A simple approach for extending up to the ultra-relativistic limit the theory of a non-relativistic Fermi gas

Luis Grave de Peralta

doi:10.12688/f1000research.123275.1

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Research Article

A simple approach for extending up to the ultra-relativistic limit the theory of a non-relativistic Fermi gas

[version 1; peer review: 3 not approved]

Luis Grave de Peralta ^1,2

PUBLISHED 08 Jul 2022

Author details Author details

¹ Department of Physics and Astronomy, Texas Tech University, Lubbock, TX, 79410, USA
² Nano Tech Center, Texas Tech University, Lubbock, TX, 79410, USA

Luis Grave de Peralta
Roles: Conceptualization, Formal Analysis, Investigation, Writing – Original Draft Preparation, Writing – Review & Editing

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Abstract

Background: The theory of a Fermi gas constitutes a useful physical model with multiple applications from solid state to stellar physics. The theory of a non-relativistic Fermi gas is relatively simple when compared with the one for a relativistic Fermi gas. Therefore, it would be useful if a rigorous theory of a relativistic Fermi gas could be constructed like the theory of a non-relativistic Fermi gas.

Methods: Such a theory, for T > 0 K, is presented here. It is based on using a recently reported formula for the energies of a relativistic spin-0 particle in a three-dimensional infinite well. The Pauli exclusion principle is used for taking care of the fermion nature of the particles forming a Fermi gas.

Results: A unified description of a Fermi gas at low temperatures is presented. This description is valid from the non-relativistic to the ultra-relativistic limits. Moreover, it is simple. The theory follows the relatively simple approach often used for constructing the theory of a non-relativistic Fermi gas.

Conclusions: The ultra-relativistic results reported here are often obtained using a different approach than the one used in this work. These results are not new, but they corroborate the correctness of the presented approach. In addition, the relative simplicity of the approach presented in this work allows for teaching the theory of a relativistic Fermi gas in introductory quantum mechanics courses.

Keywords

Quantum Mechanics, Relativistic Quantum Mechanics, Fermi Gas

Corresponding author: Luis Grave de Peralta

Competing interests: No competing interests were disclosed.

Grant information: The author(s) declared that no grants were involved in supporting this work.

Copyright: © 2022 Grave de Peralta L. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

How to cite: Grave de Peralta L. A simple approach for extending up to the ultra-relativistic limit the theory of a non-relativistic Fermi gas [version 1; peer review: 3 not approved]. F1000Research 2022, 11:761 (https://doi.org/10.12688/f1000research.123275.1) First published: 08 Jul 2022, 11:761 (https://doi.org/10.12688/f1000research.123275.1) Latest published: 08 Jul 2022, 11:761 (https://doi.org/10.12688/f1000research.123275.1)

I. Introduction

The theory of a Fermi gas constitutes a useful model with multiple applications from solid state to stellar physics.¹^–³ The theory of a non-relativistic Fermi gas is relatively simple when compared with the one for a relativistic Fermi gas.¹^–⁴ Therefore, it would be useful if a rigorous theory of a relativistic Fermi gas could be constructed like the theory of a non-relativistic Fermi gas.¹^,² Building on recently published results for a relativistic Fermi gas at T = 0 K,⁵ in this work we present such a theory for T > 0 K for the first time.

The theory of a non-relativistic Fermi gas is built from a well-known analytical expression, which gives the energies of a non-relativistic spin-0 particle in a three-dimensional infinite well (particle in a box).¹^,² The Pauli exclusion principle is used, in the non-relativistic theory of a Fermi gas at 0 K, for taking care of the fermion nature of the particles forming a Fermi gas.¹^,² However, this relatively simple theory cannot be directly extended to the relativistic domain.³^,⁴ This is because no such analytical expression was known, until recently, for the energies of a relativistic spin-0 particle in a box.⁵^–⁹ In what follows, first, in Section II is presented a discussion about the energy of a relativistic spin-0 particle in a box when the particle is in a quantum state with positive kinetic energy. In Section III, it is summarized how the energy formula presented in Section II was used for constructing the theory of a relativistic Fermi gas at T = 0 K.⁵ In Section IV, it is shown that it is possible to construct a theory of a relativistic Fermi gas at T > 0 K but using the same approach often used for constructing the theory of a non-relativistic one. We are then presenting a simple approach for constructing a theory of a Fermi gas valid from the non-relativistic to the ultra-relativistic limits. Finally, the conclusions of this work are given in Section V.

II. Positive energies of a spin-0 particle in a box

A relativistic spin-0 particle with mass m, which is completely confined in a cubic box of size L, only has kinetic energy. The possible positive values of the particle’s energy can be found by solving a special case of the spinless Salpeter equation¹⁰^,¹¹:

(1)

i ℏ \frac{\partial}{\partial t} ψ = \hat{γ} m c^{2} ψ, \hat{γ} = \sqrt{1 + \frac{{\hat{p}}^{2}}{m^{2} c^{2}}}, \hat{p} = - i ℏ \nabla .

In Eq. (1), c is the speed of the light in vacuum and the wavefunction (ψ) must satisfy null boundary conditions. As it will be shown below, after discounting the energy of the particle associated with its mass (mc²), Eq. (1) can be rewritten as the Poirier-Grave de Peralta equation⁶^,¹²:

(2)

i ℏ \frac{\partial}{\partial t} ψ = \frac{{\hat{p}}^{2}}{(\hat{γ} + 1) m} ψ .

When a particle in a box is in a stationary state, it has a constant kinetic energy value, thus a constant value of the square of its linear momentum (p²). Therefore, the solutions of Eq. (2) are exactly equal to the solution of the Grave de Peralta (GdeP) equation⁶^–⁹^,¹²:

(3)

i ℏ \frac{\partial}{\partial t} ψ = - \frac{ℏ^{2}}{(γ + 1) m} \nabla^{2} ψ, γ = \sqrt{1 + \frac{p^{2}}{m^{2} c^{2}}} .

