1 Introduction↩︎

Quantum mechanics provides the fundamental framework for describing the behavior of microscopic particles under the influence of external forces and potentials. Among the various quantum systems studied in theoretical physics, the motion of a particle subjected to a linear potential occupies a central position because of its rich mathematical structure and broad range of physical applications.

Physically, a linear potential represents the quantum analogue of a particle moving under a constant external force, such as a charged particle placed in a uniform electric field. Owing to its fundamental nature, this model appears in several branches of modern physics, including condensed matter physics, quantum information theory, and astrophysics, where the confinement and manipulation of particles by external fields play an essential role.

The dynamics of the system are governed by the time-dependent Schrödinger equation

\[i\hbar \frac{\partial }{\partial t}\psi (q,t)=H\text{ }\psi (q,t) \label{1}\tag{1}\] with the Hamiltonian

\[H\left( q,p\right) =\frac{1}{2m}p^{2}-Fq, \label{2}\tag{2}\] where \(m\) denotes the particle mass and \(F\) is the constant external force.

Unlike the harmonic oscillator, the linear potential does not possess a discrete spectrum that can be expressed through elementary functions. Instead, the stationary solutions are written in terms of Airy functions [1],

\[\psi (\xi )=C_{1}\,Ai(\xi )+C_{2}\,Bi(\xi ),\;\xi =\left( 2mF\right) ^{1/2}(q-\frac{E}{F}), \label{239}\tag{3}\] where \(Ai(\xi )\) and \(Bi(\xi )\) are Airy functions, while \(C_{1},C_{2}\) are constants determined by the boundary conditions. Since the function \(Bi(\xi )\) diverges as \(\xi \rightarrow \infty\), it violates the physical requirement that the wave function remains finite. Consequently, only \(Ai(\xi )\) corresponds to physically acceptable solutions.

The time evolution of a quantum system subjected to a spatially uniform time-dependent force has also attracted considerable attention [2]–[10]. The exact propagator for this system has long been known, as well as a class of exact solutions usually referred to as Volkov solutions. More nvestigations have focused on the derivation of exact wave-packet solutions and on the analysis of their physical properties.

The general quantum-mechanical properties of dynamical systems can be investigated using the invariant operator theory introduced by Lewis and Riesenfeld [11]. In this approach, the eigenfunctions of the invariant operator coincide with the wave functions of the Schrödinger equation up to time-dependent phase factors. The invariant formalism therefore provides a powerful framework for obtaining exact quantum solutions.

In the present work, we focus on wave solutions characterized by a discrete eigenspectrum. In particular, we investigate the case corresponding to positive frequency, for which the transformed invariant operator reduces to a harmonic oscillator with discrete eigenvalues.

2 Lewis–Riesenfeld Invariant Formalism and Discrete Spectrum Structure↩︎

According to the Lewis–Riesenfeld theory, a Hermitian operator \(I(t)\) is called an invariant if it satisfies the Liouville–von Neumann equation \[\frac{dI}{dt}=\frac{\partial I}{\partial t}+\frac{1}{i\hbar }\left[ I,H\right] =0, \label{3}\tag{4}\] if \(\varphi _{\lambda }(q,t)\) is an eigenfunction of \(I(t)\) with a time-independent eigenvalue \(\zeta _{\lambda }\),\[I(t)\varphi _{\lambda }(q,t)=\zeta _{\lambda }\text{ }\varphi _{\lambda }(q,t), \label{334}\tag{5}\] (the subscript \(\lambda\) depends on the nature of the spectrum (discrete for \(\lambda\) \(=n\)) or (continuous for \(\lambda =k\)) then a solution of the Schrödinger equation can be written as \[\psi _{\lambda }(q,t)=\exp \left[ i\alpha _{\lambda }\left( t\right) \right] \varphi _{\lambda }(q,t) \label{339}\tag{6}\] where the phase \(\alpha _{\lambda }\left( t\right)\) satisfies

\[\hbar \dot{\alpha}_{\lambda }\left( t\right) \varphi _{\lambda }=\left[ i\hbar \frac{\partial }{\partial t}-H\right] \varphi _{\lambda }. \label{4}\tag{7}\]

