Coherent States of the Conformable Quantum Oscillator

Abstract

The recently proposed conformable deformation of quantum mechanics by a fractional parameter $\alpha \in (0,1]$ has been used to construct a conformable quantum harmonic oscillator, which coincides with the standard quantum oscillator at $\alpha =1$. We argue that there is a conformable generalization of the uncertainty principle and use it to define the conformable coherent states of the conformable quantum oscillator along the general line of quantum mechanics. We investigate the fundamental physical and mathematical properties of these states in the $x^{\alpha }$-representation. In particular, we determine these states from the minimum uncertainty, compute their energy, find their conformable time-dependent form, determine the conformable translation operator, and show that conformable coherent states are eigenstates of the conformable annihilation operator. These states reproduce in the $\alpha =1$ limit of the correspondence principle the coherent states of the standard quantum harmonic oscillator.

1. Introduction

Very recently, a deformation of quantum mechanics by a fractional conformable parameter has been proposed in [1,2,3,4,5,6,7,8] that generalizes the standard quantum mechanics by assuming that the evolution in space and time takes place according to conformable (fractional) differential equations developed in [9,10,11,12,13,14]. The main objective of the conformable calculus was to generalize the fractional calculus by incorporating the Leibniz rule, a task later proved to be impossible [15,16] either in the fractional [17] or in the local fractional formulation [18]. Nevertheless, since the conformable calculus has potential applications to differential equations in a large range of fields and the conformable derivatives satisfy the basic axioms of standard calculus such as the linearity, the Leibniz rule and a (modified version of) the composition rule, several works have been dedicated to its application in physics, mainly to quantum mechanics, classical mechanics [19,20,21,22], and thermodynamics [23]. A recent interpretation of the conformable derivative was proposed in [24,25]. For more details on conformable calculus and some of its recent generalizations and applications in physics, see also [26,27,28,29].

One of the most important models of quantum mechanics, the quantum harmonic oscillator, has recently been generalized to conformable quantum mechanics in [7]. A canonical quantization formalism in terms of creation and annihilation operators has been given in [8]. The thermodynamic properties of the conformable quantum oscillator (CQO) in the canonical ensemble have been described in [23]. Since the CQO depends on the conformable parameter $\alpha \in (0,1]$, it actually describes a family of distinct oscillators whose evolution is described by differential equations of different types. The works mentioned above show that CQO shares many properties with the standard quantum harmonic oscillator that is recovered in the correspondence principle limit $\alpha =1$, which at some point simplifies the analysis. However, one should note that no physical system with conformable dynamics has been discovered yet. Nevertheless, it is important to understand the mathematical and physical properties of the CQO as functions of the parameter $\alpha$ since that system can be viewed as a deformation of the standard harmonic oscillator consistent with the axioms of quantum mechanics.

In this paper, we obtain and study the conformable coherent states (CCSs) of the CQO. Quantum coherent states, first introduced by Schrödinger [30] and later formalized by Glauber [31] in quantum optics, represent the most classical-like states of quantum systems. For the standard quantum harmonic oscillator with Hamiltonian, coherent states are uniquely characterized by three fundamental properties: (i) they are minimum uncertainty states minimizing the Heisenberg uncertainty principle; (ii) they are eigenstates of the annihilation operator; (iii) they display time evolution stability, their dynamics following classical trajectories. Also, the excitation number of coherent states follows Poisson statistics, which is guaranteed by minimizing the Heisenberg uncertainty. These properties establish coherent states as an important link between quantum and classical descriptions of oscillatory systems. Since the saturation of the Heisenberg uncertainty entails the above properties of the standard coherent states, we propose conformable coherent states defined by the minimization of the following conformable uncertainty principle:

$${\sigma }_{\alpha X}{\sigma }_{\alpha p}\ge {\displaystyle \frac{{\hslash }_{\alpha }^{\alpha }}{2}}$$

(1)

where ${\sigma }_{\alpha X}^2$ and ${\sigma }_{\alpha p}^2$ are variances under the conformable integration measure $d{\mu }_{\alpha }\left(x\right)={\left|x\right|}^{\alpha -1}dx$ in the Hilbert space. This definition generalizes the standard one while incorporating $\alpha$-dependence, and reproduces the corresponding standard uncertainty principle and measure in the limit $\alpha =1$. In this paper, we are going to establish the conformable uncertainty principle (1) and show that there are states of the conformable quantum operators that satisfy it. We are going to approach the CCS problem in the wavefunction formalism in which the conformable differential operators appear explicitly.

The paper is organized as follows. In Section 2, we briefly review the basic concepts of conformable quantum mechanics in $d=1+1$ dimensions, emphasizing the main modifications of standard quantum mechanics. In Section 3, we present the CQO in position representation. The main references for these sections are [1,2,3]. In Section 4, we derive the CCS of CQO. First, we prove that the Cauchy–Bunyakovsky–Schwarz inequality holds in the space ${\mathcal{L}}_{\alpha }^2\left(\mathbb{R}\right)$ of square-integrable functions with respect to the $\alpha$-dependent integration measure $d{\mu }_{\alpha }\left(x\right)$. Using this result, we derive the basic inequality of the conformable uncertainty principle. Next, we calculate the CCSs defined as states that minimize the conformable uncertainty analogously to the standard quantum mechanics. We derive the energy $E^{\alpha }$ of a CCS, give the conformable time evolution, and construct the conformable translation operator of CCSs. Finally, we discuss the relationship between the eigenstates of the conformable annihilation operator and the CCS. In Appendix A we list the basic properties of the conformable calculus used throughout this text, and in Appendix B, Appendix C, Appendix D and Appendix E we present our proofs for the main mathematical results, which closely follow the standard derivations from quantum mechanics.

2. Basics of Conformable Quantum Mechanics in $\mathbf{d}=\mathbf{1}+\mathbf{1}$ Dimensions

In this section, we are going to review the basic concepts of conformable quantum mechanics in $d=1+1$ dimensions. The main references used for this section are [1,2,3] to which we refer for further details. As in these papers, we are going to work in $d=1+1$ dimensions.

Fundamentals of Conformable Quantum Mechanics

Conformable quantum mechanics can be viewed as a deformation of standard quantum mechanics for which the following postulates hold:

• The states of a conformable quantum system are described by complex functions $\Psi (t,x)$. At any $t\in \mathbb{R}$, $\Psi (t,x)$ belongs to the Hilbert space ${\mathcal{L}}_{\alpha }^2\left(\mathbb{R}\right)$ of the quadratic integrable functions on $\mathbb{R}$ endowed with the inner product with respect to the integration measure $d{\mu }_{\alpha }\left(x\right)={\left|x\right|}^{\alpha -1}dx$ given by the following inner product:

  $${\langle \Psi (t,x)|\Phi (t,x)\rangle }_{\alpha }=\int {\Psi }^{*}(t,x)\Phi (t,x)d{\mu }_{\alpha }\left(x\right).$$

  (2)
• The time evolution of the conformable quantum system is described by the conformable time-dependent Schrödinger equation of the form

  $$i{\hslash }_{\alpha }^{\alpha }{D}_t^{\alpha }\Psi (t,x)={\widehat{H}}_{\alpha }\left({\widehat{x}}_{\alpha },{\widehat{p}}_{\alpha }\right)\Psi (t,x).$$

