The Gold Standard of Zero-Point Energy | Bounding a Quartic Oscillator Ground State with a Gaussian Trial Wavefunction

Consider a particle moving on the whole real line. Its potential has a quadratic part and a nonnegative quartic part, with the potential minimum chosen as the energy reference. We seek the lowest eigenenergy of its specified quantum Hamiltonian, rather than the mean energy of an arbitrary prepared state.

The distinction matters because many states have energies above the ground level. A trial state may be mathematically convenient and physically admissible while still containing contributions from several exact energy eigenstates. Its expectation value is then not an eigenenergy merely because it can be calculated accurately.

We will call the exact dimensionless ground energy e0. It remains unknown throughout the worked bound unless an equality can actually be proved. The calculations produce numbers above and below it. Keeping e0 separate from an estimate helps the reader see what each mathematical step establishes and prevents an optimisation result from acquiring a stronger interpretation than its derivation supports.

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Use h = −one-half d²/dq² + q²/two + gq⁴, with q dimensionless and g nonnegative. The wavefunction is defined for every real q, is normalised and has finite energy. The quartic term is confining when g is positive, while g = zero returns the harmonic reference problem.

The model’s zero is V(0) = zero. It does not refer to a universal absolute energy baseline. Adding a fixed constant C to the Hamiltonian would shift e0 and every trial expectation by C without changing the wavefunctions or transition differences.

The assumptions also exclude a negative quartic coefficient without further stabilising terms. For g below zero, this polynomial potential tends to minus infinity at large displacement and does not describe the stable ground-state problem treated here. A numerical stationary point of a trial expression cannot repair an unbounded physical Hamiltonian. Begin with the operator’s domain and stability before applying a familiar formula.

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One route to the dimensionless model begins with H = p²/(two m) + mω²x²/two + λx⁴, where m and ω are positive and λ is nonnegative. Define the oscillator length ℓ = √[ℏ/(mω)] and set q = x/ℓ.

Dividing H by ℏω gives the dimensionless h above, with g = λℏ/(m²ω³). The dimensionless result e becomes a physical energy E = ℏωe. The quartic coefficient therefore has a specific dimensionless meaning; it is not simply a coefficient copied from a potential written in another coordinate unit.

A length rescaling changes the numerical coefficients attached to derivative and potential terms together. Rescaling only q⁴ while leaving every other term untouched changes the model. For an actual application, record m, ω, λ and the coordinate definition before quoting g. Our fictional examples stipulate g directly and make no claim about a particular material’s force constants or measured spectrum.

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For a normalised admissible trial state ψ, expand it in the exact energy eigenstates of the stated Hamiltonian. Its energy expectation is a weighted average of eigenenergies, with nonnegative weights summing to one. None of those eigenenergies lies below e0.

Therefore e0 ≤ ⟨h⟩ψ. This is the variational principle in the form needed here. It turns a correctly evaluated trial-state expectation into an upper bound, even when the trial state is not close to the exact ground state.

The direction of the inequality should remain visible. A trial expectation of 1.6 means that the exact ground energy cannot exceed 1.6 within the model. It does not mean the ground energy is at least 1.6, and it does not specify the error of the approximation. A separate lower bound would make the interval informative on both sides. Numerical precision in the upper value alone does not supply that missing information.

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Take ψa(q) = (a/π)^(one-quarter) exp(−aq²/two), with a positive. Every member is smooth, normalised and decays at large absolute q. The adjustable a controls the width of the probability distribution.

The probability density is |ψa|² = √(a/π) exp(−aq²). Notice the factor of two between the exponent in the wavefunction and the exponent in its squared magnitude. Using the wavefunction itself as the probability density would give incorrect moments and energies.

A larger a makes the distribution narrower. A smaller a spreads it out. The Gaussian family gives us one continuous degree of freedom to balance kinetic and potential contributions. It is deliberately restrictive: the exact quartic ground state need not lie inside it. The family is useful because its normalisation, derivatives and moments can all be checked explicitly, not because a bell-shaped trial function is guaranteed to solve every smooth potential.

