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The constrained-search functional $F[n]={\min }_{\Psi \to n}⟨\Psi |T+W|\Psi ⟩$ is universal for a fixed interaction. Minimization of $F[n]+\int vn$ over admissible densities with the correct electron number gives the ground-state energy. Each trial density represents a constrained set of antisymmetric states, so minimizing over densities and their representing states recovers the ordinary variational principle. The admissible domain and possible degeneracy must be handled explicitly.

Gerald Beresford Whitham developed an averaged variational principle for slowly varying dispersive waves. The method averages a Lagrangian over a rapid wave cycle and derives equations for the slower evolution of the wave parameters. It supplies a common framework for linear and nonlinear wave modulation instead of treating every wave train as a separate perturbation calculation.^[ref]

Let $\ddot{q}=Fq$ on a Hilbert space, with a time-independent self-adjoint operator $F$ that is a bounded linear operator, and conserved quadratic energy $E=\Vert \dot{q}{\Vert }^2/2-⟨q,Fq⟩/2$. If $⟨q_0,F{q}_0⟩>0$, set $\dot{q}(0)=\alpha q_0$ with ${\alpha }^2=⟨q_0,F{q}_0⟩/\Vert q_0{\Vert }^2$ and $\alpha >0$. This gives $E=0$. For $N=\Vert q{\Vert }^2$, $N^{\prime }=2\mathrm{Re}⟨q,\dot{q}⟩$ and $N^{\prime \prime }=4\Vert \dot{q}{\Vert }^2$, soby the Cauchy-Schwarz inequality. Since $(\ln N{)}^{\prime }(0)=2\alpha$, $N(t)\ge N(0)e^{2\alpha t}$ for $t\ge 0$. Positivity of $N$ follows from $N^{\prime }(0)>0$ and $N^{\prime \prime }\ge 0$. Thus a negative direction of the restoring energy $W=-⟨q,Fq⟩/2$ yields an exponentially growing solution without a normal-mode expansion or a variational spectral theorem. If $W$ is positive definite, conserved $E$ instead controls the energy norm; a bound in the original Hilbert space norm additionally follows from a coercive lower bound on $W$. For an unbounded self-adjoint operator, the same estimate applies to sufficiently regular solutions for which the energy identity is justified; existence for arbitrary initial data must be established separately.

Use the normalized Rayleigh quotient $R(\psi )=⟨\psi ,H\psi ⟩/⟨\psi ,\psi ⟩$. Its real directional derivative in an arbitrary complex direction $\eta$ isIt vanishes for every direction precisely when $(H-R)\psi =0$, since testing $\eta =(H-R)\psi$ gives its squared norm. Conversely an eigenvector makes every derivative zero. Stationarity over all states is stronger than stationarity in a restricted trial family.

Normalize the Gaussian as ${\psi }_{\alpha }=(\alpha /\pi {)}^{1/4}{e}^{-\alpha x^2/2}$, with $\alpha >0$. Its moments are $⟨X^2⟩=1/(2\alpha )$, $⟨X^4⟩=3/(4{\alpha }^2)$ and $⟨P^2⟩={\hslash }^2\alpha /2$. The printed potential energy consequently givesPut $a=\alpha \hslash /(m\omega )$. Then $\mathcal{E}/(\hslash \omega )=a/4+1/(4a)+15/(256a^2)$, whose second derivative is $1/(2a^3)+45/(128a^4)>0$. It diverges at both positive endpoints, so its sole stationary point is the unique global minimum. Its equation isThe variational principle therefore yields the optimized upper boundThe printed minimizer is inconsistent with the printed quartic coefficient. At $a=3/2$ the derivative is $5/48>0$, so this is not the stationary Gaussian for that potential energy. It is nevertheless a legitimate trial vector and gives the simple, weaker bound $\boxed{E_0\le 109\hslash \omega /192}$. If the quartic coefficient were instead $5m^2{\omega }^3/(16\hslash )$, four times the printed value, then $a=3/2$ would indeed minimize the family and its energy would be $31\hslash \omega /48$. This possible coefficient correction is identified explicitly rather than silently used. The reusable calculation is Gaussian trial for a quadratic-plus-quartic oscillator.

