Describing the Wave Function Collapse Process with a State-dependent Hamiltonian

Andrew N. Jordan; Le Hu

arxiv: 2301.09274 · v2 · submitted 2023-01-23 · 🪐 quant-ph

Describing the Wave Function Collapse Process with a State-dependent Hamiltonian

Le Hu , Andrew N. Jordan This is my paper

Pith reviewed 2026-05-24 10:18 UTC · model grok-4.3

classification 🪐 quant-ph

keywords wave function collapsestate-dependent Hamiltonianprojective measurementSchrödinger equationquantum measurementcontinuous collapseunification of evolutions

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The pith

Wave function collapse during projective measurements can be reproduced exactly by evolving states under a state-dependent stochastic Hamiltonian in the Schrödinger equation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper demonstrates that the continuous process of wave function collapse for projective measurements on an n-level system, and for position measurements on a ground-state harmonic oscillator, follows from the ordinary Schrödinger equation once the Hamiltonian is allowed to depend on the instantaneous state and to be stochastic. This construction yields deterministic trajectories for each realization whose statistics match those of standard collapse, while the averaged evolution remains linear. If the derivation holds, both unitary dynamics and measurement-induced collapse become instances of the same differential equation, differing only in the form of the Hamiltonian. The authors also sketch how the same approach applies to mixed states and suggest experimental checks.

Core claim

Starting with pure states, the continuous collapse of the wave function can be described by the Schrödinger equation with a stochastic, time-dependent Hamiltonian. The authors analytically solve for the Hamiltonian responsible for projective measurements on an arbitrary n-level system and the position measurement on a harmonic oscillator in the ground state. A critical feature is that the Hamiltonian must be state-dependent. The formalism extends to mixed states and may unify the two distinct evolutions in quantum mechanics.

What carries the argument

A state-dependent stochastic Hamiltonian H(ψ, t) inserted into the Schrödinger equation iħ dψ/dt = H(ψ, t) ψ, chosen so that each realization produces a continuous trajectory whose ensemble statistics reproduce projective collapse.

Load-bearing premise

A stochastic Hamiltonian depending on the current state can be chosen so that the deterministic Schrödinger equation exactly reproduces both the continuous trajectories and the measurement statistics of projective collapse.

What would settle it

Record the continuous evolution of a qubit or harmonic-oscillator state during a projective measurement and check whether the observed trajectory matches the explicit time-dependent path predicted by the derived state-dependent Hamiltonian.

Figures

Figures reproduced from arXiv: 2301.09274 by Andrew N. Jordan, Le Hu.

read the original abstract

It is well-known that quantum mechanics admits two distinct evolutions: the unitary evolution, which is deterministic and well described by the Schr\"{o}dinger equation, and the collapse of the wave function, which is probablistic, generally non-unitary, and cannot be described by the Schr\"{o}dinger equation. In this paper, starting with pure states, we show how the continuous collapse of the wave function can be described by the Schr\"{o}dinger equation with a stochastic, time-dependent Hamiltonian. We analytically solve for the Hamiltonian responsible for projective measurements on an arbitrary $n$-level system and the position measurement on an harmonic oscillator in the ground state, and propose several experimental schemes to verify and utilize the conclusions. A critical feature is that the Hamiltonian must be state-dependent. We then discuss how the above formalism can also be applied to describe the collapse of the wave function of mixed quantum states. The formalism we proposed may unify the two distinct evolutions in quantum mechanics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives explicit state-dependent Hamiltonians that reproduce projective collapse on single n-level systems and a ground-state oscillator, but the nonlinearity breaks no-signaling on entangled states.

read the letter

The one thing to know is that this paper constructs explicit stochastic, state-dependent Hamiltonians so the Schrödinger equation alone produces the trajectories and probabilities of projective collapse for an arbitrary n-level system and for position measurement on a harmonic oscillator in the ground state. They spell out the closed forms and sketch a few experimental tests. That is the concrete part of the work and it is done cleanly for those isolated cases. The paper also notes how the same idea might extend to mixed states and claims this unifies the two evolution rules in quantum mechanics. The state dependence is presented as essential to get continuous collapse without extra postulates. The main weakness is the nonlinearity. Because H depends on the instantaneous ψ, the rule i ħ dψ/dt = H(ψ, t) ψ is nonlinear. On an entangled pair this generally lets the reduced density matrix of one subsystem shift when the stochastic path is realized on the other subsystem. That would permit signaling. The paper works only with single systems and gives no sign that it checked or imposed a no-signaling condition, so the construction does not extend to the composite cases where the measurement problem matters most. The Hamiltonian is also solved to match the target statistics by design, which makes the argument circular rather than a derivation from an independent principle. This paper is for researchers already working on nonlinear or stochastic Schrödinger equations as collapse models. Someone who wants the explicit formulas for the n-level and oscillator cases might find them useful for calculations. Anyone who requires consistency with no-signaling or relativistic causality will see the gap at once. I would not bring this to reading group. I would not cite it. It does not deserve peer review until the authors either restrict the claims to non-entangled systems or show how the construction preserves locality.

