Resolving the Quantum Measurement Problem through Leveraging the Uncertainty Principle

Kyoung Yeon Kim

REVIEW

Resolving the Quantum Measurement Problem through Leveraging the Uncertainty Principle

Kyoung Yeon Kim

Quantum Physics · Submitted 13 Dec 2024 · 32 references

Pith's verdict: A potentially useful numerical trick for stabilizing Wigner-Moyal simulations is wrapped in a physical interpretation—stencil width as quantum uncertainty—that is calibrated, not derived, so the measurement-problem claim does not hold up.

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arXiv ID2412.13214 (version 1)

Pith IDMFP3OP35, Pith's permanent record of this paper and its review

CitationKyoung Yeon Kim. "Resolving the Quantum Measurement Problem through Leveraging the Uncertainty Principle." arXiv:2412.13214 (2024). https://pith.science/paper/MFP3OP35

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Claim paper

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Editor's note

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The one thing to know: there is a real numerical finding buried here, but the paper's central physical claim is not supported. The genuinely new piece is the use of very high-order finite-difference stencils to widen the observation window while keeping a fine mesh, and the empirical rule N_obs = 1/(Delta x Delta k) that makes Wigner-Moyal simulations mesh-consistent and stable. If the paper were only about that, it would be a plausible contribution to phase-space transport numerics. The fact that results stabilize and converge across mesh sizes (Fig. 3d) is real evidence that the scheme works.

What the paper does well: it identifies a real instability in naive Wigner-Moyal discretization, shows that the instability worsens as Delta x Delta k shrinks, and demonstrates that expanding stencil width resolves it. The double-slit-inspired decoherence experiment (Fig. 4) is a clever numerical demonstration, even if its physical interpretation is a stretch.

Where it falls apart: the identification of stencil width with physical quantum uncertainty is asserted, not derived. Eq. (8) is calibrated against the simulation output—N_obs is chosen where coherence peaks—so the "numerical uncertainty principle" is a restatement of the observed optimum, not a derivation from Heisenberg's principle or from the Wigner-Moyal equation. A numerical stabilization condition is just that; it does not imply that the discretization parameter is a physical tuning knob. Every subsequent conclusion (decoherence for smaller windows, measurement as the zero-uncertainty limit) inherits this unjustified step. The measurement collapse argument has a separate hole: Eq. (11) is a homogeneous linear constraint, and imposing a Dirichlet value at a single point is arbitrary—it can produce any outcome, so it explains neither the definite outcome nor the Born probabilities. The "sacred timeline" and Big Bang toy model are rhetorical, not evidential. No code or data are shipped, and existing Wigner transport solvers are not compared, so the "first stable solver" claim is unsubstantiated.

Who this is for: a numerical analyst working on Wigner transport might get a usable idea from the stencil-width stabilization. A philosopher of physics should not treat this as a resolution of the measurement problem. If I were an editor, I would not desk-reject it: it deserves a serious referee on the numerics, with a clear instruction to evaluate the stabilization scheme and ignore the foundational claims. Reframed as a numerical methods paper, it could be made acceptable; as foundational physics, it is not.

Referee report

Significance

If the central identification were correct, the paper would offer a conceptual unification of classical, quantum, decoherence, and measurement phenomena within a single phase-space equation, and the reported mesh-independent stabilization of the Wigner-Moyal equation would be a practically valuable numerical result. However, the manuscript itself repeatedly marks the key step as a postulate ('We postulate that nonphysical results arise...', 'we define this range as the observation window'), and the numerical uncertainty principle is calibrated to Fig. 3c rather than derived. The physical conclusions about decoherence and measurement therefore do not follow. The paper does present a concrete finite-difference framework with high-order stencils and reports consistency across mesh sizes; if that stabilization claim can be independently validated, it could support a numerical-methods contribution, but the measurement-problem resolution is not supported by the evidence provided.

