Abstract
Paper 113 wrote that quantum mechanics is a deformation of classical mechanics by $\hbar$, and that $\hbar\to0$ returns the classical algebra. That is a statement about structures. This paper asks the neighbouring question — does the behaviour return? They are different sentences, and the second has several answers. NO NEW MATHEMATICAL THEOREM AND NO NEW PHYSICAL LAW IS CLAIMED.
First, the equations. $\langle V'(x)\rangle=V'(\langle x\rangle)$ is exact only for $V$ at most quadratic. At cubic order the gap is $\sigma^2$ and at quartic order exactly $3\mu\sigma^2$ (Section 2).
Second, the spectrum. In hydrogen the ratio of the $n\to n-1$ frequency to the classical orbital frequency runs $3.0$, $1.40625$, $1.0015020025$, approaching $1$ like $1+3/(2n)$. Not once does $\hbar$ appear in that ratio — this is a statement about large quantum number, not about small $\hbar$ (Section 3).
Third, the wavefunction. The same Bohr–Sommerfeld recipe is exact at every $n$ for the oscillator (error $0$) and never exact for the quartic, improving from $1.8\times10^{-1}$ to $2.2\times10^{-5}$ with $n$ (Section 4).
Fourth, and this is the core. The limits $\hbar\to0$ and $t\to\infty$ do not commute. Holding $t=2$, the difference dies with $\hbar$: $0.1762\to0.1050\to0.0581\to0.0307$. Holding $\hbar=0.1$ and waiting, the swing collapses from $1.5$ to $0.000195$ and then comes back to $0.664$ at $t=199.85$ — while the classical orbit has not moved from $1.4986$ (Section 5).
Fifth, the survival time is finite all the same. It grows $8.16\to10.64\to14.28\to19.20$, and only as a power law, near $\hbar^{-0.41}$ (Section 6).
Scope (what is not claimed)
NO NEW MATHEMATICAL THEOREM AND NO NEW PHYSICAL LAW IS CLAIMED — Ehrenfest's theorem, Bohr's correspondence principle, the WKB approximation, and the collapse and revival of wavepackets are all standard. No interpretation of quantum mechanics is judged. On Paper 113: that paper showed the algebra returns; this one asks whether the behaviour does — a brake, not a refutation. On Paper 154: the pattern “two limits do not commute” is already that paper's subject (the Curie–Weiss magnet). Section 5 here is a second instance, not a new pattern — what is added is that the pattern also appears at the quantum-classical boundary. What is added is setting four limits against one separator, writing the Ehrenfest gap in closed form and confirming its proportionality to $\hbar$ in numbers, measuring the rate of the correspondence principle, placing side by side one WKB recipe that is exact in one well and asymptotic in another, and measuring the collapse and revival to a precision that survives a doubling of the basis.
1. Notation and position
Throughout, $H=p^2/2+V(x)$ with unit mass. Every number here was computed for this paper, and no matrix result is quoted unless doubling the basis left it alone (values that moved are not reported). The classical orbits were integrated with my own Runge–Kutta scheme.
Position: Paper 113 wrote that modules are limits rather than walls, and joined the quantum to the classical by a deformation in $\hbar$. This paper stands next to it — the algebra returning and the behaviour returning are different sentences.
2. The equations: where Ehrenfest stops being exact
Observation 2.1 The rate of change of $\langle p\rangle$ is exactly $-\langle V'(x)\rangle$, while the classical equation reads $-V'(\langle x\rangle)$. These agree exactly when $V'$ is linear, that is when $V$ is at most quadratic.
Verification 2.2: With a Gaussian packet of mean $\mu=0.7$ and standard deviation $\sigma=0.35$.
| $V(x)$ | $V'(x)$ | gap | predicted |
|---|---|---|---|
| $x^2/2$ | $x$ | $\mathbf{0}$ | $0$ |
| $x^3/3$ | $x^2$ | $0.1225$ | $\sigma^2=0.1225$ |
| $x^4/4$ | $x^3$ | $\mathbf{0.25725}$ | $3\mu\sigma^2=\mathbf{0.25725}$ |
(QED)
Observation 2.3 What makes the gap is the width of the packet, and the width is where $\hbar$ enters — for a minimum-uncertainty packet $\sigma^2=\hbar/2$, so the gap is proportional to $\hbar$.
| $\hbar$ | $\sigma^2=\hbar/2$ | gap $3\mu\sigma^2$ |
|---|---|---|
| $1$ | $0.5$ | $1.05$ |
| $0.01$ | $0.005$ | $0.0105$ |
| $0.0001$ | $0.00005$ | $0.000105$ |
(QED)
Observation 2.4 So at a fixed time the classical equation is recovered as $\hbar\to0$. That proviso is worth remembering — Section 5 shows the same statement failing when the time is not held fixed.
