Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Proceedings of the Samahang Pisika ng Pilipinas

Uncertainty relation of the momentum components of a two-dimensional electron system under a tilted magnetic field

Rayda Gammag, Cristine Dlr. Villagonzalo

Published Oct 25, 2010• 0 citations
OpenAlex

Abstract

A two-dimensional electron gas along the xy-plane is considered. When an applied perpendicular magnetic field, directed along the zaxis, is turned on, the electrons move in cyclotron orbits and the components of the momentum along the x- and y-axes do not commute. This work shows that if the applied magnetic field is tilted at an angle theta with respect to the z-axis, the uncertainty in simultaneous measurements of the momentum components becomes dependent on theta.

Related papers

The measurability of the kinetic momentum of a charged particle in a magnetic field

W P Healy

Uncertainty relations are derived for the components of the kinetic momentum (mass times velocity) of a charged particle in a magnetic field. In constrast to the components of the canonical momentum, the components of the kinetic momentum do not all commute with each other when the field is not zero. The uncertainty relations are illustrated by two Gedanken experiments in which the kinetic momentum is measured by using (a) the Doppler shift of emitted radiation and (b) the angle of deflection of the particle by the field.

American Journal of Physics•Dec 1, 1988•0 citations

On the uncertainty relations for an electron in a constant magnetic field

Francisco Marcelo Fernández

We discuss the uncertainty relation for the azimuthal angle $ϕ$ and the $z$-component of the angular momentum $L_z$. To this end we derive the uncertainty relation for an arbitrary pair of observables and discuss the conditions for its validity. By means of a simple parameter-dependent state we illustrate the well known fact that the standard uncertainty relation for the coordinate and its conjugate momentum does not apply the pair of observables $ϕ-L_z$. This analysis is motivated by a discussion of the motion of an electron in a constant magnetic field appeared recently in this journal (Eur. J. Phys. \textbf{33} (2012) 1147) where the author assumed the validity of the standard uncertainty relation for the pair $ϕ-L_z$.

arXiv:1207.7069•Jul 30, 2012•0 citations

Parallel axis theorem for free-space electron wavefunctions

Colin Greenshields, Sonja Franke‐Arnold, Robert L. Stamps

We consider the orbital angular momentum of a free electron vortex moving in a uniform magnetic field. We identify three contributions to this angular momentum: the canonical orbital angular momentum associated with the vortex, the angular momentum of the cyclotron orbit of the wavefunction, and a diamagnetic angular momentum. The cyclotron and diamagnetic angular momenta are found to be separable according to the parallel axis theorem. This means that rotations can occur with respect to two or more axes simultaneously, which can be observed with superpositions of vortex states.

arXiv:1506.08702•New Journal of Physics•Sep 10, 2015•32 citations

Understanding quantum behaviors of an electron in a uniform magnetic field alternatively

Jin-Ming Wang, Yuan-Zao Gao, Dai-Lin Cun +1 more

Quantum mechanically, an electron moving in a uniform magnetic field forms Landau levels. A curious feature is that for states with a negative angular quantum number, the total probability current vanishes, which appears to contradict the classical picture of cyclotron motion. While a geometric interpretation based on classical orbits exists, alternative interpretations remain of interest. In this paper, we examine the probability current density and identify a critical radius that naturally partitions the plane into an inner clockwise-flow region and an outer counterclockwise-flow region. We show that the vanishing total current results from an exact cancellation between these two regions. Furthermore, by defining a partitioned kinetic angular momentum with respect to the critical radius, we reveal an intrinsic competitive structure: the electron simultaneously carries two opposing rotational components. The negative quantum number manifests in the strength of the inner counter-rotation, while the net kinetic angular momentum remains positive. This bidirectional flow picture also provides a dynamical interpretation of the infinite degeneracy of Landau levels.

cond-mat.mes-hallCross-listedarXiv:2606.13290•International Journal of Theoretical Physics•Jun 12, 2026•0 citations

Uncertainty relations for the triple components of canonical momentum

Zhi-Jie Liu, Jie Zhou, Hao-Nan Qiang +2 more

Abstract Uncertainty relations are one of the fundamental components of quantum theory, typically used to describe the limitations on the range of measurement results for two noncommuting observables. Recently, there has been growing interest in uncertainty relations concerning multiple observables. In this work, we study the uncertainty relations of the three components of the canonical momentum of an electron moving in a magnetic field. We derive the uncertainty relations for both the addition and multiplication of these components. Our results indicate that, compared to uncertainty relations involving only two components, the uncertainty relations for the three components are associated with a constant τ = 2 / 3 .

Physica Scripta•Mar 18, 2025•2 citations

The coherent states of the two-dimensional isotropic harmonic oscillator and the classical limit of the Landau theory of a charged particle in a uniform magnetic field

Wangyao Li, Kunnat J. Sebastian

Abstract In this paper we show how the classical result of a charged particle moving in a circle in the xy plane, when a uniform magnetic field is directed along the z -axis, can be derived from the Landau quantum theory using the coherent states of the two-dimensional isotropic harmonic oscillator in the xy plane. The coherent states in this case are the simultaneous eigen vectors of the annihilation operators a + and a − . We prove that the time-dependent coordinate space wave packets representing the time-dependent coherent states move in a circle with the cyclotron frequency ω c = ∣ q ∣ B m and with a radius given by the classical expression, but given in terms of the quantum mechanical expectation values. The expectation value of the energy of the particle and of the square of the radius of its circular are proportional to the square of the magnitude of the eigen value of a + in the coherent state, where as the x and y coordinates of the centre of the circle are proportional to the real and the imaginary parts of the eigen value of a − . The phase of the circular motion is the same as the phase of the complex eigen value of a + . So for a given energy of the particle or for a given radius of the circular orbit, there are an infinite number of circles which differ from each other by the x and y coordinates of the centre as well as the phase of the circular motion. The infinite degeneracy of the Landau levels is due to the invariance of the energy eigen values under spatial translations in the xy plane and rotations about the z -axis. We also show that as the magnitude of the eigen value of a + becomes much larger than one, the relative uncertainty or fluctuation in the energy and in the radius of the circular orbit becomes negligibly small as we expect for a classical state.