Major objections (4)

[Numerical Uncertainty Principle, after Eq. (8)] The central identification of the finite-difference stencil width N_obs with the physical uncertainty is asserted, not derived. The text states 'We postulate that nonphysical results arise because using excessively small mesh sizes violates the uncertainty principle' and then defines the observation window. N_obs is a discretization parameter of a derivative stencil, while ΔxΔk is a grid-resolution product; the Wigner-Moyal equation with fixed ħ has a unique continuum solution independent of the computational grid. Eq. (8), N_obs = 1/(ΔxΔk), is introduced after observing that coherence peaks near that value in Fig. 3c, so it is a calibrated stability condition rather than a physical principle. Every subsequent claim about decoherence (Eq. 9) and measurement (Eqs. 10-11) inherits this unsupported identification.
[Measurement without Uncertainty, Eqs. (10)-(11)] Even if one grants the limiting argument that C_max → ∞ makes the left-hand side negligible, Eq. (11) is a homogeneous linear constraint system. A homogeneous system has infinitely many solutions, and imposing a Dirichlet value at one point is an arbitrary regularization; it does not explain which outcome is selected or with what probability, which is precisely the core of the measurement problem. The claim in Fig. 5b that measurements are 'nearly identical regardless of the potential profile' is presented without any quantitative error measure or statistical analysis, so it does not establish collapse to a definite, potential-independent outcome.
[Numerical Uncertainty Principle, Fig. 3c and Eq. (8)] The paper's own presentation shows circularity: Fig. 3c is used to identify N_obs ≈ 1/(ΔxΔk) as the point of maximal coherence, and Eq. (8) then elevates this empirical observation to a 'universal condition' that is used to define the physical regimes said to explain Fig. 3c. No independent prediction is made from Eq. (8) and tested against data, and no derivation from the Heisenberg principle or from the structure of the Wigner-Moyal equation is supplied. This is a load-bearing circular step.
[Quantum Decoherence, Fig. 4] The decoherence claim is conditional on the same postulate: the text says 'We postulate that narrowing the observation window represents a transition from a high-uncertainty probability distribution to a low-uncertainty measurement system.' The single-point decoherence demonstration is performed for one resonant tunneling diode model with specific device parameters; no argument is given that this mimics a which-path measurement, and the interpretation of negative Wigner-function values as 'illusions' that become unobservable is asserted rather than derived from the equations. The paper does not provide the convergence or error analysis that would be needed to distinguish a genuine physical effect from a numerical artifact.

Minor objections (5)

[Abstract / Conclusion] The abstract and conclusion claim the resolution is achieved 'without introducing any additional terms,' but the numerical uncertainty principle introduces the observation window N_obs and a truncation threshold as free parameters; this tension should be addressed explicitly.
[Figure 5 caption] The caption says results are obtained by 'solving Equation (9),' but Eq. (9) is the decoherence inequality N_obs < 1/(ΔxΔk); the measurement equation is Eq. (11). The equation numbering in the caption should be corrected.
[Methods] The Methods section reuses equation numbers (11), (12), (13) that conflict with the main-text numbering for the same numbers, and the text contains the fragment 'the first stable solv' in the reader's notes; in the manuscript, the sentence about the first stable solver is incomplete.
[General] The manuscript states 'the first stable demonstration of the Wigner-Moyal equation as a stable quantum transport solver' without comparing with existing stabilized schemes in the literature; a reproducibility statement or code/data availability link would strengthen the numerical contribution.
[General] There are several typographical issues: 'unwind injection' should be 'upwind injection' in the Methods; 'a singularity occurs the system matrix' in the main text is ungrammatical; and reference [26] is to a preprint server with a misspelling. The supplementary material is referenced as 'SM. Fig. 1' and 'SM. Fig. 2' but was not included in the submission, so the referee could not verify those claims.

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The paper

The paper claims

The paper argues that the quantum measurement problem dissolves once the Heisenberg uncertainty principle is treated as a constraint on how finely a system can be observed.

The reading

The paper argues that the quantum measurement problem dissolves once the Heisenberg uncertainty principle is treated as a constraint on how finely a system can be observed. Working with the Wigner-Moyal equation, the author treats the finite-difference 'observation window' as a physical knob for uncertainty: at the window $N_{\mathrm{obs}}=1/(\Delta x\Delta k)$ the equation is stable and maximally quantum, a narrower window produces decoherence, and the zero-uncertainty limit becomes an ill-posed constraint system solved by a boundary condition—the act of measurement. If correct, classical, quantum, decoherence, and measurement are not separate phenomena but different regimes of a single equation.

