Variational Method Ground State Energy Estimator

The variational method is a technique for estimating a quantum system's ground state energy without solving the Schrödinger equation exactly. You pick a 'trial' wavefunction (often with adjustable parameters), compute its expected energy ⟨ψ|H|ψ⟩/⟨ψ|ψ⟩, and the variational principle guarantees this value is always greater than or equal to the true ground state energy.

Because the ground state energy is unknown for most real systems, having a guaranteed upper bound is extremely valuable: it lets you rank different trial wavefunctions (a lower estimate is always closer to the truth) and, with a well-chosen trial function, get remarkably accurate results without solving the full differential equation. It is the basis of many practical approximation methods across atomic, molecular, and nuclear physics.

This simple polynomial automatically satisfies the infinite square well's boundary conditions, ψ(0) = ψ(L) = 0, is smooth and single-humped like the true ground state, and, crucially, its variational integral can be worked out exactly in closed form (no numerical integration needed), which is why this calculator can guarantee an exact result for the trial energy.

Working out ⟨T⟩ = (ħ²/2m)∫(ψ')²dx / ∫ψ²dx for ψ(x) = x(L-x) gives exactly E_trial = 5ħ²/(mL²). This is an exact closed-form result for this specific trial function, not a numerical approximation.

The true ground state energy of the infinite square well is E1 = π²ħ²/(2mL²). The variational estimate E_trial = 5ħ²/(mL²) is about 1.32% higher, remarkably close for such a simple trial function with no free parameters to optimize.

No. Both E_trial and E1 scale as 1/(mL²), so their ratio, and therefore the percentage excess, is always exactly the same (about 1.32%) regardless of the well width or the particle's mass. This is because the shapes of the trial and exact wavefunctions relative to the well are identical no matter how you scale L or m.

It proves that for any normalizable trial wavefunction ψ, the expectation value ⟨ψ|H|ψ⟩/⟨ψ|ψ⟩ is always greater than or equal to the true ground state energy E1, with equality only when ψ is exactly the true ground state wavefunction. This follows from expanding any trial function in the (unknown) true energy eigenstates and noting every excited state contributes a non-negative amount of extra energy.

Yes. Adding adjustable parameters to the trial function and minimizing the resulting energy expression over those parameters (the full variational method) generally gets closer to the true ground state than a fixed-shape trial function like this one. This calculator deliberately uses the simplest textbook case, a single fixed polynomial with no parameters to optimize, so the result is exact and instantly verifiable.

Yes, the same principle, minimizing ⟨H⟩ over a family of trial wavefunctions, underlies variational calculations of the helium atom's ground state and many other many-body quantum systems where the Schrödinger equation cannot be solved exactly. The infinite square well example here is the simplest possible illustration of the same idea.

Any trial wavefunction can be written as a mixture of the system's true energy eigenstates. Since every eigenstate above the ground state contributes energy greater than or equal to the ground energy, any admixture of excited states can only raise the average energy above the true minimum, never lower it, which is exactly the content of the variational principle.

Electron, proton, and neutron mass presets are built in, or you can enter any custom mass in kilograms. Since the percentage difference from the exact result is mass-independent, changing the particle only rescales the absolute energy values in electronvolts, not the accuracy of the estimate.