Equation (3) differs from Eq. (2) in that γ is not an operator but a parameter equal to the classical value of the Lorentz factor of special theory of relativity.¹³ Clearly, Eq. (3) reduces to the Schrödinger equation when γ = 1.¹ The infinite-well potential does not interact with the spin degree of freedom; therefore, the energies of a spin-0 and spin-1/2 particle in a cubic box are the same and equal to⁵^,⁶^,¹⁴:

(4)

E_{|n|} = \frac{π^{2} ℏ^{2}}{(γ + 1) m L^{2}} {|n|}^{2} = (γ - 1) m c^{2}, n = (n_{x}, n_{y}, n_{z}), n_{x, y, z} = 1, 2, \dots

In Eq. (4), $ℏ$ is the reduced Plank constant,¹ and γ is given by the following equation⁵^–⁶:

(5)

γ = \sqrt{1 + {|n|}^{2} π^{2} {(\frac{ƛ_{C}}{L})}^{2}}, ƛ_{C} = \frac{ℏ}{mc} .

In the non-relativistic limit, the reduced Compton wavelength λ/2π << L, thus γ = 1 and Eq. (4) reduces to the well-known formula¹^,²:

(6)

E_{|n|} = \frac{π^{2} ℏ^{2}}{2 m L^{2}} {|n|}^{2} .

The theory of a non-relativistic Fermi gas, formed by N non-interacting fermions with spin-½, is based on using Eq. (6). The fermion nature of the particles is included by considering the Pauli exclusion principle.¹^,² The correct relativistic formula, Eq. (4), was only recently reported.⁵^–⁹ Consequently, it can be now used for constructing a theory of a relativistic Fermi gas like the well-known one for a non-relativistic Fermi gas.¹^,² This project was started recently.⁵

It could be helpful noting that Eq. (1) can be obtained by applying the formal first-quantization substitution¹^,³:

(7)

E^{'} \to \hat{H} = i ℏ \frac{\partial}{\partial t}, p \to \hat{p} = - i ℏ \nabla .

in the well-known formula of the energy of a classical free particle with mass m which is moving at relativistic speeds with kinetic energy (K), linear momentum p, and total energy E′¹⁵:

(8)

E^{'} = K + m c^{2} = γm c^{2} .

In Eq. (8), γ is the classical Lorentz factor given by Eq. (3). Alternatively, we could start from the following equation, which is equivalent to Eq. (8)⁶:

(9)

E^{'} = \frac{p^{2}}{(γ + 1) m} + m c^{2} .

Applying the formal first-quantization substitution to Eq. (9), we obtain⁶:

(10)

i ℏ \frac{\partial}{\partial t} Ω = - \frac{ℏ^{2}}{(\hat{γ} + 1) m} \nabla^{2} Ω + m c^{2} Ω .

In Eq. (10), γ is the operator given by Eq. (1). Now, it is well known in quantum mechanics that applying a constant energy shift to the Hamiltonian gives way to an immaterial time-evolving phase factor in the solution wavefunction. Therefore, Eq. (2) can be obtained from Eq. (10), after replacing Ω as follows⁶:

(11)

Ω (r, t) = Ψ (r, t) e^{- i \frac{m c^{2}}{ℏ} t} .

For a particle in a box, the solutions of Eq. (2) and (3) are identical. This is because a particle in a box has a constant kinetic energy value when it is in a stationary state; therefore, the particle must also have constant value of the square of its linear momentum (p²) and thus a constant value of γ. Consequently:

(12)

\hat{γ} Ψ = γΨ \Rightarrow \frac{{\hat{p}}^{2}}{(\hat{γ} + 1) m} ψ = \frac{{\hat{p}}^{2}}{(γ + 1) m} ψ = Kψ .

For instance, if a particle trapped in a one-dimensional box is in a stationary state, then γ is equal to⁶:

(13)

γ = ⟨Ψ| \hat{γ}| Ψ⟩ = \sqrt{1 + \frac{2}{m c^{2}} E_{Sch}} .

In Eq. (13), E_Sch is the well-known energy of the corresponding non-relativistic problem¹^,⁶:

(14)

E_{Sch} = \frac{p^{2}}{2 m} = \frac{ℏ^{2} k^{2}}{2 m}, k = \frac{πn}{L}, n = 1, 2, \dots

Note that for a particle in a box the possible values of k are determined by the boundary conditions; thus, k is the same for a non-relativistic and a relativistic particle in a box.⁶ Consequently, for a particle in a one-dimensional box⁶:

(15)

γ = \sqrt{1 + n^{2} π^{2} {(\frac{ƛ_{C}}{L})}^{2}}, ƛ_{C} = \frac{ℏ}{mc} .

And:

(16)

E_{n} = \frac{π^{2} ℏ^{2}}{(γ + 1) m L^{2}} n^{2} = (γ - 1) m c^{2} .

Equations (4) and (5) are the tridimensional versions of Eqs. (16) and (15), respectively.

The above discussion is enough for justifying the validity of our approach, which is based on using Eqs. (3) to (5) for constructing a theory of a relativistic Fermi gas. Moreover, it is easy to show that the GdeP equation for any time-independent scalar potential (V):

(17)

i ℏ \frac{\partial}{\partial t} ψ = - \frac{ℏ^{2}}{(γ + 1) m} \nabla^{2} ψ + Vψ .

is equivalent to the Klein-Gordon equation when K > 0. In his case E = K + V, thus:

(18)

E = (γ - 1) m c^{2} + V .

Consequently, the classical value of the Lorentz factor (γ) is in general not constant:

(19)

γ = 1 + \frac{E - V}{m c^{2}} .

Using Eqs. (7) and (19), we can rewrite the time independent GdeP equation corresponding to Eq. (17) in the following way:

(20)

[\frac{{\hat{p}}^{2}}{2 m} - (E - V) - \frac{{(E - V)}^{2}}{2 m c^{2}}] ψ = 0 .

The above equation is an appealing Schrödinger-like form of the time-independent Klein-Gordon equation. Indeed, it is easy to show that by making E′ = E - mc², Eq. (20) reduces to the well-known expression for the corresponding stationary Klein-Gordon equation (15).