What carries the argument

The load-bearing object is the 'observation window' $N_{\mathrm{obs}}$: the number of grid points used in the high-order finite-difference stencil for the potential-derivative terms of the Wigner-Moyal equation. The numerical uncertainty principle, $N_{\mathrm{obs}} = 1/(\Delta x \Delta k)$, sets the window to the spread allowed by the uncertainty relation. This single ratio controls the nonlocal power $C_j$ and the coefficient sum $A_{j,m}$: it suppresses unphysical nonlocality, turns the ill-posed zero-uncertainty equation into a well-posed one, and determines whether the system behaves classically, coherently, or as a measured collapsing state.

Extended reading notes

Core claim

The central claim is that a single equation, the Wigner-Moyal equation, contains all of quantum physics including measurement, without any added collapse postulate. The key is the 'observation window,' the range of phase-space data used in the finite-difference stencils for derivative terms. Tuning this window to match the uncertainty limit $N_{\mathrm{obs}} = 1/(\Delta x \Delta k)$ regularizes the unstable equation and produces stable quantum transport. Narrowing the window below this limit suppresses Wigner-function negativity and resonance, which the author identifies as quantum decoherence. Letting the window shrink to zero leaves an ill-posed constraint system, Eq. (11), in which all correlated degrees of freedom determine each other instantaneously; imposing a Dirichlet boundary condition at the measured point collapses the superposition and fixes the whole entangled solution. The paper thus presents measurement as an information-limited observer effect, not a new physical force.

Load-bearing premise

The paper's central claim rests on the idea that the number of grid points a numerical scheme uses to estimate a derivative is the same physical thing as the uncertainty in a measurement; if that identification is false, the rest of the argument does not follow.

Editorial extensions

If this is right

Classical mechanics and quantum mechanics become limiting cases of one phase-space equation, with no separate collapse postulate required.
Quantum decoherence is reframed as the result of a reduced observation window (low uncertainty) rather than necessarily environmental interaction, giving a new numerical model for decoherence without added scattering terms.
Measurement is instantaneous and nonlocal because in the zero-uncertainty limit the equation becomes a constraint system whose solution is pinned by boundary conditions, offering an interpretation of 'spooky action at a distance' as a byproduct of solving an ill-posed system.
The stability condition $N_{\mathrm{obs}} = 1/(\Delta x \Delta k)$ makes direct simulation of the Wigner-Moyal equation practical for large quantum-classical mixed systems, potentially improving quantum device simulation.
Schrodinger-equation-based methods that cannot tune uncertainty may be fundamentally limited for describing real-world decoherence, suggesting a shift to phase-space formulations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the observation window truly controls physical uncertainty, then varying the detector resolution (binning) in a double-slit experiment should change the interference visibility according to the predicted $1/(\Delta x \Delta k)$ scaling; this is a direct experimental test of the paper's central identification.
The 'sacred timeline' picture suggests that every measurement adds boundary conditions and branches the solution; a natural extension would be to ask whether the entropy increase of a measurement equals the number of imposed Dirichlet conditions, connecting to information-theoretic bounds like Landauer's principle.
If the framework is correct, it should reproduce known quantum correlation bounds, such as the Tsirelson bound for Bell-type experiments, when the observation window is set to the uncertainty limit; computing the maximum nonlocal correlation allowed by Eq. (10) would be a concrete unexamined check.
The identification of uncertainty with numerical stencil width could be generalized to other phase-space discretizations (spectral, wavelet, or finite-element), giving a practical criterion for choosing basis resolution in quantum simulations.