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The Gaussian integral is ∫exp(−aq²)dq = √(π/a) over the full real line for a positive. Multiplying by the density prefactor √(a/π) gives one. This establishes the stated normalisation rather than leaving an untracked scale in the expectation.

If an unnormalised trial function is used instead, the correct energy is the Rayleigh quotient: the integral of ψ* hψ divided by the integral of |ψ|². Multiplying the trial function by a nonzero constant then cancels between numerator and denominator.

A missing denominator can create an artificially small number that is not a valid variational energy. That is a particularly important failure because it can appear to beat the ground-state bound. The first check is therefore not whether the final answer looks plausible. It is whether the probability density integrates to one, or whether the quotient explicitly handles its norm. Correct normalisation preserves the meaning of every subsequent moment.

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For the stated Gaussian density, ⟨q²⟩ = one/(two a) and ⟨q⁴⟩ = three/(four a²). These moments follow by differentiating the Gaussian normalisation integral with respect to a, then dividing by the normalisation.

The fourth moment is not the square of the second moment. Their ratio is three: ⟨q⁴⟩ = three⟨q²⟩² for this Gaussian family. Replacing the fourth moment with ⟨q²⟩² would underestimate the quartic contribution and describe a different calculation.

This is a concrete example of why a mean-square displacement does not by itself determine every nonlinear potential expectation. Here the chosen family supplies the extra moment relationship. Another probability density can have the same second moment and a different fourth moment. When the potential includes q⁴, that difference matters energetically. State which shape assumption supplies the relationship before using one moment to infer another.

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Differentiate the trial wavefunction: ψa′ = −aqψa. For a sufficiently decaying real trial function, integration by parts gives the kinetic expectation as one-half ∫|ψa′|²dq. The boundary contribution vanishes for this Gaussian.

Substituting the derivative and second moment gives ⟨T⟩ = a²⟨q²⟩/two = a/four. Narrowing the distribution by increasing a raises the kinetic contribution. That is the mathematical cost of localisation in this model.

One can check the result directly using ψa″ = (a²q² − a)ψa in the original operator. Both routes give a/four. Agreement between the two evaluations is useful because it catches a derivative sign, missing half or normalisation error. The integration-by-parts route also makes the nonnegative kinetic energy transparent, while reminding us that boundary conditions are part of the calculation rather than optional details.

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The quadratic contribution is ⟨q²/two⟩ = one/(four a). The quartic contribution is g⟨q⁴⟩ = three g/(four a²). Combining them with the kinetic contribution gives U(a,g) = a/four + one/(four a) + three g/(four a²).

For every a positive, e0(g) ≤ U(a,g). The quantity U is a trial expectation and upper bound. It depends on both the physical model parameter g and the adjustable trial width a, which play different roles.

Changing a explores different trial states of the same Hamiltonian. Changing g changes the Hamiltonian itself. An optimisation over a at fixed g is therefore legitimate for finding the best bound in this family. Reducing g to obtain a smaller number would answer a different physical question. Write which quantity is held fixed when comparing two rows of a calculation ledger.

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Differentiate U with respect to a. The derivative is one-quarter − one/(four a²) − three g/(two a³). Setting it to zero and multiplying by four a³ gives a³ − a − six g = zero.

For g positive, the relevant solution is above one. On a greater than or equal to one, a³ − a increases strictly, so there is one such solution. Values between zero and one cannot solve the equation for positive g because a³ − a is negative there.

The energy diverges as a tends to zero and as a tends to infinity. Its second derivative is one/(two a³) + nine g/(two a⁴), which is positive. The stationary point is therefore the unique global minimum within the positive Gaussian-width domain. This proves the optimisation result within the family; it does not prove that the exact ground wavefunction is Gaussian.

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Choose the fictional dimensionless coupling g = ten. The stationary equation is a³ − a − sixty = zero. The positive solution a = four can be checked exactly: sixty-four minus four minus sixty equals zero.

