The Gold Standard of Zero-Point Energy | Why a Coupled-Mode Ground Energy Depends on Which Hamiltonian Terms Change

Take two dimensionless coordinates q1 and q2 with conjugate momenta p1 and p2. Within each pair, the canonical commutator is i, and operators belonging to different coordinates commute. The kinetic term is one-half times the sum of the two squared momenta.

The uncoupled reference has unit frequency for each coordinate. Its Hamiltonian is h0 = (p1² + p2² + q1² + q2²)/two. Each oscillator contributes one-half to the ground energy, so the combined reference energy is one.

We will compare this reference with a specified interacting Hamiltonian using the same energy and coordinate units. Ground state means the lowest state of the whole stated operator. It does not automatically mean that each original coordinate remains in its old independent oscillator ground state. The interaction can change both the normal-mode frequencies and the correlations between the original coordinates.

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Define hκ = h0 + κq1q2, where κ is a real dimensionless coupling parameter. The diagonal quadratic coefficients remain one while the off-diagonal interaction changes. The potential is one-half qᵀKq with K = [[one, κ], [κ, one]].

The factor convention matters. Writing κq1q2 corresponds to off-diagonal matrix entries κ because the two cross terms in qᵀKq are multiplied by the overall half. Using twice κ in one representation would change the frequencies.

This is a mathematical interaction contract. It does not say that attaching an ordinary positive spring between two fixed oscillators changes only an off-diagonal coefficient. Such a spring also changes diagonal terms, and we will examine that separate contract later. The sign and magnitude of an observed energy shift cannot be inferred from the word coupling without specifying the operator whose parameters changed.

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Set Qplus = (q1 + q2)/√two and Qminus = (q1 − q2)/√two. Define Pplus and Pminus through the same orthogonal combinations of p1 and p2. This preserves the canonical commutators and the sum of squared momenta.

The coordinate identities are q1² + q2² = Qplus² + Qminus² and q1q2 = (Qplus² − Qminus²)/two. Substituting them into hκ removes the cross term. The result is a sum of two independent harmonic Hamiltonians in the new coordinates.

The squared frequencies are one + κ and one − κ. Therefore the positive frequencies are νplus = √(one + κ) and νminus = √(one − κ), whenever both are positive. Normal modes are coordinated patterns of the original variables. They are not additional physical particles added to the system; they are coordinates that make the existing two-degree-of-freedom problem separable.

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Both quadratic mode coefficients must be positive for this confining oscillator problem. The stable domain is therefore |κ| less than one. In that domain, the potential rises in every direction and both normal-mode frequencies are real and positive.

At |κ| = one, one mode has zero restoring coefficient. On the full real coordinate line, that free direction has no normalisable oscillator ground state. The limiting energy expression can approach a finite value without establishing a normalisable ground eigenstate at the boundary.

For |κ| greater than one, one quadratic direction is unbounded below in this model. An imaginary frequency is a warning about the assumed equilibrium, not a new kind of half-quantum to insert into a real ground-energy sum. A stabilising higher-order potential would define another Hamiltonian. Keep the stable domain attached to the formula rather than extrapolating it through a point where the physical and mathematical problem changes.

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For |κ| less than one, the energy levels are e(nplus,nminus) = νplus(nplus + one-half) + νminus(nminus + one-half). The combined ground energy is e0(κ) = [√(one + κ) + √(one − κ)]/two.

At κ = zero this returns one, the uncoupled reference. Changing the sign of κ swaps the two frequencies and leaves the ground energy unchanged. The sign still changes which coordinated pattern is softer and how the original coordinates are correlated.

The calculation uses the normal-mode frequencies of the complete interacting Hamiltonian. Adding two copies of the original half-quantum would miss the change. Conversely, adding a separate interaction-energy number to the normal-mode ground sum would generally double-count the interaction, because it is already represented in the diagonalised frequencies. Use one complete energy account rather than combining pieces from incompatible representations.

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For κ nonzero inside the stable domain, the square-root function is strictly concave. The average of √(one + κ) and √(one − κ) is therefore less than the square root of their average argument, which is one. Hence e0(κ) is less than one.