To investigate the quantum dynamics of a particle subjected to the linear potential, we consider the most general quadratic Hermitian invariant \[I\left( t\right) =\;\;\left[ A\left( t\right) p^{2}+B\left( t\right) \left( pq+qp\right) +C\left( t\right) q^{2}+g\left( t\right) q+K\left( t\right) p+\gamma \left( t\right) \right] , \label{e}\tag{8}\] where the coefficients\(A\left( t\right)\),\(B\left( t\right)\), \(C(t),\) \(g\left( t\right)\), \(K\left( t\right)\)and \(\gamma \left( t\right)\)are real functions of time.

Substituting Eq. (8 ) into Eq. (4 ) yields the coupled differential system

\[\left\{ \begin{array}{c} \overset{\cdot }{A}\left( t\right) +\frac{2B\left( t\right) }{m}=0 \\ \;\overset{\cdot }{B}\left( t\right) +\frac{C\left( t_{0}\right) }{m}=0 \\ \overset{\cdot }{C}\left( t\right) =0 \\ \;\overset{\cdot }{g}\left( t\right) +2fB\left( t\right) =0 \\ \overset{\cdot }{K}\left( t\right) +\left( 2fA\left( t\right) +\frac{g\left( t\right) }{m}\right) =0 \\ \overset{\cdot }{\gamma }\left( t\right) +fK\left( t\right) =0.\end{array}\right. \label{5}\tag{9}\] The third equation implies immediately that \(C(t)\) is constant:\[C\left( t\right) =C_{0}, \label{M}\tag{10}\] where \(C_{0}\) is a constant. Integrating successively, one obtains\[\left\{ \begin{array}{c} B\left( t\right) =B_{0}-\frac{C_{0}}{m}(t-t_{0}), \\ A\left( t\right) =A_{0}-\frac{2B_{0}}{m}(t-t_{0})+\frac{C_{0}}{m^{2}}(t-t_{0})^{2}, \\ g\left( t\right) =g_{0}-2fB_{0}(t-t_{0})+\frac{C_{0}f}{m}(t-t_{0})^{2}, \\ K\left( t\right) =K_{0}-\left( 2fA_{0}+\frac{g_{0}}{m}\right) (t-t_{0})+\frac{3fB_{0}}{m}(t-t_{0})^{2}-\frac{C_{0}f}{m^{2}}(t-t_{0})^{3}, \\ \gamma \left( t\right) =\gamma _{0}-fK_{0}(t-t_{0})+f\frac{\left( 2fA_{0}+\frac{g_{0}}{m}\right) }{2}(t-t_{0})^{2}-\frac{f^{2}B_{0}}{m}(t-t_{0})^{3}+\text{ }\frac{C_{0}f^{2}}{4m^{2}}(t-t_{0})^{4}.\end{array}\right. \label{L}\tag{11}\] An important conserved quantity follows naturally from these relations:

