Trial Wavefunctions and Variational Parameters

Introduction

Exact Schrödinger solutions exist for only a few ideal systems. The variational method turns a plausible wavefunction into a controlled estimate of the ground-state energy. By adjusting parameters such as a width or decay rate, one searches for the lowest energy that a chosen trial family can represent. The quality of the answer depends on the trial family, but the direction of the energy error is known.

Core explanation

For a Hamiltonian Ĥ with ground energy E₀ and an admissible nonzero trial function ψ(α), define E(α) = ⟨ψ(α) Ĥ ψ(α)⟩/⟨ψ(α) ψ(α)⟩. The denominator is essential if the function has not been normalised. Under the usual bound-state assumptions, E(α) ≥ E₀ for every permitted α. The variational principle follows by expanding ψ in exact energy eigenstates: its expectation energy is a nonnegative weighted average of their eigenvalues, and none is below E₀. The best result within the chosen family is the minimum over α, not necessarily the exact E₀.

An adjustable parameter should have a physical role. A Gaussian width controls localisation; an exponential decay constant controls how rapidly amplitude falls with distance. The function must be square-integrable for a bound-state calculation, obey the problem's boundary conditions and make its energy integrals finite. An elegant-looking formula that violates these requirements is not a valid trial. Normalisation is convenient but optional if the quotient includes the norm.

To optimise one differentiable parameter, calculate E(α), solve dE/dα = 0, and check that the stationary point is a minimum over the allowed domain. With several parameters, set each partial derivative to zero and compare candidate minima, including any relevant boundary. A parameter with inverse-length units cannot be varied over arbitrary negative values if its negative values make an exponential grow at infinity. Dimensional and physical checks are as important as differentiation.

Adding genuine flexibility to the trial space cannot raise the best achievable variational minimum, because the old trial functions remain available within the larger set. It can lower the estimate or leave it unchanged. However, a low energy alone does not prove that every feature of the approximate wavefunction is accurate; local properties and transition amplitudes may converge differently.

Step-by-step reasoning

Identify the Hamiltonian, its physical domain and boundary conditions. Choose a normalisable trial form with parameters that control a meaningful shape. Calculate the norm, kinetic expectation and potential expectation separately, then form their quotient. Minimise over allowed parameter values, verify the nature of the stationary point, and compare the result with known limits or a simpler trial. Report the result explicitly as an upper bound for the ground-state energy.

Visual explanation

Picture a curve of calculated energy against a positive width parameter. At one extreme the state is too narrow, making its kinetic energy large. At the other it spreads into an energetically unfavourable region. The curve dips to a lowest trial energy. Draw a horizontal line at or below that dip for the unknown exact ground energy.

Real-world analogy

Imagine shaping a flexible tent over a landscape while paying a cost for both excessive tension and poor shelter. Varying one pole position finds the best tent within that design. A different design might do better. Likewise, minimising a wavefunction family finds its best member without claiming that the chosen mathematical shape spans every possible exact state.

Real-world example

For a bound electron near a nucleus, an exponential trial function exp(−αr) lets α determine the density's radial scale. A too-large α localises the electron sharply and increases kinetic energy; a too-small α weakens its attraction to the nucleus. Balancing the two gives an optimal α, a useful pattern in approximate atomic and molecular calculations.

Why?

The upper-bound property makes the method useful even without a known exact answer. A poor trial can overestimate the energy, but it cannot legitimately fall below the true ground state if the Hamiltonian and integrals are handled correctly. Thus the minimisation supplies both a number and a diagnostic: a result below a trusted exact ground energy signals an error in assumptions or arithmetic.

Common misconception

Solving dE/dα = 0 does not automatically produce the exact wavefunction. It only finds a stationary point within the proposed family, and that point might even be a maximum. Another mistake is to omit the norm denominator for an unnormalised trial. Finally, the variational bound applies to the lowest state of the allowed space; excited-state estimates need additional orthogonality constraints.

Worked example

Suppose a legitimate trial family leads to E(α) = Aα² − Bα for α > 0, where A and B are positive constants of suitable units. Differentiate: dE/dα = 2Aα − B, giving α = B/(2A). Since d²E/dα² = 2A > 0, this is a minimum. Substitution yields E var = −B²/(4A). The parameter is positive as required. The exact ground energy is at or below this value, though equality requires the exact state to be represented by the trial family.

Quick check

1. A trial energy falls when a second parameter is introduced. Does this violate the variational principle? Answer: No. A larger family may produce a lower upper bound. It must still remain at or above the true ground energy when the function and calculation are admissible.

Exam focus

Show the quotient before simplifying and state the parameter domain. Differentiate the total expectation energy, not kinetic or potential terms in isolation. Check the second derivative or endpoints. Phrase the result as E var ≥ E₀; for negative energies, “higher” means less negative than the exact ground energy.

Advanced insight

Linear combinations of fixed basis functions are variational trial states too. Their coefficients play the role of parameters. Optimising them leads to matrix equations and a secular determinant, while optimising nonlinear exponents usually requires separate parameter searches. The shared principle is minimisation of a Rayleigh quotient over an allowed subspace or family.

Summary

A trial wavefunction provides an energy expectation that cannot lie below the ground-state energy under the method's assumptions. Shape parameters are chosen by minimising the Rayleigh quotient, with normalisation and boundary conditions respected. A flexible family can improve the energy bound, but a stationary point alone is neither proof of exactness nor proof of a minimum.

Practice questions

1. Why does the quotient include ⟨ψ ψ⟩ in its denominator? Answer: It removes dependence on the arbitrary overall amplitude of an unnormalised trial function. Without it, rescaling ψ would change the apparent energy even though the physical state is the same. 2. For E(β) = 3β² − 12β + 5 in consistent energy units, find the minimum for β > 0. Answer: dE/dβ = 6β − 12 = 0 gives β = 2. The second derivative is 6 > 0, and E(2) = 12 − 24 + 5 = −7 units; this is the family's variational minimum. 3. A candidate ψ = exp(+αr) with α > 0 is proposed for an electron bound in all three-dimensional space. What is wrong? Answer: It grows without limit as r increases and is not square-integrable over infinite space. Its norm diverges, so it cannot serve as an admissible bound-state trial function.