arXiv:2607.24480

A Bohmian version of a 2-state quantum system

Matthew Dickau

quant-phSubmitted Jul 27, 2026• Announced Jul 28, 2026• 0 citations

Abstract

I construct a Bohmian version of a 2-state quantum system, where in addition to the quantum state, the system is characterized by a definite physical state which changes stochastically under the guidance of the quantum state. I argue that the probabilities produced by this process are well-defined, contrary to claims in "Why quantum mechanics cannot be formulated as a Markov process" by D. Gillespie (1994) about a related model.

Comments: 12 pages

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In reply to the preceding Comment on an article by D. T. Gillespie [Phys. Rev. A 49, 1607 (1994)], it is emphasized that the most plausible interpretation of the singly conditioned probability function implies that the evolution of the two-state quantum system being considered cannot be characterized as a Markov process on those states over any time interval. And it is pointed out that an alternative interpretation that is offered in the Comment appears to be in conflict with quantum mechanics.

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It is argued that the stochastic model of a quantum-mechanical two-state oscillator discussed in the preceding Comment by Hardy et al. [Phys. Rev. A 56, 3301 (1997)] does not constitute a valid classical stochastic process.

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We reply to suggestions made by Gillespie [Phys. Rev. A 49, 1607 (1994)] that our earlier paper [Phys. Rev. A 45, 4267 (1992)] contained mathematical inconsistencies. By use of a concrete model, we demonstrate explicitly that the behavior of a quantum-mechanical two-state oscillator may be simulated by a classical jump process.

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arXiv:2607.24480

A Bohmian version of a 2-state quantum system

Matthew Dickau

quant-phSubmitted Jul 27, 2026• Announced Jul 28, 2026• 0 citations

arXiv HTML PDF

Abstract

Comments: 12 pages

Related papers

Comment on ‘‘Why quantum mechanics cannot be formulated as a Markov process’’

Piotr Garbaczewski, Robert Olkiewicz

In the paper with the above-noted title, D. T. Gillespie [Phys. Rev. A 49, 1607 (1994)] claims that the theory of Markov stochastic processes cannot provide an adequate mathematical framework for quantum mechanics. In conjunction with the specific quantum dynamics considered there, we give a general analysis of the associated dichotomic jump processes. If we assume that Gillespie's ``measurement probabilities'' are the transition probabilities of a stochastic process, then the process must have an invariant (time independent) probability measure. Alternatively, if we demand the probability measure of the process follow the quantally implemented (via the Born statistical postulate) evolution, then we arrive at the jump process which can be interpreted as a Markov process if restricted to a suitable duration time. However, there is no corresponding Markov process consistent with the ${\mathit{Z}}_{2}$ event space assumption, if we require its existence for all times t\ensuremath{\in} ${\mathbf{R}}_{+}$ . \textcopyright{} 1996 The American Physical Society.In the paper with the above-noted title, D. T. Gillespie [Phys. Rev. A 49, 1607 (1994)] claims that the theory of Markov stochastic processes cannot provide an adequate mathematical framework for quantum mechanics. In conjunction with the specific quantum dynamics considered there, we give a general analysis of the associated dichotomic jump processes. If we assume that Gillespie's ``measurement probabilities'' are the transition probabilities of a stochastic process, then the process must have an invariant (time independent) probability measure. Alternatively, if we demand the probability measure of the process follow the quantally implemented (via the Born statistical postulate) evolution, then we arrive at the jump process which can be interpreted as a Markov process if restricted to a suitable duration time. However, there is no corresponding Markov process consistent with the ${\mathit{Z}}_{2}$ event space assumption, if we require its existence for all times t\ensuremath{\in}${\mathbf{R}}_{+}$. \textcopyright{} 1996 The American Physical Society.

