Comment on 'Two-dimensional position-dependent massive particles in the presence of magnetic fields"

TL;DR

This paper critiques and corrects the treatment of two-dimensional position-dependent mass particles in magnetic fields, specifically addressing inaccuracies in prior work and providing proper minimal coupling formulation.

Contribution

It identifies and corrects errors in previous analyses of PDM particles in magnetic fields, offering a proper treatment of minimal coupling in this context.

Findings

Corrected the treatment of minimal coupling for PDM in magnetic fields

Highlighted inaccuracies in prior solutions and provided proper formulations

Clarified the application of PDM von Roos Hamiltonian in magnetic environments

Abstract

Using the well known position-dependent mass (PDM) von Roos Hamiltonian, Dutra and Oliveira (2009 J. Phys. A: Math. Theor. 42 025304) have studied the problem of two-dimensional PDM particles in the presence of magnetic fields. They have reported exact solutions for the wavefunctions and energies. In the first part of their study "PDM-Shr\"odinger equation in two-dimensional Cartesian coordinates", they have used the so called Zhu and Kroemer's ordering {\alpha}=-1/2={\gamma} and \b{eta}=0 <cite>5</cite>. While their treatment for this part is correct beyond doubt, their treatment of second part "PDM in a magnetic field" is improper . We address these improper treatments and report the correct presentation for minimal coupling under PDM settings.

Equations

$$H=\frac{1}{2M\left(x,y\right)}{\left(\overrightarrow{p}-e\overrightarrow{A}\right)}^2+V\left(x,y\right)=\frac{1}{2M}{\overrightarrow{p}}^2-\frac{e}{2M}\left(\overrightarrow{A}\cdot \overrightarrow{p}+\overrightarrow{p}\cdot \overrightarrow{A}\right)+\frac{e^2}{2M}{\overrightarrow{A}}^2+V\left(x,y\right).$$

$$\overrightarrow{\tilde{A}}=\frac{\overrightarrow{A}\left(x,y\right)}{M\left(x,y\right)}=\frac{\overrightarrow{A}}{M},$$

$$H=\frac{1}{2M}{\overrightarrow{p}}^2-\frac{e}{2}\left(\overrightarrow{\tilde{A}}\cdot \overrightarrow{p}+\overrightarrow{p}\cdot \overrightarrow{\tilde{A}}\right)+\frac{e^{2}M}{2}{\overrightarrow{\tilde{A}}}^2+V\left(x,y\right).$$

$$\frac{1}{2M}{\overrightarrow{p}}^2+V\left(x,y\right)=-\frac{{\hslash }^2}{2M}{\overrightarrow{\nabla }}^2+\frac{{\hslash }^2}{2M^2}\left(\overrightarrow{\nabla }M\cdot \overrightarrow{\nabla }\right)+U\left(\alpha ,\gamma ,x,y\right)+V\left(x,y\right),$$

$$\frac{1}{2M}{\overrightarrow{p}}^2=-\frac{{\hslash }^2}{2M}{\overrightarrow{\nabla }}^2+\frac{{\hslash }^2}{2M^2}\left(\overrightarrow{\nabla }M\cdot \overrightarrow{\nabla }\right)+U\left(\alpha ,\gamma ,x,y\right)$$

$$\hat{O}=f\left(x\right)\hat{p}-\frac{i\hslash }{2}\left(\frac{df\left(x\right)}{dx}\right)⟹{\hat{O}}^2=\frac{1}{2M}{\hat{p}}^2⟹f\left(x\right)=\frac{1}{\sqrt{2M}}.$$

$$\hat{O}=\frac{\hat{p}}{\sqrt{2M}}=\frac{-i\hslash }{\sqrt{2M}}\left[\frac{d}{dx}-\frac{1}{4}\frac{\left(dM/dx\right)}{M}\right]⟹{\hat{p}}_x=-i\hslash \left[\frac{\partial }{\partial x}-\frac{1}{4}\frac{\left(\partial M/\partial x\right)}{M}\right]⟹{\overrightarrow{p}}_{op}=-i\hslash \left[\overrightarrow{\nabla }-\frac{\left(\overrightarrow{\nabla }M\right)}{4M}\right].$$