At a = four, the kinetic contribution is one, the quadratic contribution is one-sixteenth and the quartic contribution is thirty divided by sixty-four, or fifteen thirty-seconds. Their sum is forty-nine thirty-seconds, equal to 1.53125.

Thus e0(ten) ≤ 1.53125. The result is exact arithmetic for the chosen trial expectation and its Gaussian-family minimum. It remains an upper bound on the unknown quartic eigenenergy. The distinction between exact arithmetic and an exact physical solution is essential: a rational result with several tidy digits can still arise from a restricted approximation. No experimental measurement or numerical eigenvalue benchmark has been inserted into this example.

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The following ledger keeps g = ten fixed. All energies are dimensionless, and each row refers to a normalised member of the same trial family.

Every row provides an upper bound. The smallest row gives the tightest of these bounds, and the derivative proves that a = four is also the minimum over the entire positive Gaussian family. A table alone would only compare the sampled widths. Keep that distinction when using numerical optimisation: a low sampled value does not by itself establish a global minimum, although it can still remain a valid trial upper bound.

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An exact eigenstate satisfies hψ = eψ at every coordinate. For our Gaussian, dividing hψa by ψa gives a/two + (one − a²)q²/two + gq⁴. When g is positive, the q⁴ term remains, so this expression is not a coordinate-independent constant.

The optimised Gaussian is therefore not an exact eigenfunction of this quartic Hamiltonian. Minimising its average energy does not remove that pointwise residual. This gives a direct mathematical test of the claim that the chosen width solved the full problem.

In the harmonic case g = zero and a = one, both coordinate-dependent terms vanish. Then hψ equals one-half ψ, establishing an exact ground eigenstate. Comparing these two cases shows what equality requires. It is stronger than obtaining a stationary variational parameter or a smooth, plausible-looking probability density. A useful approximation can remain useful while its residual honestly records the missing shape information.

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For any nonnegative c, the square g(q² − c)² is nonnegative. Expanding it gives gq⁴ ≥ two gcq² − gc². Therefore h is bounded below by a harmonic Hamiltonian with quadratic coefficient one-half + two gc, shifted downward by gc².

The comparison oscillator has angular-frequency factor √(one + four gc) in the dimensionless units. Its ground energy is one-half √(one + four gc). Subtracting the constant gives L(c,g) = one-half √(one + four gc) − gc².

The operator inequality implies e0(g) ≥ L(c,g). This is a lower bound, obtained by a different argument from the trial-state upper bound. The comparison is valid over the whole coordinate domain because it began with a nonnegative square. Matching the potentials only near the origin would not supply the same global guarantee. The direction of the comparison must be retained when translating it into an energy statement.

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For the original g = ten example, choose c = one-tenth. Then one + four gc = five and gc² = one-tenth. The lower bound is √five/two minus one-tenth, approximately 1.018034.

Together with the optimised Gaussian upper bound, this gives 1.018034 ≤ e0(ten) ≤ 1.53125. The interval is mathematically justified within the stated Hamiltonian. Its width is about 0.513216. We have not claimed that the exact value sits at the midpoint or that all positions inside the interval are equally likely.

This is not a statistical confidence interval. It is a deterministic bracket based on two inequalities. If a proposed exact value fell below the lower endpoint or above the upper endpoint, it would conflict with at least one correctly established bound. The bracket can be useful even though it is not especially narrow. It tells us what has been proved and which part of the uncertainty remains due to the present analytical bounds.

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Every allowed c gives a valid lower bound, but some are stronger than others. To improve the bracket from below, maximise L(c,g), rather than minimise it. Its derivative with respect to c is g/√(one + four gc) − two gc.

For g positive, a stationary value satisfies two c√(one + four gc) = one. This condition concerns the comparison parameter c, not the Gaussian width a. The two optimisations adjust different constructions and should not be conflated.