Original abstract

The Schrodinger equation is incomplete, inherently unable to explain the collapse of the wavefunction caused by measurement; a fundamental issue known as the quantum measurement problem. Quantum mechanics is generally constrained by the uncertainty principle and, therefore, cannot interpret definite observations without uncertainty. Here, we resolve this enigma by demonstrating that in phase space quantum mechanics, particularly through the Wigner Moyal equation, uncertainty can be arbitrarily adjusted by tuning the observation window. An observation window much smaller than the uncertainty limit causes substantial nonlocality, rendering the problem ill posed. This suggests that only with sufficient uncertainty does nonlocality become bounded, resulting in a well posed universe. Conversely, in the absence of uncertainty, spacetime is warped beyond recognition, and the system exists as a superposition of numerous possible states. Measurement collapses this superposition into a unique solution, exhibiting timeless nonlocal interactions. Our framework bridges seemingly disparate concepts such as classical mechanics, quantum mechanics, decoherence, and measurement within intrinsic quantum mechanics even without invoking new theory.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,196 input tokens.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.55 · human participants 0.02 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.09 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

The Schrodinger equation is incomplete, inherently unable to explain the collapse of the wavefunction caused by measurement; a fundamental issue known as the quantum measurement problem. Quantum mechanics is generally constrained by the uncertainty principle and, therefore, cannot interpret definite observations without uncertainty. Here, we resolve this enigma by demonstrating that in phase space quantum mechanics, particularly through the Wigner Moyal equation, uncertainty can be arbitrarily adjusted by tuning the observation window. An observation window much smaller than the uncertainty limit causes substantial nonlocality, rendering the problem ill posed. This suggests that only with sufficient uncertainty does nonlocality become bounded, resulting in a well posed universe. Conversely, in the absence of uncertainty, spacetime is warped beyond recognition, and the system exists as a superposition of numerous possible states. Measurement collapses this superposition into a unique solution, exhibiting timeless nonlocal interactions. Our framework bridges seemingly disparate concepts such as classical mechanics, quantum mechanics, decoherence, and measurement within intrinsic quantum mechanics even without invoking new theory.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The Schrodinger equation is incomplete, inherently unable to explain the collapse of the wavefunction caused by measurement; a fundamental issue known as the quantum measurement problem.	background 0.85	0.04	0.03	asserted 1.00
s1	Quantum mechanics is generally constrained by the uncertainty principle and, therefore, cannot interpret definite observations without uncertainty.	background 0.97	0.04	0.03	asserted 1.00
s2	Here, we resolve this enigma by demonstrating that in phase space quantum mechanics, particularly through the Wigner Moyal equation, uncertainty can be arbitrarily adjusted by tuning the observation window.	contribution 0.95	0.86	0.77	asserted 1.00
s3	An observation window much smaller than the uncertainty limit causes substantial nonlocality, rendering the problem ill posed.	result 0.75	0.47	0.07	asserted 0.99
s4	This suggests that only with sufficient uncertainty does nonlocality become bounded, resulting in a well posed universe.	interpretation 0.96	0.39	0.30	hedged possibility 0.95
s5	Conversely, in the absence of uncertainty, spacetime is warped beyond recognition, and the system exists as a superposition of numerous possible states.	interpretation 0.59	0.26	0.07	asserted 1.00
s6	Measurement collapses this superposition into a unique solution, exhibiting timeless nonlocal interactions.	interpretation 0.57	0.46	0.20	asserted 1.00
s7	Our framework bridges seemingly disparate concepts such as classical mechanics, quantum mechanics, decoherence, and measurement within intrinsic quantum mechanics even without invoking new theory.	contribution 0.69	0.82	0.36	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.95 · evidence from proof or derivation 0.59.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Formalized claims in Lean

Claim #1: The quantum measurement problem is resolved by the Wigner-Moyal equation alone, without additional terms, using the uncertainty principle as an observation constraint.

/-- @claim 1 The quantum measurement problem is resolved by the Wigner-Moyal equation alone, without additional terms, using the uncertainty principle as an observation constraint. -/
noncomputable def single_equation_resolves_measurement : Prop :=

Claim #2: Tuning the observation window to the uncertainty limit stabilizes the Wigner-Moyal equation: whenever `N_obs = 1/(Δx Δk)` (with `Δx Δk ≠ 0`), the system is well-posed.

/-- @claim 2 Tuning the observation window to the uncertainty limit stabilizes the Wigner-Moyal equation: whenever `N_obs = 1/(Δx Δk)` (with `Δx Δk ≠ 0`), the system is well-posed. -/
noncomputable def uncertainty_window_stabilizes : Prop :=

Claim #3: In the zero-uncertainty limit (observation window shrinking to zero), the Wigner-Moyal equation becomes an ill-posed constraint system (Eq. 11), and the act of measurement is the imposition of a Dirichlet boundary condition that collapses the superposition.