$$\hat{H}=\frac{1}{4}\left[M^{\alpha }\hat{p}{M}^{\beta }\hat{p}{M}^{\gamma }+M^{\gamma }\hat{p}{M}^{\beta }\hat{p}{M}^{\alpha }\right]+V\left(x,y\right),$$

$$\hat{H}=-\frac{1}{4}\left[M^{\alpha }{\partial }_{x_j}{M}^{\beta }{\partial }_{x_j}{M}^{\gamma }+M^{\gamma }{\partial }_{x_j}{M}^{\beta }{\partial }_{x_j}{M}^{\alpha }\right]+V\left(x,y\right),j=1,2$$

$$⟨p_{{}_x}⟩=m_{\circ }\frac{d}{dt}⟨x⟩=\underset{-\infty }{\overset{\infty }{\int }}dx{\Psi }^{\ast }\left(x,t\right)\left(-i\hslash \frac{\partial }{\partial x}\right)\Psi \left(x,t\right)⟹{\hat{p}}_x=-i\hslash \partial /\partial x.$$

$$L=\frac{1}{2}M\left(x,y\right){\dot{x}}_j^2-V\left(x,y\right)$$

$$P_j\left(x,y\right)=\partial L/\partial {\dot{x}}_j=M\left(x,y\right){\dot{x}}_j,$$

$$⟨P_j\left(x,y\right)⟩=⟨M\left(x,y\right){\dot{x}}_j⟩\ne M\left(x,y\right)\frac{d}{dt}⟨x_j⟩⟹{\hat{P}}_j\left(x,y\right)\ne -i\hslash \partial /\partial x_j.$$

$$L\left(x_j,{\dot{x}}_j,t\right)=\frac{1}{2}M\left(x,y\right){\dot{x}}_j^2+e{\dot{x}}_j{A}_j\left(x,y\right)-V\left(x,y\right),$$

$$P_j\left(x,y\right)=\partial L/\partial {\dot{x}}_j=M\left(x,y\right){\dot{x}}_j+e{A}_j\left(x,y\right)=\sqrt{M\left(x,y\right)}{\Pi }_j\left(x,y\right)+e{A}_j\left(x,y\right),$$

$${\Pi }_j\left(x,y\right)=\frac{P_j\left(x,y\right)-e{A}_j\left(x,y\right)}{\sqrt{M\left(x,y\right)}}.$$

$$H\left(x_j,{\dot{x}}_j,t\right)=\frac{1}{2}M\left(x,y\right){\dot{x}}_j^2+V\left(x,y\right)=\frac{1}{2}{\Pi }_j{\left(x,y\right)}^2+V\left(x,y\right)=\frac{1}{2}{\left(\frac{P_j\left(x,y\right)-e{A}_j\left(x,y\right)}{\sqrt{M\left(x,y\right)}}\right)}^2+V\left(x,y\right).$$

$${\hat{P}}_j\left(x,y\right)=\sqrt{M\left(x,y\right)}{\hat{\Pi }}_j\left(x,y\right)=-i\hslash \left[{\partial }_{x_j}-\frac{1}{4}\left(\frac{{\partial }_{x_j}M\left(x,y\right)}{M\left(x,y\right)}\right)\right].$$

$${\hat{P}}_j\left(x,y\right)⟶{\hat{P}}_j\left(x,y\right)-e{A}_j\left(x,y\right)⟹{\left(\frac{{\hat{P}}_j\left(x,y\right)}{\sqrt{M\left(x,y\right)}}\right)}^2⟶{\left(\frac{{\hat{P}}_j\left(x,y\right)-e{A}_j\left(x,y\right)}{\sqrt{M\left(x,y\right)}}\right)}^2$$

$$\hat{H}=\frac{1}{2}{\left(\frac{{\hat{P}}_j\left(x,y\right)-e{A}_j\left(x,y\right)}{\sqrt{M\left(x,y\right)}}\right)}^2+V\left(x,y\right).$$

$$\hat{H}$$

Full text

**PACS numbers: 03.65.Ge. **, 03.65.-w

Keywords: position-dependent mass, magnetic field, Schrödinger equation.