  (3)
  Here, the conformable Hamiltonian operator ${\widehat{H}}_{\alpha }$ for a particle of conformable mass parameter $m^{\alpha }$ in the stationary potential $V\left(x\right)$ is defined by the following relation:

  $${\widehat{H}}_{\alpha }\left({\widehat{x}}_{\alpha },{\widehat{p}}_{\alpha }\right)={\displaystyle \frac{{\widehat{p}}_{\alpha }^2}{2m^{\alpha }}}+{\widehat{V}}_{\alpha }\left({\widehat{x}}_{\alpha }\right),$$

  (4)

  where the conformable linear momentum and position operators and the conformable Planck constant are defined as

  $${\widehat{x}}_{\alpha }=x,{\widehat{p}}_{\alpha }=-i{\hslash }_{\alpha }^{\alpha }{D}_x^{\alpha },{\hslash }_{\alpha }^{\alpha }={\left(2\pi \right)}^{-{\displaystyle \frac{1}{\alpha }}}h.$$

  (5)
  The conformable derivates $D_t^{\alpha }$ and $D_x^{\alpha }$ are reviewed in the Appendix A. We consider in this paper the case $s=0$, which is the original derivative proposed in [9] for a single variable and which was generalized in [13,14] for multi-variables; see Equations (A1) and (A2).

  $$\begin{array}{cc}\hfill D_t^{\alpha }f(t,x)& =\underset{ϵ\to 0}{lim}{\displaystyle \frac{f\left(t+{ϵ\left|t\right|}^{1-\alpha },x\right)-f(t,x)}{ϵ}},\hfill \end{array}$$

  (6)

  $$\begin{array}{cc}\hfill D_x^{\alpha }f(t,x)& =\underset{\eta \to 0}{lim}{\displaystyle \frac{f\left(t,x+{\eta \left|x\right|}^{1-\alpha }\right)-f(t,x)}{\eta }},\hfill \end{array}$$

  (7)

  for all $t,x\in \mathbb{R}$. The main properties of $D_x^{\alpha }f\left(x\right)$ are reviewed in Appendix A.
• The observables of the conformable system are given by Hermitian operators $\widehat{O}$ that act on the Hilbert space ${\mathcal{L}}_{\alpha }^2\left(\mathbb{R}\right)$, which are constructed from the physical quantities of the system

  $${\langle \widehat{O}\Psi |\Phi \rangle }_{\alpha }={\langle \Psi |\widehat{O}\Phi \rangle }_{\alpha }.$$

  (8)
  The eigenvalues $O_n$ of the operator $\widehat{O}$ correspond to the measured values of the observable in the eigenstates ${\Psi }_n(t,x)$. Note that the conformable systems represent just mathematical physics models that generalize the known quantum systems, with no physical system known to obey the conformable quantum mechanics up to now. Therefore, the concept of measurability should be understood in this abstract generalization sense.

In addition to the above postulates, it is also assumed that the other postulates of standard quantum mechanics hold in the conformable formalism, too. In this sense, the conformable probability density ${\rho }_{\alpha }(t,x)={|\Psi (t,x)|}^2$ has the same interpretation as its counterpart from standard quantum mechanics for the normalized state $\Psi (t,x)$ in the measure $d{\mu }_{\alpha }\left(x\right)$. The conservation of the conformable probability density is derived from the time-dependent Schrödinger equation defined in Equation (4), and it takes the form of the following conformable continuity equation:

$$D_t^{\alpha }{\rho }_{\alpha }(t,x)+D_x^{\alpha }{j}_{\alpha }(t,x)=0.$$

(9)

Here, the probability current $j_{\alpha }(t,x)$ is defined by the following relation:

$$j_{\alpha }(t,x)={\displaystyle \frac{{\hslash }_{\alpha }^{\alpha }}{2m^{\alpha }i}}\left[{\Psi }^{*}(t,x)D_x^{\alpha }\Psi (t,x)-\Psi (t,x)D_x^{\alpha }{\Psi }^{*}(t,x)\right].$$

(10)

Since the commutator has the same properties, a complete set of commutative observables is defined as in standard quantum mechanics. In particular, the position and momentum operators satisfy the following commutation relation:

$$\left[{\widehat{x}}_{\alpha },{\widehat{p}}_{\alpha }\right]=i{\hslash }_{\alpha }^{\alpha }{\left|\widehat{x}\right|}^{1-\alpha }.$$

(11)

The correspondence principle of conformable quantum mechanics states that the standard quantum mechanics is recovered at $\alpha =1$. This can be readily verified for all the quantities introduced so far.

In the case of a stationary potential, the separation of variables can be applied to Equation (3) to obtain the conformable stationary Schrödinger equation. The wavefunction $\Psi (t,x)$ is written as the product

$$\Psi (t,x)=e^{-i{\displaystyle \frac{E^{\alpha }{t}^{\alpha }}{\alpha h_{\alpha }^{\alpha }}}}\psi \left(x\right),$$

(12)

for all $t>0$. Then plugging $\Psi (t,x)$ into Equation (3) gives the following stationary Schrödinger equation:

$$\left[-{\displaystyle \frac{{\hslash }_{\alpha }^{2\alpha }}{2m^{\alpha }}}{\left(D_x^{\alpha }\right)}^2+V_{\alpha }\left(x\right)\right]\psi =E^{\alpha }\psi .$$

(13)

From Equation (12), we can see that the conformable wave–particle duality is defined by the following relations:

$$E^{\alpha }={\hslash }_{\alpha }^{\alpha }{\omega }^{\alpha },p_{\alpha }={\hslash }_{\alpha }^{\alpha }{k}^{\alpha }.$$

(14)

Here and in what follows, we use the following index convention from [3].

Note that in conformable quantum mechanics, the conformability order defined by the conformable parameter $\alpha$ appears naturally in the structure of the wavefunctions and observables, e.g., the energy, linear momentum, etc. Usually, the eigenfunction and eigenvalue problem for an operator ${\widehat{O}}_{\alpha }$ generates eigenvalues $O^{f\left(\alpha \right)}$, i.e., some $\alpha$-dependent power of the standard eigenvalue O. For example, the natural eigenvalue of the conformable Hamiltonian ${\widehat{H}}_{\alpha }$ is $E^{\alpha }$, the fractional $\alpha$-power of energy [3].

The definition of the conformable derivative can be naturally extended to negative values of arguments. For more details, see [1,2,3].

