8.1: Approximate Solution of the Schrödinger Equation

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.