\[A\left( t\right) C\left( t\right) -B^{2}\left( t\right) =\omega ^{2}, \label{P}\tag{12}\] with\[\omega ^{2}=A_{0}C_{0}-B_{0}^{2}. \label{Q}\tag{13}\] To simplify the invariant operator, we introduce the time-dependent unitary transformation \(U(t)\) = \(U_{1}(t)U_{2}(t)D^{+}\left( \alpha \left( t\right) \right)\) such that\[\varphi _{\lambda }\left( q,t\right) =D\left( \alpha \left( t\right) \right) U_{2}^{+}(t)U_{1}^{+}(t)\tilde{\varphi}_{\lambda }\left( q\right) , \label{B}\tag{14}\] where \[\left\{ \begin{array}{c} U_{1}(t)=e^{\frac{i}{4\hbar }(qp+pq)\ln (2A\left( t\right) )}, \\ U_{2}(t)=e^{\frac{i}{2\hbar }\frac{B\left( t\right) }{A\left( t\right) }q^{2}}, \\ D\left( \alpha \left( t\right) \right) =e^{-\frac{i}{\hbar }(\eta q-\mu p)}.\end{array}\right. \label{9}\tag{15}\] The displacement parameters are given by \[\left\{ \begin{array}{c} \mu \left( t\right) =\frac{A\left( t\right) g\left( t\right) -K\left( t\right) B\left( t\right) }{2\left( A\left( t\right) C\left( t\right) -B\left( t\right) ^{2}\right) } \\ \eta \left( t\right) =\frac{C\left( t\right) K\left( t\right) -g\left( t\right) B\left( t\right) }{2\left( A\left( t\right) C\left( t\right) -B\left( t\right) ^{2}\right) }\end{array}\right. \label{10}\tag{16}\] Under these transformations, the coordinate and momentum operators transform according to \[\left\{ \begin{array}{c} \begin{array}{c} \begin{array}{c} U_{1}(t)qU_{1}^{+}(t)=\sqrt{2A\left( t\right) }\text{ }q, \\ U_{1}(t)pU_{1}^{+}(t)=\frac{1}{\sqrt{2A\left( t\right) }\text{ }}p, \\ U_{2}(t)pU_{2}^{+}(t)=p\text{ }-\frac{B\left( t\right) }{A\left( t\right) }q,\end{array} \\ D^{+}\left( \alpha \left( t\right) \right) qD\left( \alpha \left( t\right) \right) =q-\mu \left( t\right) ,\end{array} \\ D^{+}\left( \alpha \left( t\right) \right) pD\left( \alpha \left( t\right) \right) =p-\eta \left( t\right) .\end{array}\right. \label{A}\tag{17}\]

After straightforward calculations, the transformed invariant operator becomes\[I^{\prime }(t)=U(t)I(t)U^{+}(t)=\frac{1}{2}\left( p^{2}+4\omega ^{2}q^{2}\right) +\Lambda \label{D}\tag{18}\]

where \[\Lambda =A\left( t\right) \eta \left( t\right) ^{2}+2B\left( t\right) \eta \left( t\right) \mu \left( t\right) +C\left( t\right) \mu \left( t\right) ^{2}-g\left( t\right) \mu \left( t\right) -K\left( t\right) \eta \left( t\right) +\gamma \left( t\right) . \label{42}\tag{19}\] i.e., \(U(t)\) brings any solution of the operator eigenvalue equation (5 ) into a solution of the operator eigenvalue equation\[I^{\prime }(t)\tilde{\varphi}_{\lambda }\left( q\right) =\zeta _{\lambda }\tilde{\varphi}_{\lambda }\left( q\right) , \label{C}\tag{20}\] The transformed invariant operator is defined by and\[\varphi _{\lambda }\left( q,t\right) =D\left( \alpha \left( t\right) \right) U_{2}^{+}(t)U_{1}^{+}(t)\tilde{\varphi}_{\lambda }\left( q\right) ; \label{8}\tag{21}\]

After straightforward calculations, the transformed invariant operator \(I^{\prime }(t)=U(t)I(t)U^{+}(t)\) takes the form

\[I^{\prime }=\frac{1}{2}\left( p^{2}+4\omega ^{2}q^{2}\right) +\Lambda\]Using the explicit expressions of the coefficients, the constant term \(\Lambda\) reduces to the explicit expressions of the coefficients, Eq. (19 ) simplifies to \[\Lambda =\frac{-A_{0}g_{0}^{2}-C_{0}K_{0}^{2}+2B_{0}g_{0}K_{0}+4A_{0}C_{0}\gamma _{0}-4B_{0}^{2}\gamma _{0}}{4\left( A_{0}C_{0}-B_{0}^{2}\right) }. \label{E}\tag{22}\] Without loss of generality, we choose \(g_{0}=K_{0}=\gamma _{0}=0\), so that so that the invariant operator finally reduces to

\[I^{\prime }(t)=\frac{1}{2}\left( p^{2}+4\omega ^{2}q^{2}\right) . \label{I39}\tag{23}\]

The eigenvalue equation associated with Eq. (23 ) is therefore \[\frac{1}{2}\left[ -\hbar ^{2}\frac{\partial ^{2}}{\partial q^{2}}+4\omega ^{2}q^{2}\right] \tilde{\varphi}_{\lambda }\left( q\right) =\zeta _{\lambda }\tilde{\varphi}_{\lambda }\left( q\right) , \label{os}\tag{24}\] which is exactly the stationary Schrödinger equation of the one-dimensional harmonic oscillator.