3. Conformable Quantum Oscillator (CQO)

In this section, we are going to review the CQO model from [7,8]. The Hamiltonian of CQO is defined by substituting the standard physical operators with their conformable counterparts:

$$\left({\displaystyle \frac{{\widehat{p}}_{\alpha }^2}{2m^{\alpha }}}+{\displaystyle \frac{{\alpha }^2}{2}}{m}^{\alpha }{\omega }^{2\alpha }{\widehat{X}}_{\alpha }^2\right)\psi \left(x\right)=E^{\alpha }\psi \left(x\right),$$

(15)

where $m^{\alpha }$ is the particle mass. In our notation, which is standard, the upper index $\alpha$ usually denotes the $\alpha$-power of the corresponding quantity with the exception of ${\hslash }_{\alpha }^{\alpha }$ and $D_x^{\alpha }$ where it is part of the symbols for the conformable Planck constant and the conformable derivative according to Equations (5) and (7). The position operator ${\widehat{X}}_{\alpha }^2$ is introduced to allow negative values for x and has the following form [3]

$${\widehat{X}}_{\alpha }\equiv X_{\alpha }\left(x\right)=\left\{\begin{array}{cc}{\displaystyle \frac{{\left(x\right)}^{\alpha }}{\alpha }},\hfill & \mathrm{if}x>0\hfill \\ {\displaystyle \frac{-{(-x)}^{\alpha }}{\alpha }},\hfill & \mathrm{if}x<0\hfill \end{array}\right..$$

(16)

One can easily verify that the position and momentum operators satisfy the following commutation relation:

$$\left[{\widehat{X}}_{\alpha },{\widehat{p}}_{\alpha }\right]=i{\hslash }_{\alpha }^{\alpha }.$$

(17)

The wavefunctions $\psi \left(x\right)$ belong to the Hilbert space ${\mathcal{L}}_{\alpha }^2\left(\mathbb{R}\right)$ of quadratically integrable functions. The index $\alpha$ reminds that ${\mathcal{L}}_{\alpha }^2\left(\mathbb{R}\right)$ is endowed with the inner product ${\langle ·|·\rangle }_{\alpha }$ with respect to the integration measure $d{\mu }_{\alpha }\left(x\right)$, which is well defined since the function ${\left|x\right|}^{\alpha -1}$ is non-vanishing almost everywhere on $\mathbb{R}$. The observables $\widehat{A}:{\mathcal{L}}_{\alpha }^2\left(\mathbb{R}\right)\to {\mathcal{L}}_{\alpha }^2\left(\mathbb{R}\right)$ are hermitian operators with respect to ${\langle ·|·\rangle }_{\alpha }$. The inner product of two arbitrary states and the hermiticity condition are given by the following natural generalization of the corresponding relations from standard quantum mechanics:

$${\langle \psi |\varphi \rangle }_{\alpha }=\int {\psi }^{*}\left(x\right)\varphi \left(x\right)d{\mu }_{\alpha }\left(x\right),{\langle \widehat{A}\psi |\varphi \rangle }_{\alpha }={\langle \psi |\widehat{A}\varphi \rangle }_{\alpha }.$$

(18)

The normalized ground state wavefunction and the $\alpha$-power of the ground energy were calculated in [8].

$${\psi }_0={\left({\displaystyle \frac{\alpha m^{\alpha }{\omega }^{\alpha }}{\pi {\hslash }_{\alpha }^{\alpha }}}\right)}^{{\displaystyle \frac{1}{4}}}exp\left(-{\displaystyle \frac{\alpha m^{\alpha }{\omega }^{\alpha }{X}_{\alpha }^2}{2{\hslash }_{\alpha }^{\alpha }}}\right),E_0^{\alpha }={\displaystyle \frac{1}{2}}\alpha {\hslash }_{\alpha }^{\alpha }{\omega }^{\alpha }.$$

(19)

In general, the eigenstates ${\psi }_n$ can be expressed in terms of Hermite or conformable Hermite polynomials. Their concrete form, which we reproduce here for completeness, was given in [8,12].

$$\begin{array}{cc}\hfill {\psi }_n& ={\left({\displaystyle \frac{\alpha m^{\alpha }{\omega }^{\alpha }}{\pi {\hslash }_{\alpha }^{\alpha }}}\right)}^{{\displaystyle \frac{1}{4}}}{\displaystyle \frac{1}{\sqrt{2^{n}n!}}}{H}_n\left(\sqrt{{\displaystyle \frac{\alpha m^{\alpha }{\omega }^{\alpha }}{{\hslash }_{\alpha }^{\alpha }}}}{X}_{\alpha }\right)exp\left(-{\displaystyle \frac{\alpha m^{\alpha }{\omega }^{\alpha }{X}_{\alpha }^2}{2{\hslash }_{\alpha }^{\alpha }}}\right),\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill {\psi }_n& ={\displaystyle \frac{1}{\sqrt{2^{n}n!}}}{\left({\displaystyle \frac{\alpha m^{\alpha }{\omega }^{\alpha }}{\pi {\hslash }_{\alpha }^{\alpha }}}\right)}^{{\displaystyle \frac{1}{4}}}exp\left(-{\displaystyle \frac{y^{2\alpha }}{2}}\right)H_n^{\alpha }\left(y\right),\hfill \end{array}$$

(21)

with the $\alpha$-power of energy in ${\psi }_n\left(x\right)$ state given by

$$E_n^{\alpha }=\alpha {\hslash }_{\alpha }^{\alpha }{\omega }^{\alpha }\left(n+{\displaystyle \frac{1}{2}}\right),n=0,1,2,\dots$$

(22)

All relations discussed above are in agreement with the correspondence principle in the sense that they reduce to their standard quantum mechanics counterparts at $\alpha =1$.

4. Conformable Coherent States (CCSs)

In this section, we define and calculate the CCSs of CQO. We also discuss their fundamental physical and mathematical properties and draw a parallel between them and the standard coherent states. As discussed in Introduction, the similarity between the CQO and the standard quantum harmonic oscillator suggests that the CCSs should be defined in the same way, that is, as states that minimize a generalized (conformable) uncertainty inequality of conformable quantum mechanics. Also, by the correspondence principle, we require that the coherent states of the standard quantum harmonic oscillator be reproduced at $\alpha =1$. In order to define the CCSs, let us first settle on some essential definitions and mathematical results.

4.1. Conformable Uncertainty

An important mathematical result is the Cauchy–Bunyakovsky–Schwarz (CBS) inequality for the functions belonging to ${\mathcal{L}}_{\alpha }^2\left(\mathbb{R}\right)$. On general grounds, one can argue that the CBS inequality holds in the conformable case since the inner product ${\langle ·|·\rangle }_{\alpha }$ has the same properties as the standard inner product in ${\mathcal{L}}^2\left(\mathbb{R}\right)$. Then, to prove the CBS inequality for ${\mathcal{L}}_{\alpha }^2\left(\mathbb{R}\right)$ we just repeat the same steps as in the standard proof, which uses only the general properties of the inner product and its associated norm [32]. The CBS inequality follows

$$\mathit{\int }|f\left(x\right)g\left(x\right)|d{\mu }_{\alpha }\left(x\right)\le {\parallel f\parallel }_{\alpha }{\parallel g\parallel }_{\alpha },$$

(23)

for any $f\left(x\right),g\left(x\right)\in {\mathcal{L}}_{\alpha }^2\left(\mathbb{R}\right)$. From it, we can also conclude that ${\mathcal{L}}_{\alpha }^2\left(\mathbb{R}\right)$ is closed under the norm

$$\begin{array}{c}\hfill {\int |f\left(x\right)+g\left(x\right)|}^{2}d{\mu }_{\alpha }\left(x\right)\le {(\parallel f\left(x\right)\parallel }_{\alpha }+{\parallel g\left(x\right)\parallel }_{\alpha }{)}^2.\end{array}$$

(24)

Although elementary, the detailed proofs are presented in the Appendices Appendix B and Appendix C.

