We compute the Bohmian trajectories of the incoming scattering plane waves for Klein's potential step in explicit form. For finite norm incoming scattering solutions we derive their asymptotic space-time localization and we compute some Bohmian trajectories numerically. The paradox, which appears in the traditional treatments of the problem based on the outgoing scattering asymptotics, is absent.

arXiv:quant-ph/0202098Journal of Physics A Mathematical and GeneralMar 26, 20019 citations

Quantum Dynamics with Bohmian Trajectories

Dirk, André Deckert

Jan 1, 20080 citations

International Journal of Theoretical Physics

Bohmian Trajectories Within Hilbert Space Based Quantum Mechanics. Solution of the Measurement Problem

Tulsi Dass

Published Mar 31, 2026• 0 citations

Publisher

DOI: 10.1007/s10773-026-06307-3

Related papers

Bohmian Mechanics, Open Quantum Systems and Continuous Measurements

Antonio Boulhosa Nassar, Salvador Miret-Artés

Springer eBooksJan 1, 201742 citations

Bohmian Trajectories Within Hilbert Space Based Quantum Mechanics. Solution of the Measurement Problem

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.

quant-pharXiv:2512.07007Dec 9, 20250 citations

Bohmian trajectories and quantum phase space distributions

Nuno Costa Dias, João Nuno Prata

arXiv:quant-ph/0208156Physics Letters ASep 1, 200225 citations

Non-differentiable Bohmian trajectories

Gebhard Grübl, Markus Penz

A solution $ψ$ to Schrödinger's equation needs some degree of regularity in order to allow the construction of a Bohmian mechanics from the integral curves of the velocity field $\hbar \Im \left( \bigtriangledown ψ/mψ\right) .$ In the case of one specific non-differentiable weak solution $Ψ$ we show how Bohmian trajectories can be obtained for $Ψ$ from the trajectories of a sequence $Ψ_{n}\rightarrow Ψ.$ (For any real $t$ the sequence $Ψ_{n}\left( t,\cdot \right) $ converges strongly.) The limiting trajectories no longer need to be differentiable. This suggests a way how Bohmian mechanics might work for arbitrary initial vectors $Ψ$ in the Hilbert space on which the Schrödinger evolution $% Ψ\mapsto e^{-iht}Ψ$ acts.A solution $ψ$ to Schrödinger's equation needs some degree of regularity in order to allow the construction of a Bohmian mechanics from the integral curves of the velocity field $\hbar \Im \left( \bigtriangledown ψ/mψ\right) .$ In the case of one specific non-differentiable weak solution $Ψ$ we show how Bohmian trajectories can be obtained for $Ψ$ from the trajectories of a sequence $Ψ_{n}\rightarrow Ψ.$ (For any real $t$ the sequence $Ψ_{n}\left( t,\cdot \right) $ converges strongly.) The limiting trajectories no longer need to be differentiable. This suggests a way how Bohmian mechanics might work for arbitrary initial vectors $Ψ$ in the Hilbert space on which the Schrödinger evolution $% Ψ\mapsto e^{-iht}Ψ$ acts.

arXiv:1011.2852Nov 12, 20101 citation

Bohmian trajectories and Klein's paradox

Gebhard Grübl, Raimund Moser, Klaus Rheinberger

arXiv:quant-ph/0202098Journal of Physics A Mathematical and GeneralMar 26, 20019 citations

Quantum Dynamics with Bohmian Trajectories

Dirk, André Deckert

Jan 1, 20080 citations

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.

A solution

ψ

to Schrödinger's equation needs some degree of regularity in order to allow the construction of a Bohmian mechanics from the integral curves of the velocity field

\hbar \Im \left( \bigtriangledown ψ/mψ\right) .

In the case of one specific non-differentiable weak solution

Ψ

we show how Bohmian trajectories can be obtained for

Ψ

from the trajectories of a sequence

Ψ_{n}\rightarrow Ψ.

(For any real

t

the sequence $Ψ_{n}\left( t,\cdot \right) $ converges strongly.) The limiting trajectories no longer need to be differentiable. This suggests a way how Bohmian mechanics might work for arbitrary initial vectors

Ψ

in the Hilbert space on which the Schrödinger evolution

% Ψ\mapsto e^{-iht}Ψ

acts.