III. Relativistic Fermi gas at T = 0 K

It was recently obtained, using Eq. (4), that the Fermi energy (E_F) of a relativistic gas formed by N fermions occupying a volume V is given by the following equation (5):

(21)

E_{F} = \frac{ℏ^{2} {(3 π^{2})}^{2 / 3}}{(γ + 1) m} {(\frac{N}{V})}^{2 / 3}, γ = \sqrt{1 + {(3 π^{2})}^{2 / 3} {ƛ_{C}}^{2} {(\frac{N}{V})}^{2 / 3}} .

In the non-relativistic limit (γ ≈ 1) Eq. (21) gives the well-known formula¹^–²:

(22)

E_{F} = \frac{ℏ^{2} {(3 π^{2})}^{2 / 3}}{2 m} {(\frac{N}{V})}^{2 / 3} .

This is hardly a surprise because Eq. (21) and Eq. (22) were obtained using the same approach.²^,⁵ However, Eq. (21) also gives the correct result in the ultra-relativistic limit (γ >>1)³^,⁴:

(23)

E_{F} = {(3 π^{2})}^{1 / 3} ℏ c {(\frac{N}{V})}^{1 / 3} .

This is a well-known formula for a relativistic Fermi gas that was originally obtained following a different approach.³^,⁴ This suggests that a correct theory of a relativistic Fermi gas can be constructing using Eq. (4) and following the relatively simple approach often used for constructing a theory of a non-relativistic Fermi gas.¹^,² We will see over and over in this work that it is always the case. For instance, the total energy of a relativistic Fermi gas at 0 K can be obtained from Eq. (4) in the following way⁵:

(24)

E_{T} = \frac{ℏ^{2}}{m} {(\frac{3 π^{2}}{V})}^{2 / 3} \int_{0}^{N} \frac{N^{2 / 3}}{γ + 1} dN .

In the non-relativistic limit Eq. (24) gives the well-known formula²^,⁵:

(25)

E_{T} = \frac{3}{5} N E_{F} .

In Eq. (25) E_F is given by Eq. (22). As expected, Eq. (24) also gives the correct result in the ultra-relativistic limit³^–⁵:

(26)

E_{T} = \frac{3}{4} N E_{F} .

In Eq. (26) E_F is given by Eq. (23). Consequently, the degeneracy pressure of a non-relativistic (nr) and an ultra-relativistic (ur) Fermi gas at 0 K are¹^–⁵:

(27)

P = - \frac{\partial E_{T}}{\partial V} = \begin{matrix} \frac{2}{3} \frac{E_{T}}{V} \propto {(\frac{N}{V})}^{\frac{5}{3}}, nr \\ \frac{1}{3} \frac{E_{T}}{V} \propto {(\frac{N}{V})}^{\frac{4}{3}}, ur \end{matrix} .

The possibility of obtaining correct non-relativistic and ultra-relativistic formulas, while using the same theoretical approach, allows for illustrating the effects of the inclusion of special theory of relativity in quantum mechanics. A dramatic example of this is the use of Eqs. (25) and (26) for explaining why Fermi gas stars gravitationally collapse when their masses surpass the Chandrasekhar mass limit.⁵^,¹⁶^,¹⁷

IV. Electronic heat capacity of a relativistic Fermi gas

Often in solid state physics, the non-relativistic theory of a Fermi gas is developed for calculating the electronic heat capacity (C_el) of solids at low temperatures, i.e., when 0 < T < E_F/K_B, where K_B is the Boltzmann constant²:

(28)

C_{el} = \frac{1}{3} π^{2} D (E_{F}) K_{B}^{2} T .

In Eq. (28), D (E_F) is the density of states at E = E_F²:

(29)

D (E_{F}) = {\frac{dN}{dE}|}_{E_{F}} .

N is obtained from Eq. (21); therefore, using Eqs. (22) and (23) in the non-relativistic and ultra-relativistic limits, respectively; we obtain:

(30)

D (E_{F}) = \begin{matrix} \frac{3}{2} \frac{N}{E_{F}}, nr \\ 3 \frac{N}{E_{F}}, ur \end{matrix} .

Therefore, from Eqs. (28) and (30) follows:

(31)

\frac{C_{el}}{N k_{B}} = \begin{matrix} \frac{π^{2}}{2} \frac{k_{B} T}{E_{F}}, nr \\ π^{2} \frac{k_{B} T}{E_{F}}, ur \end{matrix} .

Again, in both limits, Eq. (31) gives known equations.²^–⁴ Substituting Eq. (22) in Eq. (31), one obtains²^–⁴:

(32)

\frac{C_{el}}{N k_{B}} = {(\frac{π}{3})}^{2 / 3} \frac{m^{5 / 3} k_{B}}{ℏ^{2}} \frac{T}{ρ^{2 / 3}}, ρ = \frac{mN}{V} .

While substituting Eq. (23) in Eq. (31), we obtain⁴:

(33)

\frac{C_{el}}{N k_{B}} = \frac{{(3 π^{2})}^{2 / 3}}{3} \frac{m^{1 / 3} k_{B}}{ℏ c} \frac{T}{ρ^{1 / 3}} .

Therefore, in both limits C_el/NK_B is proportional to T but the dependence on the density of the gas (ρ) is different.

V. Conclusions

We presented a unified description of a Fermi gas at low temperatures. This description is rigorously valid from the non-relativistic to the ultra-relativistic limits. Moreover, it is simple. The presented theory is based on a recently reported formula for the energies of a relativistic spin-0 particle in a box. The theory follows the relatively simple approach often used for constructing the theory of a non-relativistic Fermi gas. However, the theory outlined here does not include the formation and destruction of particle-antiparticle pairs at energies large than 2mc². It is then limited to cases where the number of particles can be considered constant.

It is worth noting that the ultra-relativistic results given by Eqs. (23), (26), (27), and (31) are often obtained using a different approach than the one used in this work. These results are not new but, together with the correct deduction of the Chandrasekhar mass limit reported in Ref. [5], they corroborate that the recently reported Eq. (4) is correct and illustrate the practical utility of Eqs. (3) and (17). In addition, the approach presented in this work allows for teaching the theory of a relativistic Fermi gas in introductory quantum mechanics courses.¹⁸

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Author details Author details

¹ Department of Physics and Astronomy, Texas Tech University, Lubbock, TX, 79410, USA
² Nano Tech Center, Texas Tech University, Lubbock, TX, 79410, USA

Luis Grave de Peralta
Roles: Conceptualization, Formal Analysis, Investigation, Writing – Original Draft Preparation, Writing – Review & Editing

Competing interests

No competing interests were disclosed.