Next, consider a state $\psi$ and two conformable observables $\widehat{A}$ and $\widehat{B}$. The expectation values of $\widehat{A}$ and $\widehat{B}$ in the state $\psi$ are defined according to Equation (18) by

$${\langle \widehat{A}\rangle }_{\alpha }=\int {\psi }^{*}\left(x\right)\widehat{A}\psi \left(x\right)d{\mu }_{\alpha }\left(x\right),{\langle \widehat{B}\rangle }_{\alpha }=\int {\psi }^{*}\left(x\right)\widehat{B}\psi \left(x\right)d{\mu }_{\alpha }\left(x\right).$$

(25)

Thus, the second power of the variances of the operators $\widehat{A}$ and $\widehat{B}$ is given by the following expectation values:

$${\sigma }_{\alpha A}^2={〈{\left(\widehat{A}-{\langle \widehat{A}\rangle }_{\alpha }\right)}^2〉}_{\alpha },{\sigma }_{\alpha B}^2={〈{\left(\widehat{B}-{\langle \widehat{B}\rangle }_{\alpha }\right)}^2〉}_{\alpha }.$$

(26)

The detailed proof is given in Appendix D.

With these definitions and properties in place, one can easily derive, by following the same steps as in standard quantum mechanics [32], that the conformable generalization of the uncertainty principle takes the familiar form

$${\sigma }_{\alpha A}^2{\sigma }_{\alpha B}^2\ge {\left({\displaystyle \frac{1}{2i}}{\langle [\widehat{A},\widehat{B}]\rangle }_{\alpha }\right)}^2.$$

(27)

In particular, if $\widehat{A}={\widehat{X}}_{\alpha }$ and $\widehat{B}={\widehat{p}}_{\alpha }$, using Equation (17), the conformable uncertainty relation can be reduced to

$${\sigma }_{\alpha X}{\sigma }_{\alpha p}\ge {\displaystyle \frac{{\hslash }_{\alpha }^{\alpha }}{2}}.$$

(28)

The standard uncertainty relations can be obtained from Equations (27) and (28) at $\alpha =1$. The detailed proof can be found in Appendix E.

4.2. Conformable Coherent States

Let us analyze the existence of the CCS of the CQO. From the above results, it follows that a CCS $\psi$ of a conformable quantum system can be defined as a state in which the uncertainty relation given by Equation (27) is minimized. We introduce the following notation:

$$\begin{array}{cc}\hfill & \widehat{A}={\langle \widehat{A}\rangle }_{\alpha }+{\widehat{a}}_{\alpha },\widehat{B}={\langle \widehat{B}\rangle }_{\alpha }+{\widehat{b}}_{\alpha }.\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill & {\psi }_a={\widehat{a}}_{\alpha }\psi ,{\psi }_b={\widehat{b}}_{\alpha }\psi .\hfill \end{array}$$

(30)

Then a CCS state $\psi$ that saturates the uncertainty inequality (27) must satisfy the system of equations

$$\begin{array}{cc}\hfill {\left|{〈{\psi }_a|{\psi }_b〉}_{\alpha }\right|}^2-{〈{\psi }_a|{\psi }_a〉}_{\alpha }{〈{\psi }_b|{\psi }_b〉}_{\alpha }& =0,\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill {〈\psi \left|\left(\widehat{a}\widehat{b}+\widehat{b}\widehat{a}\right)\right|\psi 〉}_{\alpha }& =0.\hfill \end{array}$$

(32)

The two above equations imply that

$${\psi }_b=\lambda {\psi }_a,\lambda +{\lambda }^{*}=0.$$

(33)

If we take now $\widehat{A}={\widehat{X}}_{\alpha }$ and $\widehat{B}={\widehat{p}}_{\alpha }$, the relations (33) imply that

$$\begin{array}{c}\hfill \left[i{\hslash }_{\alpha }^{\alpha }{D}_x^{\alpha }+{\pi }_{\alpha }+{\displaystyle \frac{i}{\alpha }}\gamma \left(x^{\alpha }-{\xi }_{\alpha }\right)\right]\psi \left(x\right)=0,\mathrm{if}x>0,\end{array}$$

(34)

$$\begin{array}{c}\hfill \left[i{\hslash }_{\alpha }^{\alpha }{D}_x^{\alpha }+{\pi }_{\alpha }+{\displaystyle \frac{i{(-1)}^{1+\alpha }}{\alpha }}\gamma \left(x^{\alpha }-{\xi }_{\alpha }\right)\right]\psi \left(x\right)=0,\mathrm{if}x<0.\end{array}$$

(35)

where ${\xi }^{\alpha }={\langle x^{\alpha }\rangle }_{\alpha }$, ${\pi }_{\alpha }={\langle {\widehat{p}}_{\alpha }\rangle }_{\alpha }$, and $\lambda =i\gamma$, with $\gamma \in \mathbb{R}$. By using the properties of the conformable derivative from Appendix A, we transform Equations (34) and (35) into the following system

$$\begin{array}{cc}\hfill x^{1-\alpha }{\displaystyle \frac{d\psi }{dx}}-{\displaystyle \frac{i}{{\hslash }_{\alpha }^{\alpha }}}{\pi }_{\alpha }\psi & =-{\displaystyle \frac{\gamma x^{\alpha }}{\alpha {\hslash }_{\alpha }^{\alpha }}}\psi +{\displaystyle \frac{\gamma {\xi }^{\alpha }}{\alpha {\hslash }_{\alpha }^{\alpha }}}\psi ,\mathrm{if}x>0,\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill {(-x)}^{1-\alpha }{\displaystyle \frac{d\psi }{dx}}-{\displaystyle \frac{i}{{\hslash }_{\alpha }^{\alpha }}}{\pi }_{\alpha }\psi & ={\displaystyle \frac{\gamma {(-x)}^{\alpha }}{\alpha {\hslash }_{\alpha }^{\alpha }}}\psi +{\displaystyle \frac{{(-1)}^{1+\alpha }\gamma {\xi }^{\alpha }}{\alpha {\hslash }_{\alpha }^{\alpha }}}\psi ,\mathrm{if}x<0.\hfill \end{array}$$

(37)

Integrating Equations (36) and (37), we obtain

$$\begin{array}{cc}\hfill {\psi }_1\left(x\right)& ={\psi }_1\left(0\right)exp\left(\left({\displaystyle \frac{i{\pi }_{\alpha }}{{\hslash }_{\alpha }^{\alpha }}}+{\displaystyle \frac{\gamma {\xi }^{\alpha }}{\alpha {\hslash }_{\alpha }^{\alpha }}}\right){\displaystyle \frac{x^{\alpha }}{\alpha }}-{\displaystyle \frac{\gamma x^{2\alpha }}{2{\alpha }^2{\hslash }_{\alpha }^{\alpha }}}\right),\mathrm{if}x>0,\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill {\psi }_2\left(x\right)& ={\psi }_2\left(0\right)exp\left(\left({\displaystyle \frac{i{\pi }_{\alpha }}{{\hslash }_{\alpha }^{\alpha }}}+{\displaystyle \frac{{(-1)}^{\alpha +1}\gamma {\xi }^{\alpha }}{\alpha {\hslash }_{\alpha }^{\alpha }}}\right){\displaystyle \frac{{(-1)}^{\alpha +1}{x}^{\alpha }}{\alpha }}-{\displaystyle \frac{\gamma {(-x)}^{2\alpha }}{2{\alpha }^2{\hslash }_{\alpha }^{\alpha }}}\right),\mathrm{if}x<0.\hfill \end{array}$$

(39)

The continuity of the CCS wavefunction at $x=0$ requires that

$${\left.{\psi }_1\left(x\right)\right|}_{x=0}={\left.{\psi }_2\left(x\right)\right|}_{x=0}.$$

(40)

From Equation (40), we have ${\psi }_1\left(0\right)={\psi }_2\left(0\right)={\psi }_0$, which implies that

$${\left.D_x^{\alpha }{\psi }_1\left(x\right)\right|}_{x=0}={\left.D_x^{\alpha }{\psi }_2\left(x\right)\right|}_{x=0},$$

(41)

if both the left and right limits of ${\xi }^{\alpha }={\langle x^{\alpha }\rangle }_{\alpha }$ are zero.