One need not solve this extra optimisation to retain the valid c = one-tenth bound. A nonoptimal lower bound is still a lower bound. Likewise, a nonoptimal trial state can still give a valid upper bound. Optimisation sharpens a guarantee; it is not what creates the inequality in the first place. This distinction helps when a numerical search stops early: preserve the valid value already established and state clearly which best-in-family claim remains unproved.

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Set g = zero. The trial energy becomes U(a,zero) = a/four + one/(four a). Its minimum occurs at a = one and equals one-half. The lower comparison gives L(c,zero) = one-half for every allowed c.

The upper and lower bounds meet, proving e0(zero) = one-half. The Gaussian residual also becomes constant at a = one. These are three mutually consistent checks: optimisation, operator comparison and the eigenfunction condition.

A successful harmonic-limit check does not prove that the quartic calculation is exact. It establishes that the method reproduces a known case and that the definitions connect continuously to that case. Approximation schemes should be tested against limits where the answer is independently available. The lesson transfers to other subjects: recover a simple case before interpreting a more complicated numerical result, but do not let success at the simple boundary erase the new assumptions in the harder problem.

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For two couplings with g2 greater than g1, their Hamiltonians differ by the nonnegative operator (g2 − g1)q⁴. Therefore the exact ground energy cannot decrease as g increases. This conclusion follows from operator ordering, independently of the Gaussian family.

Each fixed-width trial energy also increases with g, since its quartic expectation is positive. However, comparing an upper bound at one g with a lower bound at another is the way to prove an ordering from numerical intervals. Two upper bounds alone do not generally order two unknown exact values.

For this particular model, operator ordering already supplies the direction. The separate numerical-bound caution remains useful: if U2 exceeds U1, that observation alone would not prove e02 exceeds e01 in a different problem without an ordering argument. Keep the logical source of each conclusion visible. A true statement supported by the wrong inference can become a misleading rule when transferred to a new model.

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The stationary equation is a³ − a = six g. When g is large and a is correspondingly large, the a³ term dominates the a term, suggesting a approximately (six g)^(one-third). This is a scale estimate, not an exact replacement at arbitrary coupling.

The leading kinetic and quartic trial contributions then both scale as g^(one-third), while the quadratic contribution scales as g^(minus one-third). The model therefore does not suggest a ground energy proportional to g over the whole range.

The same scaling can be seen by rescaling q with g^(minus one-sixth), which balances the kinetic and quartic terms. This agreement gives a useful structural check on the optimisation. It does not produce the exact proportionality constant for the true quartic ground energy. Distinguish a scaling law, a variational leading coefficient and an exact eigenvalue. They are related claims with different strengths, and only the first two are obtained from this short argument.

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At the original optimum a = four, the trial density has ⟨q²⟩ = one-eighth. That is a property of the chosen Gaussian state. It is not automatically the exact ground state’s variance or a measured displacement in a real specimen.

The energy bound can remain rigorous even when the trial state’s other observables have no corresponding one-sided guarantee. A good upper energy does not by itself certify that every expectation calculated from the same trial wavefunction is equally accurate.

For a physical variance, restore the length scale: ⟨x²⟩trial = ℓ²/(two a). The conversion supplies units, but it does not upgrade the approximation into an observation. If a measured variance were available, its relation to state preparation, thermal populations and the potential model would need to be addressed separately. This distinction is especially useful when a calculation produces an attractive probability picture: the picture belongs to the trial state until independent reasoning connects it more closely to the exact state.

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Our Gaussian moments were evaluated analytically. If they were instead estimated numerically, truncating the coordinate range or using an inaccurate quadrature could change the reported expectation. A numerical value below the exact ground energy would not invalidate the variational principle; it could indicate that the quantity was not evaluated as the energy of a valid normalised trial state.

A discrete matrix eigenproblem can also describe an approximation to the continuum Hamiltonian rather than an exactly equivalent problem. Boundary conditions, basis truncation and operator representation must be stated before assigning a variational guarantee to its smallest eigenvalue.