A direct check gives the same conclusion. Squaring the positive sum yields two + two√(one − κ²), which is less than four for nonzero κ. Dividing by four establishes e0² less than one without relying on a sketch of the curve.

The sum of squared frequencies stays at two, but the sum of frequencies decreases as they split. Those are different summaries. The energy uses frequencies, not squared frequencies. This simple distinction explains how a fixed trace of the restoring matrix can coexist with a lower ground-energy sum. It also supplies a transferable lesson: a conserved linear sum of inputs need not preserve a nonlinear function of those inputs.

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Take κ = 0.6. The squared frequencies are 1.6 and 0.4, giving frequencies approximately 1.264911 and 0.632456. The ground energy is half their sum, approximately 0.948683.

The uncoupled value was one, so the change is approximately minus 0.051317 in the declared energy unit. One mode became stiffer while the other became softer; their frequency changes do not cancel because of the square roots.

These numbers describe two Hamiltonians under the fixed-diagonal comparison. They are not a measured heat release or work output. To express a physical energy change, one would need the actual frequency scale and a model connecting the coupling parameter to an experimental system. The numerical example demonstrates a conditional energy ordering and a normal-mode calculation, rather than establishing how a real interaction can be created or removed.

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The following ledger holds the diagonal stiffness entries at one. Frequencies and energies use the same dimensionless reference throughout.

The last row has the same energy as the positive-coupling case but exchanges the soft and stiff coordinated patterns. Thus an energy or unordered frequency pair need not identify every feature of the interaction. Keep the mode labels and displacement patterns when the question concerns which coordinates move together, rather than only the total ground-energy contribution.

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For small |κ|, expanding the two square roots gives e0(κ) = one − κ²/eight − five κ⁴/128 plus higher even powers. The linear terms cancel because the two mode shifts have opposite signs under this comparison.

The leading energy change is therefore quadratic and negative. It is not proportional to κ near zero. This helps check both numerical calculations and the symmetry under κ changing sign. An odd-power energy term would conflict with the exact evenness of the stated model.

The expansion is an approximation with a domain of usefulness. Near the soft-mode boundary, the exact square roots should be retained; a low-order polynomial cannot reproduce all of the boundary behaviour. Success at a small κ does not establish that a truncated expansion remains accurate as κ approaches one. Compare with the exact expression rather than treating a series coefficient as a universal law for arbitrary coupling strength.

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The coupled ground state is a product of Gaussian ground states in Qplus and Qminus. When expressed through q1 and q2, it generally contains correlations between the original coordinates. The interaction changes the state as well as the frequencies.

It is therefore incorrect to evaluate κq1q2 in the old uncoupled ground state, obtain zero correlation and conclude that the ground energy cannot change. That expectation belongs to a trial state of the new Hamiltonian, not necessarily its lowest state.

The old product ground state remains an admissible trial state with expectation one. The variational principle then supplies e0(κ) no greater than one. The exact normal-mode solution gives a strictly smaller value for nonzero stable κ. This connects the present solvable model with the variational article: a trial energy can provide a correct bound while missing the correlated ground-state shape that lowers the full expectation.

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For a unit-mass normal mode of frequency ν, its ground variance is one/(two ν). Transforming back gives ⟨q1²⟩ = ⟨q2²⟩ = [one/νplus + one/νminus]/four. The cross expectation is ⟨q1q2⟩ = [one/νplus − one/νminus]/four.

For positive κ, νplus exceeds νminus, so the cross expectation is negative. The softer mode favours opposite-sign displacements in this convention. Negative κ reverses the sign of the correlation while leaving the total ground energy unchanged.

These are exact expectations of the stipulated quadratic model. They are not classical trajectories in which each coordinate has a definite simultaneous displacement and momentum. A ground state can have a stationary probability distribution with nonzero fluctuations and correlations. The numerical energy shift is connected to that distribution, rather than to two tiny classical oscillators secretly following a particular periodic path.

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The normal-mode momentum variance is ν/two. In the original coordinates, ⟨p1²⟩ = ⟨p2²⟩ = (νplus + νminus)/four. For positive nonzero κ in the fixed-diagonal model, this is below the uncoupled value one-half, while each coordinate variance is above one-half.