For normalizability, the coefficient of $x^{2\alpha }$ must be negative. By analogy with the standard harmonic oscillator, which is recovered in the $\alpha =1$ limit, and considering the Hamiltonian scaling, the coefficient $\gamma$ is determined from the equation

$${\displaystyle \frac{1}{2{\alpha }^2\gamma {\hslash }_{\alpha }^{\alpha }}}=-{\displaystyle \frac{m^{\alpha }{\omega }^{\alpha }}{2\alpha {\hslash }_{\alpha }^{\alpha }}},$$

(42)

In terms of the variable $X_{\alpha }$, the solutions given by Equations (38) and (39) take the form

$$\begin{array}{c}\hfill \psi \left(X_{\alpha }\right)=Aexp\left({\displaystyle \frac{i{\pi }_{\alpha }{X}_{\alpha }}{{\hslash }_{\alpha }^{\alpha }}}\right)exp\left(-{\displaystyle \frac{\gamma }{2{\hslash }_{\alpha }^{\alpha }}}{\left(X_{\alpha }-{\langle X_{\alpha }\rangle }_{\alpha }\right)}^2\right).\end{array}$$

(43)

The normalization constant A can be chosen to be a real and positive constant, by imposing the normalization relation

$${\int }_{-\infty }^{+\infty }\overline{\psi }\left(x\right)\psi \left(x\right){\left|x\right|}^{\alpha -1}dx=1.$$

(44)

Although the calculation of constant A is elementary, we give it here. We change variables to $z=X_{\alpha }$. For $x>0$, $z=x^{\alpha }/\alpha$, $dz=x^{\alpha -1}dx$, and ${\left|x\right|}^{\alpha -1}dx=dz$. For $x<0$, $z=-{(-x)}^{\alpha }/\alpha$, and ${\left|x\right|}^{\alpha -1}dx=-dz$ (with $z<0$). Substituting $\psi$ from Equation (43), the integral becomes

$${\int }_{-\infty }^{\infty }{\left|\psi \left(x\right)\right|}^2{\left|x\right|}^{\alpha -1}dx={\left|A\right|}^2{\int }_{-\infty }^{\infty }exp\left(-{\displaystyle \frac{m^{\alpha }{\omega }^{\alpha }\alpha }{{\hslash }_{\alpha }^{\alpha }}}{(z-{\langle X_{\alpha }\rangle }_{\alpha })}^2\right)dz.$$

(45)

This is a Gaussian integral

$${\int }_{-\infty }^{\infty }exp\left(-a{(z-{\langle X_{\alpha }\rangle }_{\alpha })}^2\right)dz=\sqrt{{\displaystyle \frac{\pi }{a}}},a={\displaystyle \frac{m^{\alpha }{\omega }^{\alpha }\alpha }{{\hslash }_{\alpha }^{\alpha }}}.$$

(46)

It follows that

$$\left|A\right|={\left({\displaystyle \frac{m^{\alpha }{\omega }^{\alpha }\alpha }{\pi {\hslash }_{\alpha }^{\alpha }}}\right)}^{1/4}.$$

(47)

Substituting (47) into (48), we obtain the final form of the CCS wavefunction

$$\begin{array}{c}\hfill \psi \left(X_{\alpha }\right)={\left({\displaystyle \frac{m^{\alpha }{\omega }^{\alpha }\alpha }{\pi {\hslash }_{\alpha }^{\alpha }}}\right)}^{1/4}exp\left({\displaystyle \frac{i{\pi }_{\alpha }{X}_{\alpha }}{{\hslash }_{\alpha }^{\alpha }}}\right)exp\left(-{\displaystyle \frac{\gamma }{2{\hslash }_{\alpha }^{\alpha }}}{\left(X_{\alpha }-{\langle X_{\alpha }\rangle }_{\alpha }\right)}^2\right).\end{array}$$

(48)

The CCS from Equation (48) is described by a Gaussian function and generalizes the coherent state of a quantum harmonic oscillator, which is recovered in the limit $\alpha =1$ as expected.

4.3. Energy Expectation Value in CCSs

It is important to determine the basic physical properties of CCSs. The first one is $\alpha$-energy given by the expectation value of the $\alpha$-power of energy $E^{\alpha }$ of the CCS that can be calculated using the following relation:

$$E^{\alpha }={\displaystyle \frac{\langle \psi |{\widehat{H}}_{\alpha }{|\psi \rangle }_{\alpha }}{{\langle \psi |\psi \rangle }_{\alpha }}}.$$

(49)

To this end, we rewrite the conformable Schrödinger Equation (15) in terms of $X_{\alpha }$ and its derivative as

$${\displaystyle \frac{d}{d{X}_{\alpha }}}\left({\displaystyle \frac{d\psi }{d{X}_{\alpha }}}\right)-{\displaystyle \frac{{\alpha }^2{m}^{2\alpha }{w}^{2\alpha }}{{\left({\hslash }_{\alpha }^{\alpha }\right)}^2}}{X}_{\alpha }^2\psi =E^{\alpha }\psi .$$

(50)

Next, we plug Equation (48) into Equation (50) and integrate, taking into account normalization Equation (44), which requires ${\langle \psi |\psi \rangle }_{\alpha }=1$. Then the energy is given by

$$E^{\alpha }={\langle {\widehat{H}}_{\alpha }\rangle }_{\alpha }={〈\psi \left|{\displaystyle \frac{{\widehat{p}}_{\alpha }^2}{2m^{\alpha }}}+{\displaystyle \frac{{\alpha }^2}{2}}{m}^{\alpha }{\omega }^{2\alpha }{\widehat{X}}_{\alpha }^2\right|\psi 〉}_{\alpha }.$$

(51)

Now, we compute the expectation values of ${\widehat{X}}_{\alpha }$ and ${\widehat{p}}_{\alpha }$. In the $z=X_{\alpha }$ representation, the probability density is Gaussian

$$\rho \left(z\right)={\left({\displaystyle \frac{m^{\alpha }{\omega }^{\alpha }\alpha }{\pi {\hslash }_{\alpha }^{\alpha }}}\right)}^{1/2}exp\left(-{\displaystyle \frac{m^{\alpha }{\omega }^{\alpha }\alpha }{{\hslash }_{\alpha }^{\alpha }}}{(z-{\langle X_{\alpha }\rangle }_{\alpha })}^2\right),$$