Use the normalized Rayleigh quotient $R(\psi )=⟨\psi ,H\psi ⟩/⟨\psi ,\psi ⟩$. Its real directional derivative in an arbitrary complex direction $\eta$ isIt vanishes for every direction precisely when $(H-R)\psi =0$, since testing $\eta =(H-R)\psi$ gives its squared norm. Conversely an eigenvector makes every derivative zero. Stationarity over all states is stronger than stationarity in a restricted trial family.

Normalize the Gaussian as ${\psi }_{\alpha }=(\alpha /\pi {)}^{1/4}{e}^{-\alpha x^2/2}$, with $\alpha >0$. Its moments are $⟨X^2⟩=1/(2\alpha )$, $⟨X^4⟩=3/(4{\alpha }^2)$ and $⟨P^2⟩={\hslash }^2\alpha /2$. The printed potential energy consequently givesPut $a=\alpha \hslash /(m\omega )$. Then $\mathcal{E}/(\hslash \omega )=a/4+1/(4a)+15/(256a^2)$, whose second derivative is $1/(2a^3)+45/(128a^4)>0$. It diverges at both positive endpoints, so its sole stationary point is the unique global minimum. Its equation isThe variational principle therefore yields the optimized upper boundThe printed minimizer is inconsistent with the printed quartic coefficient. At $a=3/2$ the derivative is $5/48>0$, so this is not the stationary Gaussian for that potential energy. It is nevertheless a legitimate trial vector and gives the simple, weaker bound $\boxed{E_0\le 109\hslash \omega /192}$. If the quartic coefficient were instead $5m^2{\omega }^3/(16\hslash )$, four times the printed value, then $a=3/2$ would indeed minimize the family and its energy would be $31\hslash \omega /48$. This possible coefficient correction is identified explicitly rather than silently used. The reusable calculation is Gaussian trial for a quadratic-plus-quartic oscillator.

A theoretical condensed matter physicist who, with Walter Kohn, established a foundation of density functional theory in 1964. Their variational principle expresses the ground-state energy of interacting electrons through the electron density, making a three-dimensional quantity central to a many-particle problem.^[ref]

His 1967 work also ruled out the relevant finite-temperature long-range order in one- and two-dimensional bosonic and fermionic systems under the theorem's assumptions. This helps explain why dimensionality matters when discussing Bose-Einstein condensation.^[ref]

A trial function is a chosen candidate used to approximate a solution or evaluate a variational principle. For an eigenvalue upper bound from a Rayleigh quotient, it must be nonzero, have finite numerator and denominator, and lie in the appropriate variational domain. In a Dirichlet problem it satisfies the fixed endpoint values. Natural boundary conditions need not always be imposed on a trial in the quadratic-form domain, but a candidate violating an essential boundary condition does not give the stated bound.

Walter Kohn and Pierre Hohenberg established a foundation for density functional theory: under their ground-state assumptions, the electron density determines the external scalar potential up to an additive constant. They formulated the ground-state energy as a universal density functional plus the external-potential contribution, with a variational principle selecting the physical density. This replaces the full many-electron wave function as the basic variable by a function of only three spatial coordinates. The theorem establishes the framework, rather than providing an explicit exact functional for practical computation.^[ref]

Whitham modulation theory describes a wave train whose amplitude, wavenumber and angular frequency change slowly compared with its wavelength and period. For a variational principle, insert a locally periodic profile into the Lagrangian, average over the fast phase and vary the resulting averaged Lagrangian. This replaces detailed oscillations by equations for their slowly changing parameters, as developed by Gerald Beresford Whitham.^[ref]

With slow variables $X,T$ and phase $\Theta (X,T)$, define $k={\Theta }_X$ and $\omega =-{\Theta }_T$. Equality of mixed partial derivatives gives the independent wave-compatibility lawVariation of the slow phase gives the Whitham modulation equation, while profile or amplitude variations give other closure relations. These equations are leading-order asymptotic expansions under scale separation; a change of modulation type or crossing of its characteristic curves can invalidate the smooth slow description even while the original oscillatory wave remains meaningful.