Grant information

The author(s) declared that no grants were involved in supporting this work.

Article Versions (1)

version 1

Published: 08 Jul 2022, 11:761

https://doi.org/10.12688/f1000research.123275.1

© 2022 Grave de Peralta L. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Grave de Peralta L. A simple approach for extending up to the ultra-relativistic limit the theory of a non-relativistic Fermi gas [version 1; peer review: 3 not approved]. F1000Research 2022, 11:761 (https://doi.org/10.12688/f1000research.123275.1)

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Reviewer Report 14 Aug 2023

E.F. El-Shamy, Physics, King Khalid University, Abha, Aseer Province, Saudi Arabia

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https://doi.org/10.5256/f1000research.135363.r171496

I have carefully reviewed this manuscript. The author presents a unified description of a Fermi gas at low temperatures that remain valid from the nonrelativistic to the ultra-relativistic regimes. The theory developed follows a relatively simple approach commonly used for constructing the theory of a non-relativistic Fermi gas. However, it should be noted that the theory outlined in this work does not account for the formation and destruction of particle-antiparticle pairs at energies greater than 2mc2 . As a result, it is limited to cases where the number of particles can be considered constant. Nevertheless, I have concerns regarding the suitability of this paper being indexed, and I have provided detailed comments below:

First of all, the work appears to be more of a straightforward mathematical exercise rather than a comprehensive exploration of the current state of knowledge. The problem addressed is relatively straightforward, close to a student exercise in an advanced quantum mechanics course.
The author's mathematical treatment simplifies the problem but overlooks the true physical intrinsic of relativistic quantum mechanics. For instance, the author's approach is based on the hypothesis that a particle in a stationary state within a box possesses a constant kinetic energy value, thus a constant value of the square of its linear momentum (p2 ). However, this assumption is not valid in the relativistic Fermi gas model.

In the Fermi gas model, it is well-established that the energy levels of particles are filled up to the Fermi energy level, which depends on the system's temperature and density. The Fermi energy represents the maximum energy level occupied by fermions at absolute zero temperature. Therefore, assuming a constant kinetic energy value (i.e., a particle in a box being in a stationary state with constant kinetic energy) would imply that all particles have the same momentum, contradicting the principles of special relativity. In the relativistic Fermi gas model, the distribution of particle momenta is determined by the Fermi-Dirac statistics and the Fermi-Dirac distribution function. These considerations account for the relativistic energy-momentum relation and the exclusion principle, enabling an accurate description of the system.

Based on the above-mentioned points, I am of the opinion that this manuscript is not suitable for indexing.

Is the work clearly and accurately presented and does it cite the current literature?

No
Is the study design appropriate and is the work technically sound?

No
Are sufficient details of methods and analysis provided to allow replication by others?

Yes
If applicable, is the statistical analysis and its interpretation appropriate?

No
Are all the source data underlying the results available to ensure full reproducibility?

No
Are the conclusions drawn adequately supported by the results?

Partly

Competing Interests: No competing interests were disclosed.

Reviewer Expertise: theoretical plasma physics and its applications

I confirm that I have read this submission and believe that I have an appropriate level of expertise to state that I do not consider it to be of an acceptable scientific standard, for reasons outlined above.

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Author Response 16 Nov 2023

Luis Grave de Peralta, Nano Tech Center, Texas Tech University, Lubbock, 79410, USA

16 Nov 2023

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The author appreciates the opinion of the reviewer, cite "the work appears to be more of a straightforward mathematical exercise rather than a comprehensive exploration of the current state of ... Continue reading The author appreciates the opinion of the reviewer, cite "the work appears to be more of a straightforward mathematical exercise rather than a comprehensive exploration of the current state of knowledge. The problem addressed is relatively straightforward, close to a student exercise in an advanced quantum mechanics course," this is because this supports the author claims about the simplicity of the proposed description of a relativistic Fermi gas.

The author disagrees with the reviewer in the following point, cite "assuming a constant kinetic energy value (i.e., a particle in a box being in a stationary state with constant kinetic energy) would imply that all particles have the same momentum, contradicting the principles of special relativity." This is because the Pauli's exclusion principle was used. Consequently, only two fermions have the same momentum at 0 K.

There is no reason for the reviewer to answer NO to the question, cite "Are all the source data underlying the results available to ensure full reproducibility? The correct answer should be "Not applicable." (See other reviewers)
The author appreciates the opinion of the reviewer, cite "the work appears to be more of a straightforward mathematical exercise rather than a comprehensive exploration of the current state of knowledge. The problem addressed is relatively straightforward, close to a student exercise in an advanced quantum mechanics course," this is because this supports the author claims about the simplicity of the proposed description of a relativistic Fermi gas.

The author disagrees with the reviewer in the following point, cite "assuming a constant kinetic energy value (i.e., a particle in a box being in a stationary state with constant kinetic energy) would imply that all particles have the same momentum, contradicting the principles of special relativity." This is because the Pauli's exclusion principle was used. Consequently, only two fermions have the same momentum at 0 K.

There is no reason for the reviewer to answer NO to the question, cite "Are all the source data underlying the results available to ensure full reproducibility? The correct answer should be "Not applicable." (See other reviewers)
Competing Interests: No competing interests were disclosed. Close
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Author Response 16 Nov 2023

Luis Grave de Peralta, Nano Tech Center, Texas Tech University, Lubbock, 79410, USA

16 Nov 2023

Author Response

The author appreciates the opinion of the reviewer, cite "the work appears to be more of a straightforward mathematical exercise rather than a comprehensive exploration of the current state of ... Continue reading The author appreciates the opinion of the reviewer, cite "the work appears to be more of a straightforward mathematical exercise rather than a comprehensive exploration of the current state of knowledge. The problem addressed is relatively straightforward, close to a student exercise in an advanced quantum mechanics course," this is because this supports the author claims about the simplicity of the proposed description of a relativistic Fermi gas.