(52)

with mean ${\langle X_{\alpha }\rangle }_{\alpha }$ and variance ${\sigma }_z^2={\displaystyle \frac{{\hslash }_{\alpha }^{\alpha }}{2m^{\alpha }{\omega }^{\alpha }\alpha }}$. It follows that

$$\begin{array}{cc}\hfill \langle {\widehat{X}}_{\alpha }\rangle & ={\int }_{-\infty }^{\infty }z\rho \left(z\right)dz={\langle X_{\alpha }\rangle }_{\alpha },\hfill \end{array}$$

(53)

$$\begin{array}{cc}\hfill \langle {\widehat{X}}_{\alpha }^2\rangle & ={\sigma }_z^2+{\langle {\widehat{X}}_{\alpha }\rangle }^2={\displaystyle \frac{{\hslash }_{\alpha }^{\alpha }}{2m^{\alpha }{\omega }^{\alpha }\alpha }}+{\langle X_{\alpha }\rangle }_{\alpha }^2.\hfill \end{array}$$

(54)

For ${\widehat{p}}_{\alpha }$, we use the minimum uncertainty condition and the commutation relation. The variances satisfy

$$\Delta {\widehat{X}}_{\alpha }\Delta {\widehat{p}}_{\alpha }={\displaystyle \frac{{\hslash }_{\alpha }^{\alpha }}{2}},\Delta {\widehat{X}}_{\alpha }^2={\displaystyle \frac{\langle {({\widehat{X}}_{\alpha }-{\langle X_{\alpha }\rangle }_{\alpha })}^2\rangle }{{\langle \psi |\psi \rangle }_{\alpha }}}={\sigma }_z^2={\displaystyle \frac{{\hslash }_{\alpha }^{\alpha }}{2m^{\alpha }{\omega }^{\alpha }\alpha }}.$$

(55)

Then

$$\Delta {\widehat{p}}_{\alpha }^2={\displaystyle \frac{{\left({\hslash }_{\alpha }^{\alpha }\right)}^2}{4\Delta {\widehat{X}}_{\alpha }^2}}={\displaystyle \frac{{\hslash }_{\alpha }^{\alpha }{m}^{\alpha }{\omega }^{\alpha }\alpha }{2}}.$$

(56)

Since $\langle {\widehat{p}}_{\alpha }\rangle ={\pi }_{\alpha }$ (by symmetry and the form of $\psi$), it follows that

$$\langle {\widehat{p}}_{\alpha }^2\rangle =\Delta {\widehat{p}}_{\alpha }^2+{\langle {\widehat{p}}_{\alpha }\rangle }^2={\displaystyle \frac{{\hslash }_{\alpha }^{\alpha }{m}^{\alpha }{\omega }^{\alpha }\alpha }{2}}+{\pi }_{\alpha }^2.$$

(57)

By substituting the above relations into Equation (51), we obtain

$$E^{\alpha }={\displaystyle \frac{1}{2m^{\alpha }}}\langle {\widehat{p}}_{\alpha }^2\rangle +{\displaystyle \frac{{\alpha }^2}{2}}{m}^{\alpha }{\omega }^{2\alpha }\langle {\widehat{X}}_{\alpha }^2\rangle ={\displaystyle \frac{\alpha }{2}}{\hslash }_{\alpha }^{\alpha }{\omega }^{\alpha }+{\displaystyle \frac{{\pi }_{\alpha }^2}{2m^{\alpha }}}+{\displaystyle \frac{{\alpha }^2}{2}}{m}^{\alpha }{\omega }^{2\alpha }{\langle X_{\alpha }\rangle }_{\alpha }^2.$$

(58)

Equation (58) represents the CCS $\alpha$-energy, which reduces to the standard energy in the $\alpha =1$ limit.

4.4. Time Evolution and Spatial Translation of CCS

A second important property is the time and space translations of CCSs. To describe the time evolution, we consider a CCS wavefunction $\Psi$ that satisfies the time-dependent conformable Schrödinger equation

$$i{\hslash }_{\alpha }^{\alpha }{D}_t^{\alpha }\Psi (X_{\alpha },t)={\widehat{H}}_{\alpha }\Psi (X_{\alpha },t).$$

(59)

For separable wavefunctions, $\Psi (X_{\alpha },t)=\psi \left(X_{\alpha }\right)T_{\alpha }\left(t\right)$, we can determine the $T\left(t\right)$ component employing the properties of $D_t^{\alpha }$, which are the same as those of $D_x^{\alpha }$. From Equation (59), we obtain the following differential equation:

$$i{\hslash }_{\alpha }^{\alpha }{\displaystyle \frac{dT}{dt}}=E^{\alpha }{t}^{\alpha -1}T.$$

(60)

It follows that

$$\Psi \left(X_{\alpha },t\right)=T\left(0\right)exp\left(-i{\displaystyle \frac{E^{\alpha }{t}^{\alpha }}{\alpha {\hslash }_{\alpha }^{\alpha }}}\right)\psi \left(X_{\alpha }\right).$$

(61)

The above Equation (61) could have been obtained directly from Equation (12) for separable wavefunctions. Substituting the CCS wavefunction from Equation (48) into the above equation, we obtain

$$\Psi (t,X_{\alpha })={\left({\displaystyle \frac{m^{\alpha }{\omega }^{\alpha }\alpha }{\pi {\hslash }_{\alpha }^{\alpha }}}\right)}^{1/4}exp\left({\displaystyle \frac{-i{E}^{\alpha }{t}^{\alpha }}{\alpha {\hslash }_{\alpha }^{\alpha }}}\right)exp\left({\displaystyle \frac{i{\pi }_{\alpha }{X}_{\alpha }}{{\hslash }_{\alpha }^{\alpha }}}\right)exp\left(-{\displaystyle \frac{\gamma }{2{\hslash }_{\alpha }^{\alpha }}}{\left(X_{\alpha }-{\langle X_{\alpha }\rangle }_{\alpha }\right)}^2\right),$$

(62)

for all $t>0$. The above equation describes the time-dependent CCS wavefunction.

To describe the spatial displacement, we note that the infinitesimal spatial interval is ${ϵ\left|x\right|}^{1-\alpha }$ rather than $ϵ$. We fix $x>0$ for definiteness. Then, one can show that in terms of $x^{\alpha }$, we have

$$\psi (x^{\alpha }-{\xi }^{\alpha })=\sum _n^{\infty }{\displaystyle \frac{{(-{\xi }^{\alpha })}^n}{{\alpha }^{n}n!}}{\left(D_x^{\alpha }\right)}^n\psi \left(x^{\alpha }\right),$$

(63)

where ${\left(D_x^{\alpha }\right)}^n=D_x^{\alpha }\stackrel{n}{\dots }{D}_x^{\alpha }$. However, it is more convenient to use the variable $X_{\alpha }$ in terms of which Equation (63) takes the known form

$$\psi \left(X_{\alpha }\right)=\sum _n^{\infty }{\displaystyle \frac{{(-{\langle X_{\alpha }\rangle }_{\alpha })}^n}{n!}}{\displaystyle \frac{d^n\psi \left(X_{\alpha }\right)}{d{X}_{\alpha }^n}}.$$

(64)

From Equation (64), we conclude that the translation operator

$${\mathcal{D}}_{\alpha }\left({\langle X_{\alpha }\rangle }_{\alpha }\right)=exp\left(-{\langle X_{\alpha }\rangle }_{\alpha }{\displaystyle \frac{d}{d{X}_{\alpha }}}\right),$$

(65)

shifts $\psi \left(X_{\alpha }\right)$ by ${\langle X_{\alpha }\rangle }_{\alpha }$. Observe that, while the mathematical equations have the familiar form from the standard quantum mechanics, the geometrical interpretation is completely different, as the system obeys the conformable dynamics. However, the known relations characterizing the standard coherent states are obtained in the limit $\alpha =1$.