For checking, compare analytical Gaussian moments with independently converged numerical integrals and retain the normalisation denominator. Separate numerical tolerance from the gap between the trial-family minimum and the true continuum eigenvalue. Making the integration error small improves evaluation of the chosen bound. It does not remove the restriction to Gaussian trial shapes. Those are different sources of approximation and require different remedies.

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Suppose a new trial family contains every original Gaussian and additional admissible shapes. Its exact minimum energy cannot exceed the old Gaussian minimum because all old choices remain available. It also cannot fall below e0 if the expectations are evaluated correctly.

This gives e0 ≤ Unew ≤ Uold. The chain says that the best upper bound improves or stays the same. It does not promise a specific rate of convergence or establish that the new family already contains the exact ground state.

Adding poorly chosen parameters can make numerical optimisation harder even while the theoretical best value improves. An unconverged numerical result may miss the old optimum unless it is explicitly retained as an available trial state. A sensible learning check is to reproduce the old family inside the new representation first. Then any claimed deterioration can be located in the optimisation or implementation rather than incorrectly attributed to the variational principle itself.

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Use the same dimensionless Hamiltonian and Gaussian family, but set g = four. Find the optimal Gaussian width, its second and fourth moments, the three energy contributions and the resulting upper bound. Check the derivative and residual rather than merely copying the original width.

For a separate lower bound, choose c = one-quarter in the comparison construction. Calculate L(c,four) and write the resulting bracket. Decide whether the two endpoints meeting would prove an exact value, and explain why they do not meet in this example.

Finally restore an unspecified energy scale ℏω. State what information would be required before expressing the bound in joules or electronvolts. The changed coupling tests the method’s structure, while the scale question tests whether a dimensionless model number has been mistaken for a physical measurement. Work through the definitions before reading the solution in the next chapter.

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With g = four, the width equation is a³ − a − twenty-four = zero. Its positive solution is a = three. The second moment is one-sixth and the fourth moment is one-twelfth. The kinetic, quadratic and quartic contributions are three-quarters, one-twelfth and one-third respectively.

Their sum is seven-sixths, approximately 1.166667, so e0(four) ≤ seven-sixths. The residual still contains the positive quartic term and is not constant, so the Gaussian is not an exact eigenfunction.

For c = one-quarter, the comparison gives one + four gc = five and gc² = one-quarter. Thus e0(four) ≥ √five/two − one-quarter, approximately 0.868034. The bracket is 0.868034 ≤ e0(four) ≤ 1.166667. To convert it to physical energy, multiply both endpoints by the specified positive ℏω. Without ω and the corresponding physical model, the numerical interval remains dimensionless and cannot honestly be labelled as a measured molecular energy.

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Barton Zwiebach’s MIT Quantum Physics II notes, dated 13 September 2013, present the variational principle and explain why minimising a trial expectation supplies a ground-energy upper bound. That institutional treatment supports the method used here.

Our g = ten and g = four examples, contribution ledgers, nonnegative-square lower comparison and changed-context checks are original teaching constructions. They are not experimental findings or quoted quartic eigenvalues. The source explains the principle; the displayed equations make the conditional model calculations reproducible.

Read MIT's Wave Mechanics notes, including the variational principle. The source was checked on 25 October 2026. Keep its distinction between a general trial state, a family minimum and an exact eigenstate when moving to another potential. A different Hamiltonian requires its own admissible trial functions and expectation calculation.

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If the aim is a tighter upper bound, enlarge the trial family while retaining the original states and evaluate the expectations reliably. If the aim is a tighter lower bound, strengthen the operator comparison or optimise its permitted parameter. These actions improve different sides of the bracket.

If the aim is an actual molecular zero-point correction, first establish an appropriate potential, coordinates, masses and relationship to measured spectral information. A clean quartic example does not supply those physical inputs. If thermal occupation is relevant, distinguish the prepared or equilibrium state’s mean energy from the Hamiltonian’s lowest eigenenergy.