The author disagrees with the reviewer in the following point, cite "assuming a constant kinetic energy value (i.e., a particle in a box being in a stationary state with constant kinetic energy) would imply that all particles have the same momentum, contradicting the principles of special relativity." This is because the Pauli's exclusion principle was used. Consequently, only two fermions have the same momentum at 0 K.

There is no reason for the reviewer to answer NO to the question, cite "Are all the source data underlying the results available to ensure full reproducibility? The correct answer should be "Not applicable." (See other reviewers)
The author appreciates the opinion of the reviewer, cite "the work appears to be more of a straightforward mathematical exercise rather than a comprehensive exploration of the current state of knowledge. The problem addressed is relatively straightforward, close to a student exercise in an advanced quantum mechanics course," this is because this supports the author claims about the simplicity of the proposed description of a relativistic Fermi gas.

The author disagrees with the reviewer in the following point, cite "assuming a constant kinetic energy value (i.e., a particle in a box being in a stationary state with constant kinetic energy) would imply that all particles have the same momentum, contradicting the principles of special relativity." This is because the Pauli's exclusion principle was used. Consequently, only two fermions have the same momentum at 0 K.

There is no reason for the reviewer to answer NO to the question, cite "Are all the source data underlying the results available to ensure full reproducibility? The correct answer should be "Not applicable." (See other reviewers)
Competing Interests: No competing interests were disclosed. Close
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Reviewer Report 14 Aug 2023

J. P. W. Diener, Department of Physics and Astronomy, Botswana International University of Science and Technology, Palapye, Central District, Botswana

Not Approved

https://doi.org/10.5256/f1000research.135363.r171501

The article aims to demonstrate an alternative approach to description of a non-relativistic and relativistic Fermi gas. In the conclusion, it is further stated that "...approach presented in this work allows for teaching the theory of a relativistic Fermi gas in introductory quantum mechanics courses."

Due to the format of the journal, this reviewer is privy to the responses of the other reviewer as well as the author's response to the previous reviewer's comments. In order to provide a constructive contribution, I will consider the implied pedagogical aim of the article.

In this regard, the effort of the author to present alternative approaches to what could be considered a standard problem is appreciated. However, in order to assist with alternative approaches to teaching the subject matter, a direct comparison between approaches would have been appreciated, thereby illuminating additional physical insights as well as the pros and cons of the different approaches (simply stating that one is "simpler" is of limited use). Unfortunately the article heavily directly references previous results (in other publications), with which the reader might or might not be familiar, with limited context given in this article. This detracts from the readability of the article.

Due to these shortcomings, I unfortunately find the article of limited pedagogical interest.

I also concur with the objections raised by the previous reviewer and note the responses by the author. However, in my opinion, the author's responses do not address the valid concerns raised by the previous reviewer.

Is the work clearly and accurately presented and does it cite the current literature?

No
Is the study design appropriate and is the work technically sound?

No
Are sufficient details of methods and analysis provided to allow replication by others?

Yes
If applicable, is the statistical analysis and its interpretation appropriate?

Not applicable
Are all the source data underlying the results available to ensure full reproducibility?

No source data required
Are the conclusions drawn adequately supported by the results?

No

Competing Interests: No competing interests were disclosed.

Reviewer Expertise: Theoretical nuclear physics, magnetised dense nuclear matter

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Author Response 16 Nov 2023

Luis Grave de Peralta, Nano Tech Center, Texas Tech University, Lubbock, 79410, USA

16 Nov 2023

Author Response

The author appreciates the reviewer opinion, cite "the effort of the author to present alternative approaches to what could be considered a standard problem is appreciated." The author also agrees ... Continue reading The author appreciates the reviewer opinion, cite "the effort of the author to present alternative approaches to what could be considered a standard problem is appreciated." The author also agrees with the reviewer in that a direct comparison between approaches would have been appreciated. However, the author consider that it is outside of the scope of this article. There are references in the manuscript (Ref. 3-4) where a reader can find the statistical approach often used for describing a relativistic Fermi gas.
The author appreciates the reviewer opinion, cite "the effort of the author to present alternative approaches to what could be considered a standard problem is appreciated." The author also agrees with the reviewer in that a direct comparison between approaches would have been appreciated. However, the author consider that it is outside of the scope of this article. There are references in the manuscript (Ref. 3-4) where a reader can find the statistical approach often used for describing a relativistic Fermi gas.
Competing Interests: No competing interests were disclosed. Close
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COMMENTS ON THIS REPORT

Author Response 16 Nov 2023

Luis Grave de Peralta, Nano Tech Center, Texas Tech University, Lubbock, 79410, USA

16 Nov 2023

Author Response

The author appreciates the reviewer opinion, cite "the effort of the author to present alternative approaches to what could be considered a standard problem is appreciated." The author also agrees ... Continue reading The author appreciates the reviewer opinion, cite "the effort of the author to present alternative approaches to what could be considered a standard problem is appreciated." The author also agrees with the reviewer in that a direct comparison between approaches would have been appreciated. However, the author consider that it is outside of the scope of this article. There are references in the manuscript (Ref. 3-4) where a reader can find the statistical approach often used for describing a relativistic Fermi gas.
The author appreciates the reviewer opinion, cite "the effort of the author to present alternative approaches to what could be considered a standard problem is appreciated." The author also agrees with the reviewer in that a direct comparison between approaches would have been appreciated. However, the author consider that it is outside of the scope of this article. There are references in the manuscript (Ref. 3-4) where a reader can find the statistical approach often used for describing a relativistic Fermi gas.
Competing Interests: No competing interests were disclosed. Close
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Reviewer Report 11 Aug 2022

Niels Benedikter, Department of Mathematics, University of Milan, Milan, Italy

Not Approved

https://doi.org/10.5256/f1000research.135363.r147128

The article considers a semi-relativistic Schrödinger equation (of square-root type) for fermionic particles in a box, computing the heat capacity in the non-relativistic and ultrarelativistic limit.

The work does not represent the state of knowledge completely: the problem considered is trivial, namely on the level of a student exercise in an advanced quantum mechanics course.