arXiv:quant-ph/9601016Physical Review AAug 1, 19964 citations

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Daniel T. Gillespie

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Tulsi Dass

de Broglie-Bohm theory (dBBT), treating quantum particles as point objects moving along well defined (Bohmian) trajectories, offers an appealing solution of the measurement problem in quantum mechanics; it has, however, problems relating to spin, relativity and lack of proper integration with the Hilbert space based framework. In this work, we present a consistent formalism which has the traditional state-observable framework integrated with the desirable features of dBBT. We adopt ensemble interpretation for the Schrodinger wave function $ψ$ . Given a Schrodinger wave function $ψ$ , we use its value $ψ_0$ at some fixed time (say, $t = 0$ ) to define the probability measure $|ψ_0|^2 {\rm d}x$ on the system configuration space $M$ ( $=\mathbb{R}^n$ ). On the resulting probability space $\mathcal{M}_0$ , we introduce a stochastic process $ξ(t)$ corresponding to the Heisenberg position operator $X_H(t)$ such that, in the Heisenberg state $|ψ_h\rangle$ corresponding to $ψ_0$ , the expectation value of $X_H(t)$ equals that of $ξ(t)$ in $\mathcal{M}_0$ . This condition leads to the de Broglie-Bohm guidance equation for the sample paths of the process $ξ(t)$ which are, therefore, Bohmian trajectories supposedly representing time-evolutions of individual members of the $ψ_0$ -ensemble. Stochastic processes and Bohmian trajectories corresponding to observables with discrete eigenvalues (in particular spin) are treated by extending the configuration space to the spectral space of the commutative algebra obtained by adding appropriate discrete observables to the position observables. Pauli's equation is treated as an example. A straightforward derivation of von Neumann's projection rule employing the Schrodinger-Bohm evolution of individual systems along their Bohmian trajectories is given. Some comments on the potential application of the formalism developed here to quantum mechanics of the universe are included.de Broglie-Bohm theory (dBBT), treating quantum particles as point objects moving along well defined (Bohmian) trajectories, offers an appealing solution of the measurement problem in quantum mechanics; it has, however, problems relating to spin, relativity and lack of proper integration with the Hilbert space based framework. In this work, we present a consistent formalism which has the traditional state-observable framework integrated with the desirable features of dBBT. We adopt ensemble interpretation for the Schrodinger wave function $ψ$. Given a Schrodinger wave function $ψ$, we use its value $ψ_0$ at some fixed time (say, $t = 0$) to define the probability measure $|ψ_0|^2 {\rm d}x$ on the system configuration space $M$ ($=\mathbb{R}^n$). On the resulting probability space $\mathcal{M}_0$, we introduce a stochastic process $ξ(t)$ corresponding to the Heisenberg position operator $X_H(t)$ such that, in the Heisenberg state $|ψ_h\rangle$ corresponding to $ψ_0$, the expectation value of $X_H(t)$ equals that of $ξ(t)$ in $\mathcal{M}_0$. This condition leads to the de Broglie-Bohm guidance equation for the sample paths of the process $ξ(t)$ which are, therefore, Bohmian trajectories supposedly representing time-evolutions of individual members of the $ψ_0$-ensemble. Stochastic processes and Bohmian trajectories corresponding to observables with discrete eigenvalues (in particular spin) are treated by extending the configuration space to the spectral space of the commutative algebra obtained by adding appropriate discrete observables to the position observables. Pauli's equation is treated as an example. A straightforward derivation of von Neumann's projection rule employing the Schrodinger-Bohm evolution of individual systems along their Bohmian trajectories is given. Some comments on the potential application of the formalism developed here to quantum mechanics of the universe are included.

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arXiv:1604.00987Fluctuation and Noise LettersJul 7, 20165 citations

{\mathit{Z}}_{2}

event space assumption, if we require its existence for all times t\ensuremath{\in}

{\mathbf{R}}_{+}

. \textcopyright{} 1996 The American Physical Society.

ψ

. Given a Schrodinger wave function

ψ

, we use its value

ψ_0

at some fixed time (say,

t = 0

) to define the probability measure

|ψ_0|^2 {\rm d}x

on the system configuration space

M

(

=\mathbb{R}^n

). On the resulting probability space

\mathcal{M}_0

, we introduce a stochastic process

ξ(t)

corresponding to the Heisenberg position operator

X_H(t)

such that, in the Heisenberg state

|ψ_h\rangle

corresponding to

ψ_0

, the expectation value of

X_H(t)

equals that of

ξ(t)

\mathcal{M}_0

. This condition leads to the de Broglie-Bohm guidance equation for the sample paths of the process

ξ(t)

which are, therefore, Bohmian trajectories supposedly representing time-evolutions of individual members of the

ψ_0

-ensemble. Stochastic processes and Bohmian trajectories corresponding to observables with discrete eigenvalues (in particular spin) are treated by extending the configuration space to the spectral space of the commutative algebra obtained by adding appropriate discrete observables to the position observables. Pauli's equation is treated as an example. A straightforward derivation of von Neumann's projection rule employing the Schrodinger-Bohm evolution of individual systems along their Bohmian trajectories is given. Some comments on the potential application of the formalism developed here to quantum mechanics of the universe are included.