Trial states give upper bounds with a precise scope

For a self-adjoint Hamiltonian bounded below, expand an admissible normalized trial state in energy eigenstates when a discrete expansion is available. Its expectation is a weighted average of the eigenvalues, with nonnegative weights summing to one. Every contribution lies at or above the ground energy, so the average does too. More generally the spectral measure gives the same lower-spectral-edge statement when continuum components are present. The trial must belong to the appropriate operator or quadratic-form domain and satisfy the physical boundary conditions.

Normalization can be handled through the Rayleigh quotient, dividing the energy matrix element by the norm squared of any nonzero admissible trial state. Multiplying that state by a constant then leaves the quotient unchanged. Equality requires support entirely in the ground eigenspace when the ground energy is an attained eigenvalue. This proof establishes an upper bound on the true ground energy; reversing the inequality would turn a useful variational calculation into an incorrect lower bound.

Choose a parameterized trial family that respects the expected symmetry and spatial decay. Minimizing its Rayleigh quotient gives the best upper bound available within that family, not necessarily the global ground state. Enlarging a nested family cannot raise its exact variational minimum, because every earlier trial remains available. Numerical integration error or an incorrectly normalized expectation can spoil that monotonicity in a computed result; such behavior should trigger a check rather than a new physical interpretation.

For a concrete one-dimensional example, use units in which ħ=m=1 and take H=p²/2+λx⁴ with positive λ. A normalized Gaussian with width parameter a has moments ⟨x²⟩=1/(2a), ⟨x⁴⟩=3/(4a²), and kinetic expectation a/4. The energy trades localization against potential cost. A very narrow Gaussian lowers its quartic potential expectation but raises its kinetic energy, while a broad Gaussian does the reverse. Minimizing this competition is a physical balance rather than an arbitrary fit.

If the first excitation gap g is known and a normalized trial has energy excess δE above the exact ground energy, its weight outside the ground eigenspace is at most δE/g. Without a known gap and a controlled ground-energy reference, the upper bound alone does not supply that overlap estimate. A small energy error can coexist with substantial mixing among nearly degenerate low states. The method also does not bound an arbitrary excited energy merely because a trial has nodes.

Excited-state bounds require orthogonality to the exact lower eigenspaces or a valid min–max construction in a trial subspace. Orthogonality to approximate lower trial states is not automatically the same condition. The quantum lab illustrates the spectral-weight average in a specified two-level model: Reset and set both controls to θ=60° and φ=0°, predicting P(1)=1/4; then Reset and set θ=0° and φ=0°, predicting zero. If the assigned basis energies are 2 and 10, compute mean energies 4 and 2 yourself. The lab displays populations, not a variational optimizer or energy estimator.