3. The spectrum: Bohr's own correspondence, and its rate
Observation 3.1 Bohr ($1913$, $1920$) said that for large quantum number the frequency radiated on $n\to n-1$ approaches the classical orbital frequency. In Rydberg units hydrogen has $E_n=-1/n^2$, so the transition frequency is $1/(n-1)^2-1/n^2$ while the classical orbital frequency at level $n$ is $2/n^3$.
Verification 3.2
| $n$ | quantum ($n\to n-1$) | classical ($2/n^3$) | ratio |
|---|---|---|---|
| $2$ | $0.75$ | $0.25$ | $3.0$ |
| $5$ | $0.0225$ | $0.016$ | $1.40625$ |
| $10$ | $0.00234567901235$ | $0.002$ | $1.17283950617$ |
| $100$ | $2.03040506071\times10^{-6}$ | $2.0\times10^{-6}$ | $1.01520253035$ |
| $1000$ | $2.00300400501\times10^{-9}$ | $2.0\times10^{-9}$ | $\mathbf{1.0015020025}$ |
| $10000$ | $2.00030004001\times10^{-12}$ | $2.0\times10^{-12}$ | $1.00015002$ |
(QED)
Verification 3.3: The approach goes like $1+3/(2n)$.
| $n$ | ratio $-1$ | $3/(2n)$ |
|---|---|---|
| $10$ | $0.1728395062$ | $0.15$ |
| $100$ | $0.01520253035$ | $0.015$ |
| $1000$ | $0.001502002503$ | $0.0015$ |
| $10000$ | $0.0001500200025$ | $0.00015$ |
(QED)
Observation 3.4 Nowhere in that ratio does $\hbar$ appear. The correspondence principle is not a statement that $\hbar$ is small but that the quantum number is large — a different sentence from the $\hbar\to0$ of Section 2.
Verification 3.5: Nor is it about a single frequency. The jump $n\to n-k$ corresponds to the $k$th harmonic of the orbit.
| transition ($n=1000$) | classical side | ratio |
|---|---|---|
| $n\to n-1$ | fundamental | $1.0015020025$ |
| $n\to n-2$ | second harmonic | $1.00300802005$ |
| $n\to n-3$ | third harmonic | $1.00451806774$ |
What corresponds is a whole Fourier series. (QED)
4. The wavefunction: one recipe, exact and asymptotic
Observation 4.1 Take the Bohr–Sommerfeld condition with the added $1/2$ (the Maslov index) and apply it in the same form to two wells.
Verification 4.2: The oscillator $H=p^2/2+x^2/2$.
| $n$ | WKB | matrix eigenvalue | error |
|---|---|---|---|
| $0$ | $0.5$ | $0.50000000000000$ | $\mathbf{0}$ |
| $1$ | $1.5$ | $1.50000000000000$ | $\mathbf{0}$ |
| $5$ | $5.5$ | $5.50000000000000$ | $\mathbf{0}$ |
| $20$ | $20.5$ | $20.50000000000000$ | $\mathbf{0}$ |
Exact at every $n$, the ground state included. (QED)
Verification 4.3: The quartic well $H=p^2/2+x^4$. Raising the basis through $100$, $200$, $400$ moves $E_0=0.66798625915578$ by only $10^{-14}$.
| $n$ | WKB | matrix eigenvalue | relative error |
|---|---|---|---|
| $0$ | $0.54626732507809$ | $0.667986259156$ | $1.822\times10^{-1}$ |
| $1$ | $2.3635614446006$ | $2.393644016466$ | $1.257\times10^{-2}$ |
| $5$ | $13.363764206125$ | $13.379336552631$ | $1.164\times10^{-3}$ |
| $10$ | $31.649312728425$ | $31.659456477223$ | $3.204\times10^{-4}$ |
| $40$ | $191.44847701265$ | $191.452604854688$ | $\mathbf{2.156\times10^{-5}}$ |
(QED)
Observation 4.4 “Semiclassical” is not one condition either. In one well the recipe is exact at every $n$; in the other it is only asymptotic in $n$. The same recipe means different things in different wells.
5. The core: $\hbar\to0$ and $t\to\infty$ do not commute
Observation 5.1 Place a packet at rest at $x_0=1.5$ in the quartic well. On the classical side $\ddot x=-4x^3$ is integrated with my own Runge–Kutta scheme.