For example, the claim "no such analytical expression was known, until recently, for the energies of a relativistic spin-0 particle in a box" is wrong. The formula is completely elementary and should be obvious to any theoretical physicist.

Equation (2) is a trivial rewriting of eq. (1). It irritates that the author tries to put his name on this equation.

Equation (3) is trivial, because an operator can always be replaced by its eigenvalue when acting on an eigenstate. Again, it irritates that the author tries to put his name on this trivial equation.

The paper continues like that.

A complete thermodynamical description may be explicitly computed looking at the grand canonical potential (as done already, e.g., by ref. 1 from the year 1942).¹

Is the work clearly and accurately presented and does it cite the current literature?

No
Is the study design appropriate and is the work technically sound?

Yes
Are sufficient details of methods and analysis provided to allow replication by others?

Yes
If applicable, is the statistical analysis and its interpretation appropriate?

Not applicable
Are all the source data underlying the results available to ensure full reproducibility?

No source data required
Are the conclusions drawn adequately supported by the results?

No

References

1. Kothari DS, Singh BN: Thermodynamics of a relativistic Fermi-Dirac gas. Proc. R. Soc. Lond. A. 1942; 180 (983). Publisher Full Text

Competing Interests: No competing interests were disclosed.

Reviewer Expertise: mathematical and theoretical physics, many-body quantum systems, Fermi gas

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Report a concern

Author Response 25 Aug 2022

Luis Grave de Peralta, Nano Tech Center, Texas Tech University, Lubbock, 79410, USA

25 Aug 2022

Author Response

The author appreciates the “Yes” response of the reviewer to the question “Is the study design appropriate and is the work technically sound? This is because as stated in the ... Continue reading The author appreciates the “Yes” response of the reviewer to the question “Is the study design appropriate and is the work technically sound? This is because as stated in the Conclusions, the author does not claim that the results presented in this work are originals.

The author also appreciates the reviewer opinion that “the problem considered is … on the level of a student exercise in an advanced quantum mechanics course.” This is because, as stated in the Conclusions, the author considers that “the relative simplicity of the approach presented in this work allows for teaching the theory of a relativistic Fermi gas in introductory quantum mechanics courses,” which is not a common practice due to the relative complexity of relativistic quantum mechanics.

The author respectfully disagrees with the reviewer in the following: Eqs. (4) and (5) give the energies of a relativistic spin-0 particle completely confined in a cubic box. These formulas are a three-dimensional generalization of the one-dimensional infinite well case. The formulas corresponding to the one-dimensional case were recently published for the first time (see Refs. (6) to (8)). Certainly, Eqs. (4) and (5) are simple; thus, it could be surprising that they were reported recently. The reason for this is that there are a lot of technical difficulties for solving the Klein-Gordon and the Dirac equations for a one-dimensional infinite well (see for instance, P. Alberto, C. Fiolhais, and V. M. S. Gil, Eur. J. Phys, 17, 19 (1996)). The originality of the approach presented in this work is that starting from Eqs. (4) and (5), we can arrive to well-known relativistic results using the same approach commonly followed in Introductory Quantum Mechanics and Solid-State courses for studying a non-relativistic Fermi gas (see for instance Refs. (1) and (2)).

The author also respectfully disagrees with the reviewer in the following: at a first look, Eq. (3) could be seemed like a trivial relativistic extension of the Schrödinger equation, but Eq. (3) is a very interesting and reach equation that only recently has been intensively studied (see for instance Refs. (6), (7), and (12)). Note that in many interesting cases (for instance, the relativistic harmonic oscillator and the Coulomb potential) the eigenfunctions of the energy operator are not eigenfunctions of the γ operator. Fortunately, as the reviewer cleverly pointed out, the eigenfunctions of the energy operator are eigenfunctions of the γ operator for a relativistic spin-0 particle completely confined in a cube. This simplifies the solution of Eq. (3) for this specific case (see Refs. (6) to (8)). Also note that Eq. (3) describes a constant number (one) of spin-0 particles; therefore, in contrast with the Klein-Gordon equation, there are not paradoxes associated to Eq. (3) for an infinite well.

Equation (2) is as least as interesting and reach as Eq. (3). The author suggests to the readers to go carefully over Refs. (6) and (12). There is a lot of original work there.
The author appreciates the “Yes” response of the reviewer to the question “Is the study design appropriate and is the work technically sound? This is because as stated in the Conclusions, the author does not claim that the results presented in this work are originals.

The author also appreciates the reviewer opinion that “the problem considered is … on the level of a student exercise in an advanced quantum mechanics course.” This is because, as stated in the Conclusions, the author considers that “the relative simplicity of the approach presented in this work allows for teaching the theory of a relativistic Fermi gas in introductory quantum mechanics courses,” which is not a common practice due to the relative complexity of relativistic quantum mechanics.

The author respectfully disagrees with the reviewer in the following: Eqs. (4) and (5) give the energies of a relativistic spin-0 particle completely confined in a cubic box. These formulas are a three-dimensional generalization of the one-dimensional infinite well case. The formulas corresponding to the one-dimensional case were recently published for the first time (see Refs. (6) to (8)). Certainly, Eqs. (4) and (5) are simple; thus, it could be surprising that they were reported recently. The reason for this is that there are a lot of technical difficulties for solving the Klein-Gordon and the Dirac equations for a one-dimensional infinite well (see for instance, P. Alberto, C. Fiolhais, and V. M. S. Gil, Eur. J. Phys, 17, 19 (1996)). The originality of the approach presented in this work is that starting from Eqs. (4) and (5), we can arrive to well-known relativistic results using the same approach commonly followed in Introductory Quantum Mechanics and Solid-State courses for studying a non-relativistic Fermi gas (see for instance Refs. (1) and (2)).

