= Solution
The <quantum variational principle> turns the ground-state problem into minimization of a <Rayleigh quotient>:
$$
E[\psi]=\frac{\langle\psi|H|\psi\rangle}{\langle\psi|\psi\rangle}\geq E_0.
$$
The <Hamiltonian> must be a <self-adjoint operator> bounded below, and the nonzero trial state must have finite form <energy>. Expanding a normalized trial state in exact <energy eigenstates> gives $E[\psi]=\sum_n|c_n|^2E_n$, a weighted average no smaller than $E_0$. Equality requires support only in the ground-state eigenspace. The spectral version also accommodates continuous <spectrum>; a lowest normalizable <eigenstate> is not required merely to bound the spectral infimum.
The <variational method> chooses a trial family with the correct <boundary conditions>, <fermionic exchange symmetry> and other required <quantum numbers>, evaluates its <energy> and minimizes over its parameters. It gives an \b[upper bound on <ground-state energy>]. Enlarging a nested trial family can only improve or preserve its optimal bound. A low <energy> does not by itself establish a highly accurate <wavefunction>, especially near degeneracy; <energy> errors are second order in small admixtures orthogonal to a nondegenerate exact <ground state>.
For example, a normalized Gaussian trial for a one-dimensional <quantum harmonic oscillator> is $\psi_\beta=(\beta/\pi)^{1/4}e^{-\beta x^2/2}$, with $\beta>0$. Its <energy> is
$$
E(\beta)=\frac{\hbar^2\beta}{4m}+\frac{m\omega^2}{4\beta},\qquad
\boxed{\beta_{\rm opt}=m\omega/\hbar,\quad E_{\min}=\hbar\omega/2.}
$$
The two terms balance kinetic localization cost against the potential cost of spreading out. This <Gaussian variational energy for the harmonic oscillator> happens to attain the exact answer because the family contains the exact <ground state>.
\Image[/past-exam-of-the-advanced-quantum-physics-course-of-the-university-of-cambridge/2015/paper-1-variational.png]
{title=Gaussian variational energy and its exact harmonic-oscillator minimum}
A trial expansion $\psi=\sum_j c_j\chi_j$ in a finite, possibly nonorthogonal basis gives the <generalized eigenvalue problem> $H\mathbf c=ES\mathbf c$, with $H_{ij}=\langle\chi_i|H|\chi_j\rangle$ and $S_{ij}=\langle\chi_i|\chi_j\rangle$. Its lowest Ritz value minimizes the quotient in that subspace. For excited states, the <excited-state variational constraint> imposes exact <orthogonality> to the lower <eigenstates> to obtain a corresponding upper bound; the <Courant–Fischer min-max principle> yields upper bounds for ordered Ritz <eigenvalues> from a common subspace. <orthogonality> only to an independently approximate <ground state> does not automatically provide an excited-state bound: residual true-ground-state contamination can lower the trial <energy> below the desired excited <energy>. Symmetry can impose exact <orthogonality> when the relevant states have different symmetry sectors.
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