterwards. With such a \(\phi\), \(\mathcal{H}\phi\) is a measurable function; \(\nabla^2\phi \in C^\infty_0(\mathbb{R}^3)\), and \(V\phi\) is a measurable function (since products of measurable functions are measurable). Requiring \(\mathcal{H}\phi\) to be actually in \(L^2\) might be too strong, so let us consider the energy expectation value, \[\mathcal{E}(\phi) = \frac{\left\langle\phi,\mathcal{H}\phi\right\rangle}{\left\langle\phi,\phi\right\rangle}.\] This is now a well-defined function on all nonzero elements of \(C^\infty_0(\mathbb{R}^3)\). To see this, note that the denominator is certainly well-defined. The numerator is \[\left\langle\phi,\mathcal{H}\phi\right\rangle = \int \frac{-1}{2} \overline{\phi} \nabla^2 \phi + \overline{\phi} V\phi.\] Since \(V\phi\) is measurable, \[|\int |\phi|^2 V | \leq \|\phi\|_\infty^2 \int_K |V| < + \infty,\] where \(K\subset \mathbb{R}^3\) is the support of \(\phi\). Thus, the numerator is well-defined as well.

We now define the ground-state energy of \(\mathcal{H}\) to be \[E_0 = \inf \{ \mathcal{E}(\phi) \mid \phi \in C^\infty_0(\mathbb{R}^3), \phi \neq 0 \}\] where the infimum is the greatest lower bound of the function, i.e., \(E_0\) is the largest number such that there is no \(\phi\) that gives \(\mathcal{E}(\phi) < E_0\).

As chemists, we know that the elctronic wavefunction has cusps. Thus, the space \(C^\infty_0\) cannot be large enough to capture our eigenfunctions. We need to extend the energy function to a properly large space.

Using integration by parts, we write the kinetic energy as \[-\frac{1}{2}\left\langle\phi,\nabla^2\phi\right\rangle = \frac{1}{2} \left\langle\nabla\phi,\nabla\phi\right\rangle.\] This is one out of two locations where the term weak formulation is relevant, because we no longer differentiate twice, but only once. The second location is the following: We would like to define \(\mathcal{E}\) on a complete vector space, because we would like to say something about existence of a minimizer, i.e., existence of the eigenfunction of the energy, e.g., since we know about cusps, and also that the exact eigenfunction does not suddenly come zero, such as is the case with \(C^\infty_0\). Our enlarged space should be a subspace of \(L^2\) due to the fact that we want a probability interpretation. (See also de denominator.) The largest subspace of \(L^2\) where the kinetic energy is finite is the Sobolev space \(H^1\). This space is a space of weak derivatives.

So our candidate space is \(H^1\). What assumptions do we need on \(V\) in order to make the potential energy well-defined? We also need to make sure that if we find a minimizer of \(\mathcal{H}\), this will be an eigenfunction of a self-adjoint operator \(\hat{H} : D(\hat{H}) \to L^2\), because this is needed to define quantum mechanics.

Suppose \(V\) is such that it is relatively form-bounded from below by kinetic energy, in the following manner: Suppose that there exists \(\epsilon \in[0,1[\) and a \(C_\epsilon \geq 0\), such that for all \(\phi\in H^1\),

\[ |\left\langle\phi,V\phi\right\rangle| \leq \epsilon \frac{1}{2}\left\langle\nabla\phi,\nabla\phi\right\rangle + C_\epsilon \|\phi\|^2. \tag{20.2}\]

Then we see that \[\left\langle\phi,\mathcal{H}\phi\right\rangle \geq \frac{1}{2}(1-\epsilon) \left\langle\nabla\phi,\nabla\phi\right\rangle - C_\epsilon \|\phi\|^2.\] Thus, \[\mathcal{E}(\phi) \geq \frac{1}{2}(1-\epsilon) \frac{\left\langle\nabla\phi,\nabla\phi\right\rangle}{\left\langle\phi,\phi\right\rangle} - C_\epsilon \geq - C_\epsilon,\] and therefore the function \(\mathcal{E}\) is defined on \(H^1\) and bounded from below.

Not only that, but we also get \[\left\langle\phi,\mathcal{H}\phi\right\rangle \leq \frac{1}{2}(1 + \epsilon)\|\nabla\phi\|^2 + C_\epsilon \|\phi\|^2 \leq C_\epsilon'\|\phi\|_{H^1}^2,\] i.e., that \(\mathcal{E}\) is also bounded above.

It turns out that the following is true:

Lemma 20.1 (Form boundedness of a class of potentials) Let \(V \in L^2(\mathbb{R}^3) + L^\infty(\mathbb{R}^3)\). Then \(V\) is relatively form bounded in the sense of Equation 20.2.

There is a powerful representation theorem for quadratic forms such as \(h(\phi) = \left\langle\phi,\mathcal{H}\phi\right\rangle\). Note that we have proven (or at least outlined the proof) that \(|h(\phi)| \leq C \|\phi\|_{H^1}\) for some constant \(C\). Them, the representation theorem says that there is a unique self-adjoint operator \(\hat{H} : D(\hat{H})\to L^2\) such that \(D(\hat{H})\subset H^1\) (a dense subset), and such that for all \(\phi \in D(\hat{H})\), \[h(\phi) = \left\langle\phi,\hat{H}\phi\right\rangle.\] We are almost done, because now we can differentiate the function \(\mathcal{E} : H^1 \to \mathbb{R}\) to form the eigenvalue equation \[\hat{H}\psi = E\psi, \quad E = \mathcal{E}(\psi).\] The question is: what is \(D(\hat{H})\)? A remarkable fact is that for \(V \in L^2 + L^\infty\), then \(D(\hat{H}) = H^2(\mathbb{R}^3)\).

Note that even if we have a weak formulation of the Schrödinger equation, that only assumes one weak derivative in the variational principle, we actually get eigenfunctions that are twice weakly differentiable!

We summarize as a theorem:

Theorem 20.1 (Weak formulation of the Schrödinger equation) Let \(V \in L^2 + L^\infty\) (which includes the Coulomb potential of the hydroge atom). Then the operator \(\hat{H} : H^2\to L^2\) given by \[\hat{H} = -\frac{1}{2}\nabla^2 + V\] is self-adjoint. Its eigenfunctions are critical points of the energy function \(\mathcal{E} : H^1 \to \mathbb{R}\) given by \[\mathcal{E}(\phi) = \frac{\frac{1}{2}\left\langle\nabla\phi,\nabla\phi\right\rangle + \left\langle\phi,V\phi\right\rangle}{\left\langle\phi,\phi\right\rangle}.\]

We note in passing, that even for functions \(\phi \in H^1\) we can defined \(\nabla^2\phi\) as a distribution, i.e., a generalized function. This distribution is such that \(\left\langle\phi,-\nabla^2\phi\right\rangle = \left\langle\nabla\phi,\nabla\phi\right\rangle\).

We also note that the construction can be generalized to \(N\)-electron Hamiltonians with Coulomb interactions between pairs of electrons and between electrons and nuclei.