The author also respectfully disagrees with the reviewer in the following: at a first look, Eq. (3) could be seemed like a trivial relativistic extension of the Schrödinger equation, but Eq. (3) is a very interesting and reach equation that only recently has been intensively studied (see for instance Refs. (6), (7), and (12)). Note that in many interesting cases (for instance, the relativistic harmonic oscillator and the Coulomb potential) the eigenfunctions of the energy operator are not eigenfunctions of the γ operator. Fortunately, as the reviewer cleverly pointed out, the eigenfunctions of the energy operator are eigenfunctions of the γ operator for a relativistic spin-0 particle completely confined in a cube. This simplifies the solution of Eq. (3) for this specific case (see Refs. (6) to (8)). Also note that Eq. (3) describes a constant number (one) of spin-0 particles; therefore, in contrast with the Klein-Gordon equation, there are not paradoxes associated to Eq. (3) for an infinite well.

Equation (2) is as least as interesting and reach as Eq. (3). The author suggests to the readers to go carefully over Refs. (6) and (12). There is a lot of original work there.
Competing Interests: No competing interests were disclosed. Close
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Reviewer Response 09 Sep 2022

Niels Benedikter, Department of Mathematics, University of Milan, Milan, Italy

09 Sep 2022

Reviewer Response

Equation (1) and its trivial rewriting (2) have already been given by Dirac, Klein, and Fock. As early as 1964, this equation has been given in the textbook on relativistic ... Continue reading Equation (1) and its trivial rewriting (2) have already been given by Dirac, Klein, and Fock. As early as 1964, this equation has been given in the textbook on relativistic quantum mechanics by Bjorken and Drell.

Mathematically it does not present any difficulty, as may be learned for example from the textbook "Analysis" by Lieb and Loss. In fact, the equation has been widely studied in mathematical physics, and its solution as discussed in the paper is obvious to any researcher with some basic knowledge of Fourier analysis.

Equation (3) constitutes a naive and uncontrolled approximation, that, except for the treated trivial case, cannot be expected to yield a correct result.
Equation (1) and its trivial rewriting (2) have already been given by Dirac, Klein, and Fock. As early as 1964, this equation has been given in the textbook on relativistic quantum mechanics by Bjorken and Drell.

Mathematically it does not present any difficulty, as may be learned for example from the textbook "Analysis" by Lieb and Loss. In fact, the equation has been widely studied in mathematical physics, and its solution as discussed in the paper is obvious to any researcher with some basic knowledge of Fourier analysis.

Equation (3) constitutes a naive and uncontrolled approximation, that, except for the treated trivial case, cannot be expected to yield a correct result.
Competing Interests: No competing interests were disclosed. Close
Report a concern
Respond or Comment

COMMENTS ON THIS REPORT

Author Response 25 Aug 2022

Luis Grave de Peralta, Nano Tech Center, Texas Tech University, Lubbock, 79410, USA

25 Aug 2022

Author Response

The author appreciates the “Yes” response of the reviewer to the question “Is the study design appropriate and is the work technically sound? This is because as stated in the ... Continue reading The author appreciates the “Yes” response of the reviewer to the question “Is the study design appropriate and is the work technically sound? This is because as stated in the Conclusions, the author does not claim that the results presented in this work are originals.

The author also appreciates the reviewer opinion that “the problem considered is … on the level of a student exercise in an advanced quantum mechanics course.” This is because, as stated in the Conclusions, the author considers that “the relative simplicity of the approach presented in this work allows for teaching the theory of a relativistic Fermi gas in introductory quantum mechanics courses,” which is not a common practice due to the relative complexity of relativistic quantum mechanics.

The author respectfully disagrees with the reviewer in the following: Eqs. (4) and (5) give the energies of a relativistic spin-0 particle completely confined in a cubic box. These formulas are a three-dimensional generalization of the one-dimensional infinite well case. The formulas corresponding to the one-dimensional case were recently published for the first time (see Refs. (6) to (8)). Certainly, Eqs. (4) and (5) are simple; thus, it could be surprising that they were reported recently. The reason for this is that there are a lot of technical difficulties for solving the Klein-Gordon and the Dirac equations for a one-dimensional infinite well (see for instance, P. Alberto, C. Fiolhais, and V. M. S. Gil, Eur. J. Phys, 17, 19 (1996)). The originality of the approach presented in this work is that starting from Eqs. (4) and (5), we can arrive to well-known relativistic results using the same approach commonly followed in Introductory Quantum Mechanics and Solid-State courses for studying a non-relativistic Fermi gas (see for instance Refs. (1) and (2)).

The author also respectfully disagrees with the reviewer in the following: at a first look, Eq. (3) could be seemed like a trivial relativistic extension of the Schrödinger equation, but Eq. (3) is a very interesting and reach equation that only recently has been intensively studied (see for instance Refs. (6), (7), and (12)). Note that in many interesting cases (for instance, the relativistic harmonic oscillator and the Coulomb potential) the eigenfunctions of the energy operator are not eigenfunctions of the γ operator. Fortunately, as the reviewer cleverly pointed out, the eigenfunctions of the energy operator are eigenfunctions of the γ operator for a relativistic spin-0 particle completely confined in a cube. This simplifies the solution of Eq. (3) for this specific case (see Refs. (6) to (8)). Also note that Eq. (3) describes a constant number (one) of spin-0 particles; therefore, in contrast with the Klein-Gordon equation, there are not paradoxes associated to Eq. (3) for an infinite well.

Equation (2) is as least as interesting and reach as Eq. (3). The author suggests to the readers to go carefully over Refs. (6) and (12). There is a lot of original work there.
The author appreciates the “Yes” response of the reviewer to the question “Is the study design appropriate and is the work technically sound? This is because as stated in the Conclusions, the author does not claim that the results presented in this work are originals.

The author also appreciates the reviewer opinion that “the problem considered is … on the level of a student exercise in an advanced quantum mechanics course.” This is because, as stated in the Conclusions, the author considers that “the relative simplicity of the approach presented in this work allows for teaching the theory of a relativistic Fermi gas in introductory quantum mechanics courses,” which is not a common practice due to the relative complexity of relativistic quantum mechanics.