Over the last few decades, both classical and quantum mechanical particles endowed with position-dependent mass (PDM) have attracted much research attention (c.f., e.g., 1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7 ; 8 and references cited therein). The study of a classical PDM-nonlinear oscillator by Mathews and Lakshmanan 8 has sparked and inspired a large number of research in different fields of study. It was only recently, to the best of our knowledge, that the effect of a uniform magnetic field on a PDM quantum particle in two dimensions is studied by Dutra and Oliveira 1 . They have reported exact wavefunctions and energies for such a problem. Whilst the first part of their ”PDM-Shrödinger equation in two-dimensional Cartesian coordinates” follows Zhu and Kroemer’s ordering $\alpha=-1/2=\gamma$ and $\beta=0$ 5 (their Eq. (24)) and is correct beyond doubt, their second part ”PDM in a magnetic field” suffers improper and conflicting treatments that inspired the current obligatory comments.

In their attempt to analyze and discuss the quantum mechanical effect of a uniform magnetic field on a charged particle endowed with position-dependent mass in two dimensions, Dutra and Oliveira 1 have started (in their section 3) with the classical Hamiltonian

[TABLE]

Where $e$ is the electric charge, $M\equiv M\left(x,y\right)$ is the PDM, $\overrightarrow{A}\equiv\overrightarrow{A}\left(x,y\right)$ is the vector potential, and $V\left(x,y\right)$ is a scalar potential (their equation (37)). Next, with the substitution ((38) in 1 )

[TABLE]

they have obtained the classical Hamiltonian

[TABLE]

At this point, they have suggested that the ordering of the kinetic energy operator is discussed in their section 2 and hence the terms without magnetic interaction (i.e., $\vec{A}\left(x,y\right)=0$) in (3) are given by their Eq.(9) as

[TABLE]

where $U\left(\alpha,\gamma,x,y\right)$ is given by their Eq.(8). But then, to deal with the operator linear in $\overrightarrow{p}$ (i.e., $\overrightarrow{\tilde{A}}\cdot\overrightarrow{p}+\overrightarrow{p}\cdot\overrightarrow{\tilde{A}}$, the 2nd term in (3) or their 2nd term of (39)) they have used $\overrightarrow{p}$ $=-i\hbar\overrightarrow{\nabla}$ to obtain their Eq.(50). Next, they used $\psi\left(x,y\right)=M^{1/2}\chi\left(x,y\right)$ in (50) to get their Eq.(51) and proceeded with their solution considering the well known Zhu and Kroemer’s ordering $\alpha=-1/2=\gamma$ and $\beta=0$ 5 (to get rid of the differential forms of the PDM terms).

In the light of our experience and practical contact with this paper 1 , we feel obligated to pin point our observations that are in order:

1- In handling (3), they have used two conflicting/inconsistent definitions for the momentum operator. Having used

[TABLE]

in (4) (their (39) to obtain (50)) necessarily suggests that $\overrightarrow{p}$ $\neq-i\hbar\overrightarrow{\nabla\text{ }}$ (as documented in their Eq.s (43) to (46)). One should notice that $\overrightarrow{p}$ in (1) is the PDM canonical momentum and should correspond to PDM-momentum operator in (4). In fact, by a factorization recipe 3 , equation (5) immediately implies that the PDM-momentum operator should look like that in their equation (42) as

[TABLE]

Which will, consequently, strictly determine the ordering parameters as $\alpha=\gamma=-1/4$ and $\beta=-1/2$ (known as MM-ordering 3 ). Moreover, it suggests that

[TABLE]

Which, in turn, collapses into $\overrightarrow{p}_{op}=-i\hbar\overrightarrow{\nabla}$ for constant mass settings as it should.