The sign of \(\omega ^{2}\) determines the spectral structure of the system:

\(\cdot\) \(\omega ^{2}\) \(>0\): discrete spectrum,

\(\cdot\) \(\omega ^{2}\) \(=0\): continuous spectrum,

\(\cdot\) \(\omega ^{2}\) \(<0\): continuous spectrum.

In the present work, we focus exclusively on the physically relevant regime \(\omega ^{2}\) \(>0\), where the transformed invariant describes an oscillatory system with quantized eigenvalues. The normalized eigenfunctions are then given by

\[\tilde{\varphi}_{n}\left( x\right) =\left( \frac{2\omega }{\hbar }\right) ^{\frac{1}{4}}\frac{1}{\sqrt{\sqrt{\pi }2^{n}n!}}e^{-\frac{\omega x^{2}}{\hbar }}H_{n}\left( \left( \frac{2\omega }{\hbar }\right) ^{\frac{1}{2}}q\right) ,\qquad\]with eigenvalues\[\zeta _{n}=2\hbar \omega \left( n+\frac{1}{2}\right) ,\text{ \;\;\;\;\;}n=0,1,2,...\]where \(H_{n}\) are the Hermite polynomials.

The displacement parameters introduced through the unitary transformation \(D\left( \alpha \left( t\right) \right)\) satisfy

\(\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\)\[\left\{ \begin{array}{c} \mu \left( t\right) =\mu _{0}-\frac{\eta _{0}}{m}(t-t_{0})+\frac{f}{2m}(t-t_{0})^{2} \\ \eta \left( t\right) =\eta _{0}-f(t-t_{0}).\end{array}\right. \label{m}\tag{25}\] Differentiation gives \[\overset{\cdot }{\mu }\left( t\right) =\frac{\eta \left( t\right) }{m}\text{, \;\;\;}\overset{\cdot }{\eta }\left( t\right) =-f\text{\;,\;\;\;} \label{mm}\tag{26}\] which reproduce exactly the classical equations of motion for a particle subjected to a constant external force. Hence, the classical dynamics emerge naturally from the invariant operator formalism and the associated unitary transformations.

3 Conclusion↩︎

In this work, we studied the quantum dynamics of a particle subjected to a linear potential using the Lewis–Riesenfeld invariant operator method. Starting from the most general quadratic Hermitian invariant, we derived the complete set of differential equations governing the invariant coefficients and obtained their exact analytical solutions.

Through a sequence of time-dependent unitary transformations, the invariant operator was reduced to the Hamiltonian form of a harmonic oscillator. This transformation revealed the existence of a conserved quantity \(\omega ^{2}\) \(>0\), whose sign determines the spectral nature of the system.

Special attention was devoted to the case \(\omega ^{2}\) \(>0\), corresponding to a discrete spectrum. In this regime, the transformed invariant admits harmonic oscillator eigenfunctions and quantized eigenvalues. Explicit expressions for the displacement parameters and associated phase structure were also obtained.

The present formulation provides an exact and elegant framework for studying quantum systems subjected to constant external forces and establishes a direct correspondence between linear-potential dynamics and harmonic oscillator quantization through invariant theory.

3.0.0.0.1 \(\circ\) Author Contribution declaration in the manuscript

The study was conceived and schemed by Maamache Mustapha.

The mathematical evaluations in the text were performed by Aymen Bendjoudi and Maamache Mustapha .

The paper was written by Mustapha Maamache and the final correction of the paper was done by Aymen Bendjoudi

3.0.0.0.2 \(\circ\) Conflicts of interests

.The authors declare no conflict of interest.

\(\circ\) Funding Declaration

No funding Declaration

References↩︎