Their product is [(one/νplus + one/νminus)(νplus + νminus)]/sixteen. The ratio terms ensure that it is at least one-quarter, with equality only when the frequencies coincide. The uncertainty relation has not been violated by the reduction in one variance.

This provides a useful check on a tempting interpretation: lower total ground energy does not mean every fluctuation becomes smaller. Different observables can change in different directions. A statement about squeezing or uncertainty must name the coordinate, momentum convention and reference state. The ground-energy sum alone does not determine every local measurement or justify treating the original coordinates as independent after the interaction is present.

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For the coupled ground state, the total kinetic expectation is T = (νplus + νminus)/four. The original diagonal potential expectation is Vdiag = [one/νplus + one/νminus]/four. The interaction expectation is Vint = κ⟨q1q2⟩.

Their sum must equal the normal-mode ground energy. At κ = 0.6, T is approximately 0.474342, Vdiag approximately 0.592927 and Vint approximately minus 0.118585. Together they give approximately 0.948683.

The diagonal part T + Vdiag is actually above the old reference value one. The negative interaction expectation more than offsets that increase. This ledger explains why adding the old independent energies to a separately guessed interaction term can be misleading: the state used to evaluate every contribution has changed. Reconstruct all terms in one state, or use the already complete normal-mode spectrum, and check that the two representations agree.

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Differentiating the exact ground energy gives de0/dκ = [one/√(one + κ) − one/√(one − κ)]/four. This is exactly the cross expectation ⟨q1q2⟩ derived from the ground-state variances.

The agreement is an instance of the Hellmann–Feynman relationship for this smooth nondegenerate ground state: the derivative of the energy with respect to a Hamiltonian parameter equals the expectation of the corresponding derivative operator. Here ∂hκ/∂κ = q1q2.

For positive κ, the derivative is negative, matching the energy decrease and the negative correlation. This is an independent consistency route within the solved model, not an extra measurement. A fitted energy curve and a correlation calculated from the same assumed Hamiltonian should not be counted as independent experimental evidence. Their agreement checks the algebra and interpretation of the model’s linked predictions.

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Now define a different Hamiltonian hλ = h0 + λ(q1 − q2)²/two, with λ nonnegative. Expanding the square adds λ to both diagonal stiffness entries and adds an off-diagonal entry minus λ. The diagonal terms are no longer held fixed.

The symmetric mode has squared frequency one, while the antisymmetric mode has squared frequency one + two λ. The ground energy is [one + √(one + two λ)]/two. It increases with positive λ.

For λ = 0.5, the result is (one + √two)/two, approximately 1.207107. That does not contradict the earlier decrease. The two comparisons change different terms of the Hamiltonian. A physical description such as connecting the oscillators with a spring must be translated into the complete stiffness matrix before choosing which result applies. The mathematical control matters as much as the word coupling.

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The fixed-diagonal model changes only an off-diagonal coefficient. The added-spring model changes that coefficient and the self-restoring terms together. Their different energy trends follow from those distinct paths through parameter space.

A general quadratic interaction can also change masses, kinetic couplings or external confinement. Each variation changes the normal-mode problem. There is no universal rule that coupling must raise or must lower the combined ground energy under every possible control.

For a learner’s explanation, write the before and after matrices explicitly. Identify which entries changed and which physical reference remains common. Then calculate the eigenvalues and their square roots. This sequence prevents a familiar special case from becoming an overgeneralised slogan. It also makes the comparison reviewable: another reader can check whether the stated physical change actually produces the matrix used in the calculation.

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Consider K = [[one + d, κ], [κ, one − d]], where d describes a diagonal detuning. The trace remains two. The eigenvalues are one plus and minus r, with r = √(d² + κ²).

Stability now requires r less than one. The ground energy is [√(one + r) + √(one − r)]/two. It depends on the combined splitting r, not on d and κ separately. The earlier equal-diagonal model is the case d = zero.

This extension shows another observation limit. A measured pair of mode frequencies can identify eigenvalues without uniquely identifying the entries of the stiffness matrix in a specified coordinate basis. Displacement patterns or other independent information can be needed to distinguish detuning from off-diagonal coupling. The spectrum is informative, but the inverse problem has more unknown structure than two frequency values necessarily reveal.

