8.1: Approximate Solution of the Schrödinger Equation
- Page ID
- 28788
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.