4.5. Commentaries on Mathematical and Physical Properties of CCSs

As viewed in the previous subsections, CCSs generalize the standard quantum harmonic oscillator coherent states using conformable derivatives and a modified inner product structure in the Hilbert space. These states are parameterized by the fractional parameter $\alpha \in (0,1]$, with $\alpha =1$ recovering standard quantum mechanics. To resume, the key results are the CCS wavefunction

$${\psi }_{\alpha }\left(x\right)={\left({\displaystyle \frac{m^{\alpha }{\omega }^{\alpha }\alpha }{\pi {\hslash }_{\alpha }^{\alpha }}}\right)}^{1/4}exp\left(-{\displaystyle \frac{m^{\alpha }{\omega }^{\alpha }\alpha }{2{\hslash }_{\alpha }^{\alpha }}}{(X_{\alpha }-{\langle X_{\alpha }\rangle }_{\alpha })}^2+i{\displaystyle \frac{{\pi }_{\alpha }}{{\hslash }_{\alpha }^{\alpha }}}{X}_{\alpha }\right),$$

(66)

where $X_{\alpha }={\displaystyle \frac{x^{\alpha }}{\alpha }}$ for $x>0$ and $X_{\alpha }=-{\displaystyle \frac{{(-x)}^{\alpha }}{\alpha }}$ for $x<0$, and the CCS energy

$$E^{\alpha }={\displaystyle \frac{\alpha }{2}}{\hslash }_{\alpha }^{\alpha }{\omega }^{\alpha }+{\displaystyle \frac{{\pi }_{\alpha }^2}{2m^{\alpha }}}+{\displaystyle \frac{{\alpha }^2{m}^{\alpha }{\omega }^{2\alpha }}{2}}{\langle X_{\alpha }\rangle }_{\alpha }^2.$$

(67)

We examine their mathematical structure, physical properties, and compare them to the standard coherent states ($\alpha =1$). The first property to be noted is that the CCS wavefunction exhibits a Gaussian structure in the conformable coordinate $X_{\alpha }$, but has nontrivial modifications: (i) the state depends on the conformable coordinate$X_{\alpha }\sim {\displaystyle \frac{x^{\alpha }}{\alpha }}$ rather than x, introducing nonlinear spatial distortion for $\alpha \ne 1$; (ii) the normalization procedure uses the $\alpha$-dependent inner product with the conformable measure $d{\mu }_{\alpha }\left(x\right)={\left|x\right|}^{\alpha -1}dx$, modifying the probability density to ${\rho }_{\alpha }\left(x\right)=|{\psi }_{\alpha }{\left(x\right)|}^2{\left|x\right|}^{\alpha -1}$, implying that the probability density is concentrated near $x=0$ for $\alpha <1$; (iii) the normalization constant $A_{\alpha }={\left({\displaystyle \frac{m^{\alpha }{\omega }^{\alpha }\alpha }{\pi {\hslash }_{\alpha }^{\alpha }}}\right)}^{1/4}$ reduces to ${\left({\displaystyle \frac{m\omega }{\pi \hslash }}\right)}^{1/4}$ at $\alpha =1$, matching standard coherent states in the correspondence principle limit $\alpha \to 1$. The CCS wavefunctions belong to ${\mathcal{L}}_{\alpha }^2\left(\mathbb{R}\right)$ with inner product

$${\langle \varphi |\psi \rangle }_{\alpha }=\int {\varphi }^{*}\left(x\right)\psi \left(x\right){\left|x\right|}^{\alpha -1}dx,$$

which also imply a modification of the orthogonality relations compared to standard quantum mechanics [8]. The overcompleteness is preserved but with respect to the conformable measure that satisfies $\int |\alpha \rangle \langle \alpha |d{\mu }_{\alpha }\left(\alpha \right)=\mathbb{I}$. The new mathematical structure implies physical properties of CCSs that differ from their standard counterparts. As discussed in the Introduction, the CCSs saturate the conformable position–momentum uncertainty $\Delta {\widehat{X}}_{\alpha }\Delta {\widehat{p}}_{\alpha }={\displaystyle \frac{{\hslash }_{\alpha }^{\alpha }}{2}}$, maintaining minimum uncertainty. As we have seen, the position variance $\Delta {\widehat{X}}_{\alpha }^2={\displaystyle \frac{{\hslash }_{\alpha }^{\alpha }}{2m^{\alpha }{\omega }^{\alpha }\alpha }}$ shows an explicit $\alpha$-dependence, reducing to the standard form ${\displaystyle \frac{\hslash }{2m\omega }}$ at $\alpha =1$. Also, the fractionality of CCS states determine a squeezing effect: for $\alpha <1$, the spatial probability distribution ${\left|\psi \left(x\right)\right|}^2{\left|x\right|}^{\alpha -1}$ exhibits asymmetric squeezing toward the origin. Under the conformable time-dependent Schrödinger equation

$$i{\hslash }_{\alpha }^{\alpha }{D}_t^{\alpha }\psi ={\widehat{H}}_{\alpha }\psi ,$$

the wavepacket maintains its Gaussian form in $X_{\alpha }$, analogous to standard coherent states. The energy spectrum $E^{\alpha }$ presents an expected scaling. The zero-point energy ${\displaystyle \frac{\alpha }{2}}{\hslash }_{\alpha }^{\alpha }{\omega }^{\alpha }$ scales differently with energy due to fractional parametrization, but again, the standard energy ${\displaystyle \frac{1}{2}}\hslash \omega$ is obtained in the limit $\alpha =1$. The kinetic term ${\displaystyle \frac{{\pi }_{\alpha }^2}{2m^{\alpha }}}$ describes a modified inertial response in fractional space. A similar property was observed in several fractional particle models at the classical level in [33,34,35]. The potential term ${\displaystyle \frac{{\alpha }^2}{2}}{m}^{\alpha }{\omega }^{2\alpha }{X}_0^2$ shows nonlinear coupling between $\alpha$ and the oscillator parameters.

It is useful to summarize comparatively the basic properties of the CCSs and standard coherent states in

Table 1. Comparison of coherent state properties.

Property	Standard ($\mathit{\alpha }=1$)	Conformable ($\mathit{\alpha }\ne 1$)
Coordinate	x	$X_{\alpha }={\displaystyle \frac{x^{\alpha }}{\alpha }},x>0$
		$X_{\alpha }={\displaystyle \frac{-{(-x)}^{\alpha }}{\alpha }}$, $x<0$
Measure	$dx$	${\left|x\right|}^{\alpha -1}dx$
Annihilation eigenstate	$\widehat{a}|\alpha \rangle =\alpha |\alpha \rangle$	${\widehat{a}}_{\alpha }{|\alpha \rangle }_{\alpha }=\alpha {|\alpha \rangle }_{\alpha }$
Uncertainty product	$\Delta x\Delta p=\hslash /2$	$\Delta X_{\alpha }\Delta p_{\alpha }={\hslash }_{\alpha }^{\alpha }/2$
Zero-point energy	${\displaystyle \frac{1}{2}}\hslash \omega$	${\displaystyle \frac{\alpha }{2}}{\hslash }_{\alpha }^{\alpha }{\omega }^{\alpha }$
Normalization	${\left({\displaystyle \frac{m\omega }{\pi \hslash }}\right)}^{1/4}$	${\left({\displaystyle \frac{m^{\alpha }{\omega }^{\alpha }\alpha }{\pi {\hslash }_{\alpha }^{\alpha }}}\right)}^{1/4}$

.