2- The von Roos operator they have used (their Eq.(1)) as

[TABLE]

caused all the confusion, in our opinion. In the von Roos 2 proposal, this Hamiltonian operator is given in a differential form as

[TABLE]

where the summation runs over the repeated index, $\partial_{x_{j}}=\partial/\partial x_{j}$ and the ordering parameters satisfy the von Roos constraint $\alpha+\beta+\gamma=-1$ (to recover the constant mass settings when $M\left(x,y\right)=m_{\circ}$). However, one should be aware that the canonical PDM-momentum $\overrightarrow{p}$ in (1) would lead to a PDM-momentum operator, as discussed in point 1 above.. To make this point more clear, perhaps it is wise to go back to the very fundamentals of Quantum mechanics by S. Gasiorowicz 9 and recollect that, the canonical momentum for a constant mass is $p_{{}_{x}}=m_{\circ}\dot{x}=$ $m_{\circ}dx/dt$ and the corresponding quantum mechanical momentum operator is determined through

[TABLE]

Whereas, the canonical PDM-momentum for a classical PDM-Lagrangian

[TABLE]

is given by

[TABLE]

and hence the PDM-momentum operator would be found through the same recipe as

[TABLE]

3- As long as classical mechanics is concerned, the presentation of the kinetic energy term in (1) is correct. However, when quantum mechanics is in point, this presentation is improper. To find the proper presentation one would recollect that the PDM Lagrangian with magnetic interaction is given by

[TABLE]

to imply the PDM canonical momentum

[TABLE]

where $\Pi_{j}\left(x,y\right)=\sqrt{M\left(x,y\right)}\dot{x}_{j}$ is the $j$the component of a pseudo-mechanical-momentum (which is shown to be a conserved quantity for a quasi-free PDM case 4 , i.e., for $V\left(x,y\right)=0=A_{j}\left(x,y\right)$). This would consequently imply that

[TABLE]

Therefore, the corresponding classical PDM-Hamiltonian reads

[TABLE]

Of course, it looks the same as that in (1) for a classical particle but this presentation is the one to be used for a PDM quantum particle in a magnetic interaction, with $\widehat{P}_{j}\left(x,y\right)$ as the $j$the component of the PDM-momentum operator (7) and takes the differential form 6

[TABLE]

Under such PDM-settings, the minimal coupling for the PDM-Schrödinger Hamiltonian should look like

[TABLE]

Hence, their equation (37) should correspond to a PDM-Schrödinger Hamiltonian

[TABLE]

Quantum mechanically speaking, this Hamiltonian is not the same as the one they have used in (1).

4- In connection with the above mentioned points, their Eq.(39) should look exactly like

[TABLE]

for the quantum mechanical treatment. This is not the same as the Hamiltonian operator they have used in (3) (their (39)).

5- Yet, in-between their (41) and (42) they have used a constraint $\alpha+\beta=1$ to obtain (42). This would, in turn, imply that $\gamma=-2$ (using the von Roos constraint $\alpha+\beta+\gamma=-1)$. However, they have used $\alpha=\gamma=-1/2$ and $\beta=0$ (i.e. Zhu and Kroemer ordering 5 ) to obtain (54). Using two parametric orderings at the same time is confusing and improper. Two different parametric orderings necessarily mean two different kinetic energy operator presentations, as obvious from (9) above.

Finally, we believe that it is necessary and vital to put PDM quantum particles in magnetic field on the most proper track. This will open a new window for PDM research community to make the relevant theoretical progress not only for PDM in magnetic fields but also for PDM in electromagnetic and laser fields.