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Take one matrix with d = zero and κ = 0.6. Take another with d = 0.6 and κ = zero. Both have eigenvalues 1.6 and 0.4 and therefore the same normal-mode frequencies and ground-energy sum.

The first matrix couples the original coordinates and has symmetric and antisymmetric mode patterns. The second is already diagonal: the original coordinates are independent oscillators with different stiffnesses. Their correlations in that coordinate basis differ even though the unordered spectra agree.

This is an exact original counterexample to the claim that the two frequencies alone establish a particular coupling strength. If d is independently known to be zero, the frequencies can constrain |κ| under that model. Without that information, they identify r. Even with d fixed at zero, changing κ’s sign swaps the mode patterns, so an unordered pair does not establish its sign. State the independently supplied structure before interpreting a spectral splitting as one microscopic interaction parameter.

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An orthogonal rotation of a stiffness matrix preserves its eigenvalues. It changes how its normal modes are represented through chosen laboratory coordinates. A frequency list therefore lacks the eigenvector information needed to reconstruct an arbitrary symmetric matrix in a fixed basis.

The determinant also gives νplus²νminus² = one − κ² in the equal-diagonal model. This identifies a magnitude combination while leaving the sign of κ unresolved without mode information.

The basis is not merely a mathematical decoration when q1 and q2 refer to physically identified displacements. A mode shape measurement concerns how those particular coordinates contribute to a mode. It can add information that the frequency pair alone does not supply.

For the two-parameter trace-fixed example, a justified mode orientation can separate d from κ up to appropriate conventions. For a more general system, more coefficients and observations may be needed. The present model does not supply an experimental reconstruction protocol. It identifies why the missing information is structural and which kind of observation could address it, rather than assuming that a more precise frequency measurement alone resolves every ambiguity.

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As κ approaches one from below in the equal-diagonal model, νminus tends to zero while νplus tends to √two. The ground-energy expression tends to √two/two. Meanwhile the soft-mode coordinate variance one/(two νminus) diverges.

A finite limiting energy therefore coexists with an increasingly broad probability distribution along the soft direction. At the boundary, the normalisable oscillator ground state is lost on the full real line. The limit of a sequence of ground energies is not automatically the energy of an attained ground eigenstate at the limiting parameter.

This distinction is useful when reading mode-softening claims. A small frequency can signal that neglected nonlinear terms or constraints become important for a real system. Our quadratic model does not determine what happens beyond its stability range. A quartic stabilisation would create a new problem requiring a different ground-state calculation, such as the variational methods developed in the existing quartic-oscillator article.

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Comparing ground energies of two Hamiltonians does not specify a process that carries one ground state into the other. A sufficiently slow, controlled change may permit adiabatic following under appropriate gap and dynamical conditions. A sudden parameter change generally leaves the state initially unchanged while changing the Hamiltonian used to evaluate its energy.

The old uncoupled ground state has expectation one under hκ because its cross expectation is zero. The new ground energy is lower for stable nonzero κ. After a sudden change, the system is therefore not automatically in that new ground state.

Any relaxation, control work and energy exchanged with other degrees of freedom require a complete process model. A difference between two ground-energy numbers does not establish how much useful work a specified operation delivers. The state, parameter path, timescale and external controller all belong in that question. The static calculation remains useful precisely because its conclusion is kept separate from an unmodelled dynamical claim.

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If a parameter-controlled interaction changes the ground energy, forces or work can be associated with the control coordinate in a fully specified system. Returning that system to its original Hamiltonian and state requires the corresponding reverse account, along with any dissipation and external inputs.

The fixed-diagonal energy decrease is therefore not a perpetual energy source. It describes a configuration-dependent quantum ground contribution under a declared reference. A complete cycle cannot repeatedly count the same initial-to-final lowering while omitting the restoration step.

For a paper exercise, label the initial Hamiltonian, final Hamiltonian and state at each stage. If a proposed work ledger contains only the favourable energy difference, ask where the parameter reset and state preparation appear. This is not an additional constraint invented for quantum mechanics; it is ordinary energy bookkeeping applied to the whole controlled system. The exact normal-mode result does not remove the need for that full boundary.