A practical learner route is to reconstruct normalisation, derive moments, calculate one width, optimise, then add the independent lower comparison. If an error appears, return to the earliest step that no longer agrees with its definition. This sequence makes the reasoning transferable: a final number becomes useful because its operator, state, reference and inequality direction remain attached to it.

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At the optimum width, differentiating the scaled Gaussian energy implies T = V2 + two V4, where T is the kinetic expectation and V2 and V4 are the quadratic and quartic potential expectations. For g = ten and a = four, the ledger gives T = one, V2 = one-sixteenth and V4 = fifteen thirty-seconds. One-sixteenth plus twice fifteen thirty-seconds is exactly one.

This agrees with the virial balance expected for a stationary bound state in this polynomial potential. Yet the optimised Gaussian has already failed the pointwise eigenfunction test. The balance is therefore a useful necessary consistency check, not a sufficient certificate of an exact eigenstate.

The reason is that the width optimisation explores a particular rescaling direction. A trial state can be stationary against that direction while still improving under changes of shape that the family does not contain. A single scalar balance does not replace the full differential equation. This is a valuable general lesson: satisfying one consequence of a model is weaker than satisfying every defining relationship. Use the check to find errors while preserving its logical limit.

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Suppose, in a different supplied calculation, the gap between the exact ground level and every excited level is known to be at least a positive number Δ. If a normalised trial state has ground-state probability p0, its mean energy satisfies U ≥ e0 + Δ(one − p0).

Consequently the excited-state weight is at most (U − e0)/Δ. If only a lower bound L on e0 is known, the weaker but usable conclusion is one − p0 ≤ (U − L)/Δ, capped at one. This requires an independently justified gap. The quartic calculation above has not supplied such a gap, so no numerical fidelity claim follows merely from its upper energy.

Even with a known gap, the conclusion concerns the trial state’s overlap in the specified energy basis, not a laboratory preparation that has not been characterised. It also does not bound every observable automatically. This extension helps explain why a small energy error can sometimes support a strong state statement, while a narrow-looking energy number without a gap estimate cannot do the same job. Identify the extra information before strengthening the interpretation.

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Is the optimised value the exact zero-point energy?

Not generally. It is the lowest expectation within the chosen Gaussian family and therefore an upper bound. In the quartic examples the coordinate-dependent residual shows that the Gaussian is not an exact eigenfunction. The harmonic limit is exact because the upper and lower bounds meet and the residual becomes constant.

Can a lower trial energy violate the ground energy?

A correctly evaluated energy expectation of an admissible normalised state cannot lie below the exact ground energy. A reported violation calls for checking normalisation, the operator, domain conditions and numerical evaluation. It does not show that a trial state has discovered a state below the defined ground state.

Is the bracket a confidence interval?

No. It is a deterministic consequence of two mathematical inequalities for the specified Hamiltonian. It supplies no probability distribution for the exact value inside the interval. The midpoint is not automatically a better physical estimate than another interior point.

Does a better energy guarantee every trial observable is better?

No. The variational principle directly bounds the energy expectation. Other observables need their own error analysis or evidence. A trial variance is useful information about that trial state, but it is not automatically a measured or rigorously bounded exact ground-state variance.

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Return to Zero-Point Energy for the harmonic spectrum, reference-energy distinctions, spectroscopy and the limits of energy-extraction claims. The present calculation develops its anharmonicity caveat into a specific bound rather than replacing the root’s broader explanation.

For state, spectrum and expectation foundations, use Quantum Mechanics. For the relationship between molecular motion and a potential model, continue with Molecular Physics. For the distinction between measured transitions and inferred ground corrections, use Spectroscopy. These guides supply different foundations rather than independent measurements of the fictional quartic example.

Continue through Learning That Transfers into Life and How X Works. A successful explanation identifies the Hamiltonian, normalises the trial state, reproduces the moment calculation and keeps upper bound, lower bound and exact eigenenergy distinct.

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