To conclude this section, we summarize the reduction of CCSs to standard coherent states in the limit $\alpha \to 1$:

• Coordinate: $X_{\alpha }\to x$ since ${lim}_{\alpha \to 1}{\displaystyle \frac{x^{\alpha }}{\alpha }}=x$.
• Measure: $d{\mu }_{\alpha }\left(x\right)\to dx$ since ${\left|x\right|}^{\alpha -1}\to 1$.
• Operators: ${\widehat{p}}_{\alpha }=-i{\hslash }_{\alpha }^{\alpha }{x}^{1-\alpha }{\displaystyle \frac{d}{dx}}\to -i\hslash {\displaystyle \frac{d}{dx}}$ and ${\hslash }_{\alpha }^{\alpha }\to \hslash$.
• Wavefunction:

  $${\psi }_{\alpha }\left(x\right)\to {\left({\displaystyle \frac{m\omega }{\pi \hslash }}\right)}^{1/4}exp\left(-{\displaystyle \frac{m\omega }{2\hslash }}{(x-x_0)}^2+i{\displaystyle \frac{p_{0}x}{\hslash }}\right)$$

  (68)
• Energy: $E^{\alpha }\to {\displaystyle \frac{1}{2}}\hslash \omega +{\displaystyle \frac{p_0^2}{2m}}+{\displaystyle \frac{1}{2}}m{\omega }^2{x}_0^2$.

Here, we have used the notations ${lim}_{\alpha \to 1}{\langle X_{\alpha }\rangle }_{\alpha }=x_0$ and ${\pi }_{\alpha \to 1}=p_0$.

4.6. CCSs as Eigenvectors of Annihilation Operator

Our approach to the CCSs of CQO was in the $x^{\alpha }$-representation, or in the analytic approach, since in this framework the conformable calculus can be explicitly applied. This raises the question of the algebraic approach to the formulation of the CCSs. Due to the similarities with standard coherent states, one can infer that one can describe CCSs in terms of conformable creation and annihilation operators, which were given in [8]. In order to make this claim more precise, let us consider these operators given by the following relations:

$${\widehat{a}}_{\alpha }={\displaystyle \frac{\left(\alpha m^{\alpha }{\omega }^{\alpha }{\widehat{X}}_{\alpha }+i{\widehat{p}}_{\alpha }\right)}{\sqrt{2m^{\alpha }\alpha {\hslash }_{\alpha }^{\alpha }{\omega }^{\alpha }}}},{\widehat{a}}_{\alpha }^{†}={\displaystyle \frac{\left(\alpha m^{\alpha }{\omega }^{\alpha }{\widehat{X}}_{\alpha }-i{\widehat{p}}_{\alpha }\right)}{\sqrt{2m^{\alpha }\alpha {\hslash }_{\alpha }^{\alpha }{\omega }^{\alpha }}}},$$

(69)

where ${\widehat{a}}_{\alpha }$ and ${\widehat{a}}_{\alpha }^{†}$ satisfy the standard commutation relations

$$\left[{\widehat{a}}_{\alpha },{\widehat{a}}_{\alpha }^{†}\right]=1.$$

(70)

From the eigenvalue and eigenstate equation of ${\widehat{a}}_{\alpha }$

$${\widehat{a}}_{\alpha }|{\sigma }_{\alpha }\rangle ={\sigma }_{\alpha }|{\sigma }_{\alpha }\rangle ,$$

(71)

one can conclude that the states $|{\sigma }_{\alpha }\rangle$ satisfy the CCS relations given in Equation (33), if

$${\sigma }_{\alpha }={\displaystyle \frac{\alpha m^{\alpha }{\omega }^{\alpha }{\langle {\widehat{X}}^{\alpha }\rangle }_{\alpha }+i{〈{\widehat{p}}_{\alpha }〉}_{\alpha }}{\sqrt{2\alpha m^{\alpha }{\hslash }_{\alpha }^{\alpha }{\omega }^{\alpha }}}}.$$

(72)

Again, even if Equation (72) has a strong similarity with the expression of the eigenvalue of the standard operator $\widehat{a}={\widehat{a}}_{\alpha =1}$, the interpretation is different since the eigenvalue ${\sigma }_{\alpha }$ does not, in general, label the position of the corresponding classical operator in the phase space. This result is another manifestation of the lack of a geometric interpretation of the conformable derivative and shows that, in general, $D_x^{\alpha }f\left(x\right)$ cannot automatically substitute $f^{\prime }\left(x\right)$ in conformable quantum mechanics.

5. Discussion

In this paper, we have presented several new results in conformable quantum mechanics, which is a deformation of standard quantum mechanics with the conformable parameter $\alpha$. In particular, we have established the standard mathematical properties of the Hilbert space of the CQO, which are necessary to generalize the uncertainty principle, and we have given this generalization. Also, we have obtained the CCSs of the CQO as states that minimize the uncertainty inequality, as in standard quantum mechanics, and have determined the oscillator $\alpha$-power of energy, the conformable time evolution, and the conformable translation operator. We do not claim any originality on the computations, most of which are performed along the same line as in standard quantum mechanics, which is recovered at $\alpha =1$. However, since the conformable derivative does not have a simple geometrical interpretation, one cannot claim that the results obtained here have the same physical interpretation as their counterparts from standard quantum mechanics. For example, it is not easy to construct a (classical and) quantum phase space since the conformable momentum is not tangent to the curves $x\left(t\right)$.

The CCSs obtained here generalize the standard coherent states through conformable calculus and a modified inner product in Hilbert space. In this paper, we have shown that CCSs have a Gaussian structure in conformable coordinates $X_{\alpha }$ with $\alpha$-dependent normalization, minimal uncertainty in conformable position and momentum operators, and explicit $\alpha$-scaling in zero-point energy and dynamics. The limit $\alpha =1$ recovers standard quantum mechanics. Thus, these states provide an interesting theoretical framework for exploring fractional quantum systems and their correspondence principles. The results obtained in this paper are preliminary and open new directions for research. An important problem is to establish the relationship between the CCSs that minimize the uncertainty and the classical dynamics of the conformable oscillator. This problem is non-trivial due to the challenges discussed above as well as to the complex dynamics of the conformable oscillator. Another important problem concerns the symmetry structure of the CCSs. The standard coherent states arise from the Heisenberg–Weyl group, while conformable states connect to deformed symmetry algebras of derivatives. Therefore, it is important to understand the relationship between the CCSs and the symmetry algebras. Also, a deeper analysis of the displacement operators and the overcomplete basis is in order.

While there is no physical system known to date that has conformable dynamics, it is certainly interesting from the mathematical physics point of view to explore further the structure of the conformable quantum mechanics and the properties of the CCSs of the CQO.