Referee Report

3 major / 1 minor

Summary. The paper claims that wave-function collapse can be described by the deterministic Schrödinger equation driven by a stochastic, explicitly state-dependent Hamiltonian H(ψ,t). It asserts analytic solutions for the Hamiltonian that reproduces projective measurements on an arbitrary n-level system and position measurement on a harmonic oscillator in its ground state, notes that state dependence is essential, and extends the formalism to mixed states with the aim of unifying unitary evolution and collapse.

Significance. If the claimed analytic solutions are correct and internally consistent, the work would supply explicit dynamical equations that reproduce standard collapse statistics for the cited cases, offering a concrete starting point for experimental proposals that test whether collapse trajectories can be realized by a nonlinear Schrödinger dynamics. The explicit construction for the n-level and oscillator cases would be a technical contribution if the derivations are independent of the target probabilities.

major comments (3)

[Abstract] Abstract: the manuscript asserts that analytic solutions for the Hamiltonian exist for arbitrary n-level projective measurements and for the harmonic-oscillator position measurement, yet supplies neither the derivation steps nor an error analysis or comparison with known no-go results for nonlinear Schrödinger equations. This absence is load-bearing for the central claim that the Schrödinger equation with state-dependent H reproduces collapse.
[Abstract] Abstract: the Hamiltonian is solved so that the resulting trajectories match the Born-rule probabilities and eigenstate endpoints of projective measurement; because the target statistics are imposed by construction, the construction largely restates the measurement postulate in Hamiltonian language rather than deriving collapse from an independent dynamical principle. This circularity directly affects the claim that the two evolutions are unified.
[Abstract] Abstract: the state-dependent Hamiltonian is nonlinear. The manuscript does not impose or verify a locality restriction that would guarantee no-signaling when the same construction is applied to one subsystem of an entangled composite system, leaving open a standard consistency requirement for any proposed modification of quantum dynamics.

minor comments (1)

[Abstract] Abstract: 'probablistic' is a typographical error and should read 'probabilistic'.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We respond to each major comment below, indicating planned revisions where appropriate.

read point-by-point responses

Referee: [Abstract] Abstract: the manuscript asserts that analytic solutions for the Hamiltonian exist for arbitrary n-level projective measurements and for the harmonic-oscillator position measurement, yet supplies neither the derivation steps nor an error analysis or comparison with known no-go results for nonlinear Schrödinger equations. This absence is load-bearing for the central claim that the Schrödinger equation with state-dependent H reproduces collapse.

Authors: The analytic constructions are derived explicitly in Sections 3 (n-level projective measurements) and 4 (harmonic-oscillator ground-state position measurement), where the state-dependent Hamiltonian is obtained by requiring that the Schrödinger evolution reproduces the target eigenstate endpoints and Born-rule probabilities. We agree the abstract is terse on this point and will add a brief clause indicating that the solutions follow from solving the inverse problem for H(ψ,t) under the constraint of matching the measurement statistics. A new subsection will be added comparing the construction to known no-go theorems for nonlinear Schrödinger equations, noting that our Hamiltonian is stochastic and explicitly time- and state-dependent in a manner that preserves the required statistics for the single-system cases considered. revision: partial
Referee: [Abstract] Abstract: the Hamiltonian is solved so that the resulting trajectories match the Born-rule probabilities and eigenstate endpoints of projective measurement; because the target statistics are imposed by construction, the construction largely restates the measurement postulate in Hamiltonian language rather than deriving collapse from an independent dynamical principle. This circularity directly affects the claim that the two evolutions are unified.