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MIT’s Normal Modes chapter explains how matrices and coordinate transformations separate linear multi-degree-of-freedom oscillations. Its harmonic-oscillator notes describe the quantum half-quantum spectrum and phonon modes. Together they provide the foundations for diagonalising a positive quadratic Hamiltonian and summing the resulting ground contributions.

Our fixed-diagonal interaction, added-spring comparison, detuning counterexample and all numerical ledgers are original teaching constructions. They are not measurements or coefficients taken from a specific material. The energy, correlation and inverse statements follow from the explicitly displayed matrices and canonical model.

Read MIT's normal-mode chapter and its quantum harmonic-oscillator notes, checked on 25 October 2026. Retain the model’s stability, units and coordinate definitions when comparing another system.

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Use the fixed-diagonal Hamiltonian with κ = 0.8. Calculate its two frequencies, ground energy and change relative to the uncoupled reference. State which coordinated pattern is softer. Repeat the energy calculation for κ = minus 0.8 and explain what changes besides the unordered frequency pair.

Then compare a positive added spring with λ = 1.5. Write its full stiffness matrix and calculate the two frequencies and ground energy. Explain why it is not the same calculation as setting κ = minus 1.5 in the fixed-diagonal model.

Finally construct a diagonal detuning matrix with the same spectrum as the κ = 0.8 model. Decide whether its original coordinates are coupled. These changed contexts test the matrix contract, stability and inverse interpretation, rather than only recalling a square-root formula from the first example.

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For κ = 0.8, the squared frequencies are 1.8 and 0.2. Their roots are three/√five and one/√five, so e0 = two/√five, approximately 0.894427. The change relative to one is approximately minus 0.105573. The antisymmetric mode is softer for positive κ.

Changing κ to minus 0.8 swaps the soft and stiff coordinated patterns and reverses the cross correlation while retaining the energy. For λ = 1.5, the added-spring matrix has diagonal entries 2.5 and off-diagonal entries minus 1.5. Its eigenvalues are one and four, so its frequencies are one and two and its ground energy is 1.5.

Setting κ = minus 1.5 while holding diagonal entries at one would instead be unstable and would not describe this positive-spring model. A diagonal matrix with entries 1.8 and 0.2 is isospectral to the κ = 0.8 matrix but has no off-diagonal coupling in the stated original basis. The frequencies identify the splitting, not the whole physical structure.

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Record the kinetic convention, the coordinate basis, the stiffness matrix and the energy reference. Check positive eigenvalues before assigning real harmonic frequencies. Keep eigenvectors when the question concerns mode character, not just eigenvalues.

Calculate the energy in normal coordinates, then reconstruct it from original-coordinate expectations in at least one manageable case. Check the uncoupled limit and any symmetry under changing a parameter’s sign. These tests reveal missing factors and double-counted terms before a physical claim is made.

For an inverse question, list which matrix entries are known independently and which observations are available. An energy sum, an unordered frequency pair and a mode-shape record supply different information. A clear account identifies the combination constrained by each. This keeps a precise spectral measurement from being overstated as a unique microscopic reconstruction when multiple Hamiltonians remain compatible with it.

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Our teaching coordinates have an identity kinetic matrix. With physical coordinates x and a positive mass matrix M, the equations instead involve Kphysical u = ω² M u. The symmetric mass-weighted stiffness is M^(minus one-half) Kphysical M^(minus one-half).

Its positive eigenvalues supply squared physical frequencies. Diagonalising Kphysical alone would generally give the wrong frequencies if the masses are unequal. The coordinate transformation must preserve the corresponding kinetic and canonical structure, rather than treating a stiffness entry as a frequency without checking its mass factor.

For a simple uncoupled check, one coordinate with stiffness k and mass m has frequency √(k/m). Doubling m while keeping k fixed reduces that frequency by √two. A numerical stiffness matrix with no declared mass convention cannot therefore determine a physical half-quantum energy by itself. Our dimensionless model deliberately fixes that convention so the matrix arithmetic is transparent. Restoring units means restoring the physical relationships, not merely appending hertz to the dimensionless eigenvalues.