European Journal of Physics•Mar 26, 2018•4 citations
A two-dimensional electron gas along the xy-plane is considered. When an applied perpendicular magnetic field, directed along the zaxis, is turned on, the electrons move in cyclotron orbits and the components of the momentum along the x- and y-axes do not commute. This work shows that if the applied magnetic field is tilted at an angle theta with respect to the z-axis, the uncertainty in simultaneous measurements of the momentum components becomes dependent on theta.
Uncertainty relations are derived for the components of the kinetic momentum (mass times velocity) of a charged particle in a magnetic field. In constrast to the components of the canonical momentum, the components of the kinetic momentum do not all commute with each other when the field is not zero. The uncertainty relations are illustrated by two Gedanken experiments in which the kinetic momentum is measured by using (a) the Doppler shift of emitted radiation and (b) the angle of deflection of the particle by the field.
We discuss the uncertainty relation for the azimuthal angle ϕϕϕ and the zzz-component of the angular momentum LzL_zLz​. To this end we derive the uncertainty relation for an arbitrary pair of observables and discuss the conditions for its validity. By means of a simple parameter-dependent state we illustrate the well known fact that the standard uncertainty relation for the coordinate and its conjugate momentum does not apply the pair of observables ϕ−Lzϕ-L_zϕ−Lz​. This analysis is motivated by a discussion of the motion of an electron in a constant magnetic field appeared recently in this journal (Eur. J. Phys. \textbf{33} (2012) 1147) where the author assumed the validity of the standard uncertainty relation for the pair ϕ−Lzϕ-L_zϕ−Lz​.
We consider the orbital angular momentum of a free electron vortex moving in a uniform magnetic field. We identify three contributions to this angular momentum: the canonical orbital angular momentum associated with the vortex, the angular momentum of the cyclotron orbit of the wavefunction, and a diamagnetic angular momentum. The cyclotron and diamagnetic angular momenta are found to be separable according to the parallel axis theorem. This means that rotations can occur with respect to two or more axes simultaneously, which can be observed with superpositions of vortex states.
Quantum mechanically, an electron moving in a uniform magnetic field forms Landau levels. A curious feature is that for states with a negative angular quantum number, the total probability current vanishes, which appears to contradict the classical picture of cyclotron motion. While a geometric interpretation based on classical orbits exists, alternative interpretations remain of interest. In this paper, we examine the probability current density and identify a critical radius that naturally partitions the plane into an inner clockwise-flow region and an outer counterclockwise-flow region. We show that the vanishing total current results from an exact cancellation between these two regions. Furthermore, by defining a partitioned kinetic angular momentum with respect to the critical radius, we reveal an intrinsic competitive structure: the electron simultaneously carries two opposing rotational components. The negative quantum number manifests in the strength of the inner counter-rotation, while the net kinetic angular momentum remains positive. This bidirectional flow picture also provides a dynamical interpretation of the infinite degeneracy of Landau levels.
Abstract Uncertainty relations are one of the fundamental components of quantum theory, typically used to describe the limitations on the range of measurement results for two noncommuting observables. Recently, there has been growing interest in uncertainty relations concerning multiple observables. In this work, we study the uncertainty relations of the three components of the canonical momentum of an electron moving in a magnetic field. We derive the uncertainty relations for both the addition and multiplication of these components. Our results indicate that, compared to uncertainty relations involving only two components, the uncertainty relations for the three components are associated with a constant τ = 2 / 3 .
Abstract In this paper we show how the classical result of a charged particle moving in a circle in the xy plane, when a uniform magnetic field is directed along the z -axis, can be derived from the Landau quantum theory using the coherent states of the two-dimensional isotropic harmonic oscillator in the xy plane. The coherent states in this case are the simultaneous eigen vectors of the annihilation operators a + and a − . We prove that the time-dependent coordinate space wave packets representing the time-dependent coherent states move in a circle with the cyclotron frequency ω c = ∣ q ∣ B m and with a radius given by the classical expression, but given in terms of the quantum mechanical expectation values. The expectation value of the energy of the particle and of the square of the radius of its circular are proportional to the square of the magnitude of the eigen value of a + in the coherent state, where as the x and y coordinates of the centre of the circle are proportional to the real and the imaginary parts of the eigen value of a − . The phase of the circular motion is the same as the phase of the complex eigen value of a + . So for a given energy of the particle or for a given radius of the circular orbit, there are an infinite number of circles which differ from each other by the x and y coordinates of the centre as well as the phase of the circular motion. The infinite degeneracy of the Landau levels is due to the invariance of the energy eigen values under spatial translations in the xy plane and rotations about the z -axis. We also show that as the magnitude of the eigen value of a + becomes much larger than one, the relative uncertainty or fluctuation in the energy and in the radius of the circular orbit becomes negligibly small as we expect for a classical state.