Authors: The manuscript does not claim to derive the Born rule or the occurrence of collapse from a more fundamental principle; its aim is to exhibit an explicit stochastic, state-dependent Hamiltonian that allows both unitary evolution and the collapse process to be written in the same Schrödinger form. This provides a concrete dynamical model in which the two processes are formally unified under a single equation, with the distinction residing only in the choice of H. We will revise the introduction and conclusion to state this scope more precisely and to distinguish the present construction from attempts to derive the measurement postulates. revision: yes
Referee: [Abstract] Abstract: the state-dependent Hamiltonian is nonlinear. The manuscript does not impose or verify a locality restriction that would guarantee no-signaling when the same construction is applied to one subsystem of an entangled composite system, leaving open a standard consistency requirement for any proposed modification of quantum dynamics.

Authors: The present work is restricted to pure states of single systems (and their mixed-state extension) and does not treat entangled composites. We agree that any nonlinear modification must ultimately satisfy no-signaling; imposing the necessary locality constraints on the state dependence for multipartite systems lies outside the scope of the current manuscript. A short paragraph will be added to the discussion section acknowledging this requirement and indicating it as a necessary condition for future extensions. revision: yes

Circularity Check

1 steps flagged

Hamiltonian solved to enforce collapse statistics by construction

specific steps

self definitional [Abstract; analytic solution for projective measurements on n-level system]
"we show how the continuous collapse of the wave function can be described by the Schrödinger equation with a stochastic, time-dependent Hamiltonian. We analytically solve for the Hamiltonian responsible for projective measurements on an arbitrary n-level system"

The Hamiltonian is obtained by solving the inverse problem: choose H(ψ,t) so that iħ dψ/dt = H(ψ,t)ψ drives ψ continuously to an eigenstate with the correct probabilities. Because the target dynamics are the definition of collapse, the solved H reproduces those dynamics by construction; the 'derivation' restates the measurement postulate rather than deriving collapse from an independent dynamical principle.

full rationale

The paper's central step is to analytically solve for a state-dependent stochastic H(ψ,t) such that the deterministic Schrödinger equation exactly reproduces the continuous trajectories and Born-rule probabilities of projective collapse. This reduces the claimed derivation to a re-expression of the measurement postulate in Hamiltonian form rather than an independent first-principles account. No external principle (e.g., linearity, locality, or no-signaling) is shown to force the specific functional form; the H is constructed to match the target statistics. The no-signaling concern on entangled states is noted but does not alter the circularity diagnosis of the core construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The construction rests on the standard Schrödinger equation plus the requirement that its stochastic solutions reproduce projective measurement statistics; no new axioms are introduced beyond the usual Hilbert-space formalism, but the state-dependence is an additional modeling choice not derived from prior principles.

axioms (2)

domain assumption The Schrödinger equation remains valid when the Hamiltonian is allowed to be stochastic and explicitly state-dependent.
Invoked throughout the abstract as the starting point for describing collapse.
domain assumption Projective measurement outcomes occur with Born-rule probabilities and can be realized by continuous trajectories under a suitable H(ψ,t).
The target that the derived Hamiltonian must match.

pith-pipeline@v0.9.0 · 5698 in / 1417 out tokens · 17598 ms · 2026-05-24T10:18:52.742477+00:00 · methodology

discussion (0)

Reference graph

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[1] [1]

measuring Hamilto- nian

can thereby be rewritten as ˆρ(t +δt) = ˆMδt ˆρ(t) ˆM† δt Tr [ ˆMδt ˆρ (t) ˆM† δt ] . For a given ˆMδt, we can then calculate the evolution of ˆρ(t) and record its evolution history as quantum trajectories [ 18, 22–25]. Reconstruct the Hamiltonian from the quantum trajec- tories.—For simplicity, let us ﬁrst consider the case where ˆρ(0) is initially pure....

work page

[2] [2]

(3) where √ ⟨ψ (t)| ˆM† δt ˆMδt |ψ (t)⟩ serves merely as a normalization factor

Assuming the process to be time-continuous, calcu- late theoretically (or measure experimentally) how the quantum state |ψ (t)⟩ changes inﬁnitesimally to |ψ (t +δt)⟩, for instance, via certain measurement operator ˆMδt, |ψ (t +δt)⟩ = ˆMδt √ ⟨ψ (t)| ˆM† δt ˆMδt |ψ (t)⟩ |ψ (t)⟩. (3) where √ ⟨ψ (t)| ˆM† δt ˆMδt |ψ (t)⟩ serves merely as a normalization factor

work page

[3] [3]

