Research-level question in a solvable setting
Why does minimizing a trial energy produce an upper bound, and how does the optimal quantum wavepacket respond to stronger confinement?
A Gaussian trial wavefunction yields a closed-form upper bound on the ground energy of an anharmonic oscillator. The optimum width changes with interaction strength. Compare it with a separate truncated harmonic-oscillator diagonalization to distinguish a variational bound from a numerical reference with its own basis error.
Prerequisites: quantum-state normalization, Schrödinger time evolution, matrix spectral problems, and the corresponding section of the English lecture notes. The mathematical explanation below is integral to the interactive exercise.
Quantitative predictions
Computed observables
Trial energy versus spatial width
For a normalized Gaussian, kinetic energy penalizes very narrow widths and potential energy penalizes very broad ones.
Variational energy of a Gaussian wavepacket versus width parameter sigma
The orange point is the selected width; the optimum of Etrial is marked against an independently diagonalized finite-basis energy.
Independent numerical-basis convergence
The energy obtained by Rayleigh–Ritz diagonalization in increasingly large harmonic-oscillator subspaces is nonincreasing as the space expands.
Ground-energy estimate versus finite Fock-basis dimension
Every finite-basis value is itself an upper bound on the exact ground energy. It need not be below a separately optimized Gaussian bound at small basis size.
How the optimum adjusts to coupling
A stronger quartic term favors a narrower spatial state, balanced against the increasing kinetic energy cost.
Optimal Gaussian width as a function of quartic coupling
The stationarity condition is σ⁴+6λσ⁶=1. The harmonic reference is σ=1.
Derive and compare
Mathematical model
Use dimensionless units ℏ=m=ω=1 and H=p²/2+x²/2+λx⁴ for λ≥0. The normalized Gaussian trial wavefunction is ψσ(x)=(πσ²)−1/4exp[−x²/(2σ²)]. It has 〈x²〉=σ²/2, 〈p²〉=1/(2σ²), and 〈x⁴〉=3σ⁴/4. Therefore
Setting ∂σEtrial=0 yields σ⁴+6λσ⁶=1 and defines a unique minimum for λ≥0. The variational principle guarantees E₀≤Etrial(σ) for every allowed σ; optimizing can improve the bound but cannot determine the exact ground state unless the exact wavefunction happens to be in the trial family. At λ=0, ψσ=1 is the exact oscillator ground state and E=1/2.
For an independent comparison, diagonalize H in the first N harmonic-oscillator Fock states. The harmonic part is diagonal with energies n+1/2. Quartic matrix elements are 〈n|x⁴|n〉=3(2n²+2n+1)/4, 〈n+2|x⁴|n〉=(2n+3)√[(n+1)(n+2)]/2, and 〈n+4|x⁴|n〉=√[(n+1)(n+2)(n+3)(n+4)]/4, with Hermitian transposes. A symmetric Jacobi diagonalizer extracts the lowest finite-basis eigenvalue.
The Fock cutoff is a numerical approximation with a variational property of its own: enlarging a nested basis cannot raise its minimum Rayleigh quotient. Comparing σ-optimization against basis convergence is more meaningful than assuming any one finite diagonalization is exact.
Test an exactly soluble physical limit
At λ=0, both independent methods reproduce the exact ground energy 1/2 (with σ=1). In the strong quartic regime the variational optimal width decreases and the λx⁴ contribution balances the kinetic term, producing a characteristic λ^(1/3) energy scaling rather than perturbatively linear growth.
Assumptions, uncertainties, and reliability
Single one-dimensional bound particle, quartic confinement λ≥0, dimensionless m=ℏ=ω=1, Gaussian trial family with no nodes and Fock-space spectral approximations of finite N. The numerical reference must be checked by changing N, especially for large λ.
Explore independently
Three worked investigations
- Recover the exact point. Set λ=0, vary σ and identify σ=1. Derive each expectation value and confirm the minimum 1/2.
- Watch the optimum move. Increase λ while comparing the selected σ against σopt. Explain the kinetic-versus-potential balance by differentiating Etrial.
- Challenge the numerical reference. Choose λ>0 and vary N. Confirm Rayleigh–Ritz monotonicity and compare against the optimized Gaussian bound rather than assuming the first finite answer is exact.
Self-check: Which statement is guaranteed for every normalized trial wavefunction?
Extended exercise: derive one prediction independently, identify the model approximation or physical symmetry that supports it, and test its numerical convergence. AI guidance is useful for generating further test cases after the derivation has been checked.
Interpretation
What survives beyond the model
The variational method is a mathematically controlled way to trade an infinite quantum problem for a physically motivated family of trial states. A second convergent basis calculation reveals how to assess a bound honestly, including its dependence on numerical truncation.
Lecture notes: 04_variational_method.tex ↗.