/-- @claim 3 In the zero-uncertainty limit (observation window shrinking to zero), the Wigner-Moyal equation becomes an ill-posed constraint system (Eq. 11), and the act of measurement is the imposition of a Dirichlet boundary condition that collapses the superposition. -/
noncomputable def zero_uncertainty_ill_posed_collapse : Prop :=

References

Works cited (32)

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Fein, Yaakov Y., et al. "Quantum superposition of molecules beyond 25 kDa." Nature Physics 15.12 (2019): 1242-1245

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Feynman, Richard P., et al. "The feynman lectures on physics; vol. i." American Journal of Physics 33.9 (1965): 750-752

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Zeilinger, Anton. "Experiment and the foundations of quantum physics." Reviews of Modern Physics 71.2 (1999): S288

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Ballentine, Leslie E. "The statistical interpretation of quantum mechanics." Reviews of modern physics 42.4 (1970): 358

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Stapp, Henry Pierce. "The copenhagen interpretation." American journal of physics 40.8 (1972): 1098- 1116

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Schrödinger, Erwin. "An undulatory theory of the mechanics of atoms and molecules." Physical review 28.6 (1926): 1049

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Wigner, Eugene P. "The problem of measurement." American Journal of Physics 31.1 (1963): 6-15

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Zurek, Wojciech Hubert. "Decoherence, einselection, and the quantum origins of the classical." Reviews of modern physics 75.3 (2003): 715

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Joos, Eric, and H. Dieter Zeh. "The emergence of classical properties through interaction with the environment." Zeitschrift für Physik B Condensed Matter 59 (1985): 223-243

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Zurek, Wojciech H. "Decoherence and the transition from quantum to classical." Physics today 44.10 (1991): 36-44

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Adler, Stephen L. "Why decoherence has not solved the measurement problem: a response to PW Anderson." Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 34.1 (2003): 135-142

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Schlosshauer, Maximilian. "Decoherence, the measurement problem, and interpretations of quantum mechanics." Reviews of Modern physics 76.4 (2004): 1267-1305

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Wigner, Eugene. "On the quantum correction for thermodynamic equilibrium." Physical review 40.5 (1932): 749

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Moyal, José E. "Quantum mechanics as a statistical theory." Mathematical Proceedings of the Cambridge Philosophical Society. Vol. 45. No. 1. Cambridge University Press, 1949

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Ancona, M. G., and G. J. Iafrate. "Quantum correction to the equation of state of an electron gas in a semiconductor." Physical Review B 39.13 (1989): 9536

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Gardner, Carl L. "The quantum hydrodynamic model for semiconductor devices." SIAM Journal on Applied Mathematics 54.2 (1994): 409-427

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Heisenberg, Werner. "Ü ber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik." Zeitschrift für Physik 43.3 (1927): 172-198

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Kennard, Earle H. "Zur quantenmechanik einfacher bewegungstypen." Zeitschrift für Physik 44.4 (1927): 326-352

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K. Y. Kim, et al. “Quantum Transport through a Constriction in Nanosheet Gate -all-around Transistor.” Researchsqaure (2024)

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View on Pith 2009

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

Donati, O., G. P. Missiroli, and G. Pozzi. "An experiment on electron interference." American Journal of Physics 41.5 (1973): 639-644

View on Pith 1973

[2] [2]

Fein, Yaakov Y., et al. "Quantum superposition of molecules beyond 25 kDa." Nature Physics 15.12 (2019): 1242-1245

View on Pith 2019

[3] [3]

Feynman, Richard P., et al. "The feynman lectures on physics; vol. i." American Journal of Physics 33.9 (1965): 750-752

View on Pith 1965

[4] [4]

Zeilinger, Anton. "Experiment and the foundations of quantum physics." Reviews of Modern Physics 71.2 (1999): S288

View on Pith 1999

[5] [5]

Ballentine, Leslie E. "The statistical interpretation of quantum mechanics." Reviews of modern physics 42.4 (1970): 358

View on Pith 1970

[6] [6]