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The example fixes the potential minimum at zero for every supplied κ. Suppose another physical model includes an additional coordinate-independent term C(κ). It shifts every energy level by that term and changes the ground energy to e0(κ) + C(κ), while leaving normal-mode transition spacings unchanged.

An unordered frequency pair therefore does not determine an arbitrary parameter-dependent baseline contribution. If the question concerns a force conjugate to κ or work during a parameter change, the derivative C′(κ) belongs in the account. It cannot be omitted merely because it does not alter spectral gaps.

This does not undermine the fixed-reference calculation above. It states the reference contract that made that calculation specific. Adding one constant independent of κ shifts all compared energies equally and leaves the difference unchanged. Adding a function of κ changes the physical comparison. A claim about configuration-dependent work must specify the whole energy function, including contributions outside the two observed oscillator modes. Spectroscopy can constrain mode spacings while leaving other parts of that larger energy ledger unresolved.

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In the detuning model, suppose a measured eigenvalue splitting establishes r = 0.6 and the trace is independently fixed at two. The pairs d = zero, κ = 0.6; d = 0.36, κ = 0.48; and d = 0.6, κ = zero all satisfy d² + κ² = 0.36.

They therefore share the frequencies √1.6 and √0.4 and the same harmonic ground-energy sum. The middle matrix has diagonal entries 1.36 and 0.64 and off-diagonal entries 0.48. It is neither the equal-diagonal example nor the uncoupled detuned example, yet its spectrum is identical.

These are three points on a full circle of parameter pairs, subject to the chosen basis and sign conventions. More precise versions of the same two eigenvalues narrow the splitting uncertainty but do not select an angle on that circle. A mode orientation or an independently known diagonal difference would add a different relationship. This distinction between precision and identifiability is the practical lesson: repeating one kind of observation can sharpen a constrained combination while leaving the structural trade-off intact.

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Does coupling always lower the ground energy?

No. It lowers the energy in the stable fixed-diagonal cross-term model derived here. Adding a positive coupling spring changes the diagonal terms as well and raises the ground energy in the separate example. Write the complete Hamiltonian before selecting a trend.

Why does a fixed sum of squared frequencies not fix the energy?

The harmonic ground contribution uses the frequencies themselves, which are square roots of stiffness eigenvalues in the stated mass convention. A nonlinear square-root sum can change even when the eigenvalue sum remains fixed.

Does the soft-mode boundary have an ordinary ground state?

Not in the full-line quadratic model used here when one restoring frequency becomes zero. The limiting energy is finite, but the corresponding soft coordinate loses a normalisable oscillator ground state. Additional confinement would define a different problem.

Can two frequencies determine a coupling uniquely?

Only with enough independently established structural assumptions. Detuning and off-diagonal interaction can produce the same eigenvalue splitting. Mode patterns and known matrix entries can add information that an unordered frequency pair lacks.

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Return to Zero-Point Energy and Quantum Mechanics for the spectrum and state foundations. For collective vibrations in a material, use how vibrations carry heat through a solid, keeping its transport question distinct from this static two-mode energy calculation.

For a different Hamiltonian whose ground energy is not obtained by a simple quadratic diagonalisation, compare Gaussian variational bounds for a quartic oscillator. Its bounds and residual checks become useful when a soft-mode model requires nonlinear stabilisation.

For molecular mode assignments and their observation, use Molecular Physics and Spectroscopy. A measured frequency splitting and a measured displacement pattern contribute different information. Preserve that distinction when relating this deliberately small matrix model to a real material, where masses, mode identity and the assumed interaction require their own evidence before a coupling coefficient is assigned.

For the distinction between an actual preparation and the ground energy of its Hamiltonian, continue with mean oscillator energy and ground-state population bounds. For molecular spacings beyond an exactly quadratic spectrum, read inferring an anharmonic ground correction from assigned transitions. These routes keep preparation, mode spectrum and potential-model assumptions separate.

Continue through Learning That Transfers into Life and How X Works. A useful final explanation names the interaction, verifies stability and adds half-quanta only after the full normal-mode problem has been solved.

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