8.1: Approximate Solution of the Schrödinger Equation

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If we can’t find an analytic solution to the Schrödinger equation, a trick known as the variational principle allows us to estimate the energy of the ground state of a system. We choose an unnormalized trial function \(\Phi (a_n)\) which depends on some variational parameters, \(a_n\) and minimise

\[E[a_n] = \frac{\langle \Phi |\hat{H} |\Phi \rangle}{\langle \Phi |\Phi \rangle} \nonumber\]

with respect to those parameters. This gives an approximation to the wavefunction whose accuracy depends on the number of parameters and the clever choice of \(\Phi (a_n)\). For more rigorous treatments, a set of basis functions with expansion coefficients \(a_n\) may be used.

The proof is as follows, if we expand the normalised wavefunction

\[|\phi (a_n) \rangle = \Phi (a_n)/ \langle \Phi (a_n)|\Phi (a_n) \rangle^{1/2} \nonumber\]

in terms of the true (unknown) eigenbasis \(|i\rangle\) of the Hamiltonian, then its energy is

\[E[a_n] = \sum_{ij} \langle\phi |i \rangle \langle i|\hat{H} |j \rangle \langle j|\phi \rangle = \sum_i |\langle\phi |i \rangle |^2 E_i = E_0 + \sum_i |\langle\phi |i\rangle |^2 (E_i − E_0) \geq E_0 \nonumber\]

where the true (unknown) ground state of the system is defined by \(\hat{H} |i_0 \rangle = E_0|i_0 \rangle\). The inequality arises because both \(|\langle \phi |i\rangle |^2\) and \((E_i − E_0)\) must be positive.

Thus the lower we can make the energy \(E[a_i ]\), the closer it will be to the actual ground state energy, and the closer \(|\phi \rangle\) will be to \(|i_0\rangle\).

If the trial wavefunction consists of a complete basis set of orthonormal functions \(|\chi i\rangle\), each multiplied by \(a_i : |\phi \rangle = \sum_i a_i |\chi_i \rangle\) then the solution is exact and we just have the usual trick of expanding a wavefunction in a basis set. Alternately, we might just use an incomplete set with a few low-energy basis functions to get a \(|\Phi \rangle\) close to the ground state \(|i_0 \rangle\). In practice, this is how most quantum mechanics problems are solved.