Stapp, Henry Pierce. "The copenhagen interpretation." American journal of physics 40.8 (1972): 1098- 1116

View on Pith 1972

[7] [7]

Schrödinger, Erwin. "An undulatory theory of the mechanics of atoms and molecules." Physical review 28.6 (1926): 1049

View on Pith 1926

[8] [8]

Wigner, Eugene P. "The problem of measurement." American Journal of Physics 31.1 (1963): 6-15

View on Pith 1963

[9] [9]

Zurek, Wojciech Hubert. "Decoherence, einselection, and the quantum origins of the classical." Reviews of modern physics 75.3 (2003): 715

View on Pith 2003

[10] [10]

Joos, Eric, and H. Dieter Zeh. "The emergence of classical properties through interaction with the environment." Zeitschrift für Physik B Condensed Matter 59 (1985): 223-243

View on Pith 1985

[11] [11]

Zurek, Wojciech H. "Decoherence and the transition from quantum to classical." Physics today 44.10 (1991): 36-44

View on Pith 1991

[12] [12]

Adler, Stephen L. "Why decoherence has not solved the measurement problem: a response to PW Anderson." Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 34.1 (2003): 135-142

View on Pith 2003

[13] [13]

Schlosshauer, Maximilian. "Decoherence, the measurement problem, and interpretations of quantum mechanics." Reviews of Modern physics 76.4 (2004): 1267-1305

View on Pith 2004

[14] [14]

Wigner, Eugene. "On the quantum correction for thermodynamic equilibrium." Physical review 40.5 (1932): 749

View on Pith 1932

[15] [15]

Moyal, José E. "Quantum mechanics as a statistical theory." Mathematical Proceedings of the Cambridge Philosophical Society. Vol. 45. No. 1. Cambridge University Press, 1949

View on Pith 1949

[16] [16]

Ancona, M. G., and G. J. Iafrate. "Quantum correction to the equation of state of an electron gas in a semiconductor." Physical Review B 39.13 (1989): 9536

View on Pith 1989

[17] [17]

Gardner, Carl L. "The quantum hydrodynamic model for semiconductor devices." SIAM Journal on Applied Mathematics 54.2 (1994): 409-427

View on Pith 1994

[18] [18]

Steuernagel, Ole, Dimitris Kakofengitis, and Georg Ritter. "Wigner flow reveals topological order in quantum phase space dynamics." Physical Review Letters 110.3 (2013): 030401

View on Pith 2013

[19] [19]

Heisenberg, Werner. "Ü ber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik." Zeitschrift für Physik 43.3 (1927): 172-198

View on Pith 1927

[20] [20]

Kennard, Earle H. "Zur quantenmechanik einfacher bewegungstypen." Zeitschrift für Physik 44.4 (1927): 326-352

View on Pith 1927

[21] [21]

Boltzmann, Ludwig. Weitere studien über das wärmegleichgewicht unter gasmolekülen. Vol. 166. Aus der kk Hot-und Staatsdruckerei, 1872

View on Pith

[22] [22]

Godunov, Sergei K., and I. Bohachevsky. "Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics." Matematičeskij sbornik 47.3 (1959): 271- 306

View on Pith 1959

[23] [23]

Fornberg, Bengt. "Generation of finite difference formulas on arbitrarily spaced grids." Mathematics of computation 51.184 (1988): 699-706

View on Pith 1988

[24] [24]

Oppenheim, Jonathan, and Stephanie Wehner. "The uncertainty principle determines the nonlocality of quantum mechanics." Science 330.6007 (2010): 1072-1074

View on Pith 2010

[25] [25]

Stettler, Mark A., et al. "Industrial TCAD: modeling atoms to chips." IEEE Transactions on Electron Devices 68.11 (2021): 5350-5357

View on Pith 2021

[26] [26]

K. Y. Kim, et al. “Quantum Transport through a Constriction in Nanosheet Gate -all-around Transistor.” Researchsqaure (2024)

View on Pith 2024

[27] [27]

Pastawski, Horacio M., LEF Foa Torres, and Ernesto Medina. "Electron –phonon interaction and electronic decoherence in molecular conductors." Chemical physics 281.2-3 (2002): 257-278

View on Pith 2002

[28] [28]

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