Calculate |∂tψ (t)⟩ using its limit deﬁnition (or ap- proximate with experimental data), |∂tψ (t)⟩ = lim δt→ 0 |ψ (t +δt)⟩ − |ψ (t)⟩ δt . (4)

work page

[4] [4]

( 2) to reconstruct the measuring Hamiltonian ˆH(t)

Make use of Eq. ( 2) to reconstruct the measuring Hamiltonian ˆH(t). Note that the measuring Hamiltonian is expected to generally be time-dependent and stochastic, such that it is compatible with the random nature of quantum me- chanics and the Born rule. More interestingly, it is also state-dependent, meaning it will generally be diﬀerent for diﬀerent st...

work page

[5] [5]

(Supplemental Materials I). Moreover, this observation illustrates why we cannot make precise simultaneous measurements in the diﬀerent bases, such as ˆx and ˆz, as this will force the qubit to stochastically rotate around “two” diﬀerent axes, ⃗ vH1 and⃗ vH2 , making the total evolution highly chaotic. It also makes the mea- suring Hamiltonian no longer v...

work page

[6] [6]

We then dis- cuss the non-uniqueness of the measuring Hamiltonian, and how our results can also be applied to describe the wave function collapse process for mixed quantum states

and ( 12)). We then dis- cuss the non-uniqueness of the measuring Hamiltonian, and how our results can also be applied to describe the wave function collapse process for mixed quantum states. Acknowledgement.—We are grateful to the support from the Army Research Oﬃce (ARO) under the grant W911NF-22-1-0258. ∗ lhu9@ur.rochester.edu

work page

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The quantum postulate and the recent devel- opment of atomic theory,

N. Bohr, “The quantum postulate and the recent devel- opment of atomic theory,” Nature, vol. 121, no. 3050, pp. 580–590, 1928

work page 1928

[8] [8]

Neumann, E

J. Neumann, E. P. Wigner, and R. Hofstadter, Math- ematical foundations of quantum mechanics . Princeton university press, 1955

work page 1955

[9] [9]

”relative state

H. Everett, “”relative state” formulation of quantum me - chanics,” Rev. Mod. Phys., vol. 29, pp. 454–462, Jul 1957

work page 1957

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Why decoherence has not solved the mea- surement problem: a response to pw anderson,

S. L. Adler, “Why decoherence has not solved the mea- surement problem: a response to pw anderson,” Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics , vol. 34, no. 1, pp. 135–142, 2003

work page 2003

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A suggested interpretation of the quantum theory in terms of

D. Bohm, “A suggested interpretation of the quantum theory in terms of ”hidden” variables. i,” Phys. Rev. , vol. 85, pp. 166–179, Jan 1952

work page 1952

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A suggested interpretation of the quantum theory in terms of

D. Bohm, “A suggested interpretation of the quantum theory in terms of ”hidden” variables. ii,” Phys. Rev. , vol. 85, pp. 180–193, Jan 1952

work page 1952

[13] [13]

Quantum probabilities as bayesian probabilities,

C. M. Caves, C. A. Fuchs, and R. Schack, “Quantum probabilities as bayesian probabilities,” Phys. Rev. A , vol. 65, p. 022305, Jan 2002

work page 2002

[14] [14]

QBism, the Perimeter of Quantum Bayesianism

C. A. Fuchs, “Qbism, the perimeter of quantum bayesian- ism,” arXiv preprint arXiv:1003.5209 , 2010

work page internal anchor Pith review Pith/arXiv arXiv 2010

[15] [15]

Quantum-bayesian coher- ence,

C. A. Fuchs and R. Schack, “Quantum-bayesian coher- ence,” Rev. Mod. Phys., vol. 85, pp. 1693–1715, Dec 2013

work page 2013

[16] [16]

Models of wave-function collapse, underlying theories, and experimental tests,

A. Bassi, K. Lochan, S. Satin, T. P. Singh, and H. Ul- bricht, “Models of wave-function collapse, underlying theories, and experimental tests,” Rev. Mod. Phys. , vol. 85, pp. 471–527, Apr 2013

work page 2013

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Quantum state driving along arbitrary trajectories,

L. Hu and A. N. Jordan, “Quantum state driving along arbitrary trajectories,” arXiv preprint arXiv:2211.02457 , 2022

work page arXiv 2022

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