Verification 5.2: Hold the time at $t=2$ and lower $\hbar$.
| $\hbar$ | quantum $\langle x\rangle(2)$ | classical $x(2)$ | difference |
|---|---|---|---|
| $0.4$ | $0.640102912023$ | $0.463905500086$ | $1.762\times10^{-1}$ |
| $0.2$ | $0.568908324699$ | $0.463905500086$ | $1.050\times10^{-1}$ |
| $0.1$ | $0.521994255976$ | $0.463905500086$ | $5.809\times10^{-2}$ |
| $0.05$ | $0.494576155187$ | $0.463905500086$ | $\mathbf{3.067\times10^{-2}}$ |
Halving $\hbar$ halves the difference, in agreement with the closed form of Section 2. Every row moved by only $10^{-14}$ when the basis was doubled. (QED)
Verification 5.3: Now hold $\hbar=0.1$ and wait.
| $t$ | quantum swing | classical swing |
|---|---|---|
| $0$ | $1.500000$ | $1.500000$ |
| $10$ | $0.482629$ | $1.499910$ |
| $20$ | $0.014062$ | $1.499426$ |
| $40$ | $\mathbf{0.001587}$ | $1.499486$ |
| $120$ | $0.022298$ | $1.498760$ |
| $200$ | $\mathbf{0.664054}$ | $1.498946$ |
| $240$ | $0.289268$ | $1.498613$ |
The deepest collapse was $0.000195$ at $t=27.50$ and the strongest revival $0.664054$ at $t=199.85$. (QED)
Observation 5.4 The swing comes back. The classical orbit neither collapses nor revives and has not moved from $1.4986$. So this is not a small correction to an orbit. Both the collapse and the revival come from the level spacing not being uniform — and nothing on the classical side answers to that.
Observation 5.5 Side by side.
| order taken | result |
|---|---|
| fix $t$, then $\hbar\to0$ | classical (Ver. 5.2) |
| fix $\hbar$, then $t\to\infty$ | not classical (Ver. 5.3) |
This pattern is not new to this corpus. Paper 154 takes “two limits do not commute” as its subject and sets $0$ against $0.957504$ in the Curie–Weiss magnet. All that is added here is that the same pattern appears at the quantum-classical boundary.
6. And yet the survival time is finite
Verification 6.1: Measure how long it takes for the swing to fall below a tenth of its initial value.
| $\hbar$ | survival time | moved on doubling the basis |
|---|---|---|
| $0.4$ | $8.160$ | $4.5\times10^{-11}$ |
| $0.2$ | $10.640$ | $2.7\times10^{-11}$ |
| $0.1$ | $14.280$ | $1.4\times10^{-11}$ |
| $0.05$ | $\mathbf{19.200}$ | $1.5\times10^{-11}$ |
(QED)
Observation 6.2 It grows, and only as a power law (the measured log-log slope is near $\hbar^{-0.41}$). Making $\hbar$ small postpones the classical orbit's failure; it does not prevent it.
Observation 6.3 “Ehrenfest time” is itself one name for two behaviours. The short $\log(1/\hbar)$ often quoted belongs to chaotic systems, where the spreading is exponential. The well here is integrable, and measurement gives a power law. The exponent above is a measurement on one packet in one well, not a derived exponent.
7. The census
Observation 7.1
| what becomes classical | name | when it holds | number here |
|---|---|---|---|
| the equations | Ehrenfest | exact up to quadratic | $3\mu\sigma^2$ |
| the spectrum | Bohr's correspondence | large $n$ (no $\hbar$) | $1+3/(2n)$ |
| the wavefunction | WKB, semiclassical | depends on the well | error $0$ vs $2.2\times10^{-5}$ |
| the state | decoherence | superposition goes | Papers 55, 138 |
Observation 7.2 These four are not a staircase of strength. Decoherence makes the state classical without making the equations classical (as Paper 55 wrote, a diagonal density matrix does not say which outcome occurs). Conversely the equations can be classical while superposition survives. What settles it is what is being asked to be classical.
8. Conclusion
- Ehrenfest is exact only up to quadratic (Ver. 2.2): the gap is exactly $3\mu\sigma^2=0.25725$.
- The gap is proportional to $\hbar$ (Obs. 2.3).
- No $\hbar$ appears in the correspondence principle (Obs. 3.4): the ratio is $1+3/(2n)$.
- One WKB recipe is exact at every $n$ for the oscillator (Ver. 4.2) and asymptotic for the quartic (Ver. 4.3).
- The two limits do not commute (Ver. 5.2, 5.3): the difference dies with $\hbar$, yet waiting collapses the swing to $0.000195$ and returns it to $0.664$.
- The survival time grows and stays finite (Ver. 6.1): $8.16\to19.20$.
References
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Reference
Yamagishi, Y. (2026). “The Classical Limit” Is Not One Limit [Paper 180]. YOI: https://www.yamagishiyuuki.com/yoi/180
DOI: 10.5281/zenodo.22099046
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 180. This paper is a preprint and has not been peer reviewed.