The variational method is based on the Variational principle which says that a wavefunction that is not the true wavefunction will always yield a value for the energy that is greater than the true ground state energy of the system. This principle can be proven using the superposition theorem that was previously discussed. Theorem \(\PageIndex{1}\): Variational Method Assume a trial wavefunction \(\psi(x)\) describing a particle in a box, that can be expressed as a linear combination of the normal particle in a box wavefunctions. \[\psi (x)=\sum _{n}c_{n} \phi _{n} (x)\nonumber \] Assuming \(\psi(x)\) is normalized, the expectation value of energy \(\langle E \rangle \) is obtained from the expression \[\left\langle E\right\rangle =\int \psi (x)\, \hat{H}\, \psi (x)\, d\tau \nonumber\] Substituting the expression for \(\psi(x)\) from above \[\left\langle E\right\rangle =\int \left(\sum _{m}c_{m} \phi _{m} \right)\hat{H}\left(\sum _{n}c_{n} \phi _{n} \right) d\tau \nonumber \] Noting that \[\hat{H}\phi _{n} =E_{n} \phi _{n}\nonumber \] Substitution yields \[\left\langle E\right\rangle =\int \left(\sum _{m}c_{m} \phi _{m} \right)\left(\sum _{n}c_{n} E_{n} \phi _{n} \right) d\tau\nonumber \] Gathering terms, one obtains \[\begin{align*} \left\langle E\right\rangle &= \int \left(\sum _{m}\sum _{n}c_{m} c_{n} E_{n} \phi _{m} \phi _{n} \right) d\tau \\[4pt] &= \sum _{m}\sum _{n}c_{m} c_{n} E_{n} \int \left(\phi _{m} \phi _{n} \right) d\tau \\[4pt] &= \sum _{m}\sum _{n}c_{m} c_{n} E_{n} \delta _{mn} \end{align*} \] The Kronecker delta will destroy one of the summations since it will pick out only one value to be non-zero. \[\begin{align*} \left\langle E\right\rangle &= \sum _{m}\sum _{n}c_{m} c_{n} E_{n} \delta _{mn} \\[4pt] &=\sum _{n}c_{n}^{2} E_{n} \end{align*} \] Thus if any components of the linear combination have a non-zero contribution (\(c_ {n} \neq 0\) for \(n > 1\)) the expectation value has to be larger than \(E_ {1}\). The Variational principle can be used to determine reasonable trial wavefunctions (\(\Psi\)) based on a set of approximate wavefunctions (\(\phi_ {n}\)). This is done by assuming the trial wavefunction can be expressed as a linear combination of the approximate wavefunctions \[\Psi =\sum _{n}c_{n} \phi _{n}\nonumber \] and then determining the contribution to the trial function by minimizing the energy with respect to the coefficients (\(c_ {n}\)) in the expansion. \[\dfrac{\partial }{\partial c_{n} } \left\langle E\right\rangle =0\nonumber \] This will produce n equations with \(n\) unknown values of \(c_ {n}\) which can be simultaneously solved to yield the optimal values of \(c_ {n}\). This methodology is used to a great extent in computational chemistry methods. Example \(\PageIndex{1}\) What is \(\langle E \rangle\) for a system with the following wavefunction that approximates \(\psi_{1}\) (x) for a particle in a box? \[\psi (x)=\sqrt{\dfrac{30}{a^{5} } } \cdot x\cdot \left(a-x\right) \nonumber \] Solution The wavefunction is a reasonable, but not perfect, approximation of the \(n=1\) level of a particle in a box. \[\phi_1( x )=\sqrt{\frac{2}{ a }} \cdot \sin \left(\frac{\pi \cdot x }{ a }\right) \nonumber\] The expectation value of energy is found in the usual manner. \[\begin{align*} \left\langle E\right\rangle &= \int _{0}^{a}\psi {\kern 1pt} \hat{H}{\kern 1pt} \psi \, d\tau \\[4pt] &=-\dfrac{\hbar ^{2} }{2m} \dfrac{30}{a^{5} } \int _{0}^{a}\left(ax-x^{2} \right)\dfrac{d^{2} }{dx^{2} } \left(ax-x^{2} \right)dx =-\dfrac{15\hbar ^{2} }{ma^{5} } \int _{0}^{a}\left(ax-x^{2} \right)\left(-2\right)dx \\[4pt] &=\dfrac{30\hbar ^{2} }{ma^{5} } \left[\dfrac{ax^{2} }{2} -\dfrac{x^{3} }{3} \right]_{0}^{a} =\dfrac{30\hbar ^{2} }{ma^{5} } \left(\dfrac{a^{3} }{6} \right) \\[4pt] &= \dfrac{5\hbar ^{2} }{ma^{2} } \end{align*}\nonumber \] This result is slightly larger than \(\dfrac{h^ {2}}{ 8ma ^{2}}\) since \(\dfrac{5}{(2\pi) ^{2}} = 0.127\) and \(\dfrac{1}{8} = 0.125\). In the variational method, an approximate form of a wave function can be used