The author respectfully disagrees with the reviewer in the following: Eqs. (4) and (5) give the energies of a relativistic spin-0 particle completely confined in a cubic box. These formulas are a three-dimensional generalization of the one-dimensional infinite well case. The formulas corresponding to the one-dimensional case were recently published for the first time (see Refs. (6) to (8)). Certainly, Eqs. (4) and (5) are simple; thus, it could be surprising that they were reported recently. The reason for this is that there are a lot of technical difficulties for solving the Klein-Gordon and the Dirac equations for a one-dimensional infinite well (see for instance, P. Alberto, C. Fiolhais, and V. M. S. Gil, Eur. J. Phys, 17, 19 (1996)). The originality of the approach presented in this work is that starting from Eqs. (4) and (5), we can arrive to well-known relativistic results using the same approach commonly followed in Introductory Quantum Mechanics and Solid-State courses for studying a non-relativistic Fermi gas (see for instance Refs. (1) and (2)).

The author also respectfully disagrees with the reviewer in the following: at a first look, Eq. (3) could be seemed like a trivial relativistic extension of the Schrödinger equation, but Eq. (3) is a very interesting and reach equation that only recently has been intensively studied (see for instance Refs. (6), (7), and (12)). Note that in many interesting cases (for instance, the relativistic harmonic oscillator and the Coulomb potential) the eigenfunctions of the energy operator are not eigenfunctions of the γ operator. Fortunately, as the reviewer cleverly pointed out, the eigenfunctions of the energy operator are eigenfunctions of the γ operator for a relativistic spin-0 particle completely confined in a cube. This simplifies the solution of Eq. (3) for this specific case (see Refs. (6) to (8)). Also note that Eq. (3) describes a constant number (one) of spin-0 particles; therefore, in contrast with the Klein-Gordon equation, there are not paradoxes associated to Eq. (3) for an infinite well.

Equation (2) is as least as interesting and reach as Eq. (3). The author suggests to the readers to go carefully over Refs. (6) and (12). There is a lot of original work there.
Competing Interests: No competing interests were disclosed. Close
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Reviewer Response 09 Sep 2022

Niels Benedikter, Department of Mathematics, University of Milan, Milan, Italy

09 Sep 2022

Reviewer Response

Equation (1) and its trivial rewriting (2) have already been given by Dirac, Klein, and Fock. As early as 1964, this equation has been given in the textbook on relativistic ... Continue reading Equation (1) and its trivial rewriting (2) have already been given by Dirac, Klein, and Fock. As early as 1964, this equation has been given in the textbook on relativistic quantum mechanics by Bjorken and Drell.

Mathematically it does not present any difficulty, as may be learned for example from the textbook "Analysis" by Lieb and Loss. In fact, the equation has been widely studied in mathematical physics, and its solution as discussed in the paper is obvious to any researcher with some basic knowledge of Fourier analysis.

Equation (3) constitutes a naive and uncontrolled approximation, that, except for the treated trivial case, cannot be expected to yield a correct result.
Equation (1) and its trivial rewriting (2) have already been given by Dirac, Klein, and Fock. As early as 1964, this equation has been given in the textbook on relativistic quantum mechanics by Bjorken and Drell.

Mathematically it does not present any difficulty, as may be learned for example from the textbook "Analysis" by Lieb and Loss. In fact, the equation has been widely studied in mathematical physics, and its solution as discussed in the paper is obvious to any researcher with some basic knowledge of Fourier analysis.

Equation (3) constitutes a naive and uncontrolled approximation, that, except for the treated trivial case, cannot be expected to yield a correct result.
Competing Interests: No competing interests were disclosed. Close
Report a concern

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Version 1

VERSION 1 PUBLISHED 08 Jul 2022

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Reviewer Reports

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	1	2	3
Version 1 08 Jul 22	read	read	read

Niels Benedikter, University of Milan, Milan, Italy
J. P. W. Diener, Botswana International University of Science and Technology, Palapye, Botswana
E.F. El-Shamy, King Khalid University, Abha, Saudi Arabia

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Reviewer Report

7 Views

14 Aug 2023 | for Version 1

E.F. El-Shamy, Physics, King Khalid University, Abha, Aseer Province, Saudi Arabia

7 Views Cite this report Responses(1)

Not Approved

First of all, the work appears to be more of a straightforward mathematical exercise rather than a comprehensive exploration of the current state of knowledge. The problem addressed is relatively straightforward, close to a student exercise in an advanced quantum mechanics course.
The author's mathematical treatment simplifies the problem but overlooks the true physical intrinsic of relativistic quantum mechanics. For instance, the author's approach is based on the hypothesis that a particle in a stationary state within a box possesses a constant kinetic energy value, thus a constant value of the square of its linear momentum (p2 ). However, this assumption is not valid in the relativistic Fermi gas model.

Is the work clearly and accurately presented and does it cite the current literature?

No
Is the study design appropriate and is the work technically sound?

No
Are sufficient details of methods and analysis provided to allow replication by others?

Yes
If applicable, is the statistical analysis and its interpretation appropriate?

No
Are all the source data underlying the results available to ensure full reproducibility?

No
Are the conclusions drawn adequately supported by the results?

Partly

Competing Interests

No competing interests were disclosed.

Reviewer Expertise

theoretical plasma physics and its applications

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Responses (1)

Author Response

16 Nov 2023

Luis Grave de Peralta, Nano Tech Center, Texas Tech University, Lubbock, 79410, USA

The author appreciates the opinion of the reviewer, cite "the work appears to be more of a straightforward mathematical exercise rather than a comprehensive exploration of the current state of knowledge. The problem addressed is relatively straightforward, close to a student exercise in an advanced quantum mechanics course," this is because this supports the author claims about the simplicity of the proposed description of a relativistic Fermi gas.

The author disagrees with the reviewer in the following point, cite "assuming a constant kinetic energy value (i.e., a particle in a box being in a stationary state with constant kinetic energy) would imply that all particles have the same momentum, contradicting the principles of special relativity." This is because the Pauli's exclusion principle was used. Consequently, only two fermions have the same momentum at 0 K.

There is no reason for the reviewer to answer NO to the question, cite "Are all the source data underlying the results available to ensure full reproducibility? The correct answer should be "Not applicable." (See other reviewers)

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Competing Interests

No competing interests were disclosed.

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Reviewer Report

5 Views

14 Aug 2023 | for Version 1

J. P. W. Diener, Department of Physics and Astronomy, Botswana International University of Science and Technology, Palapye, Central District, Botswana

5 Views Cite this report Responses(1)

Not Approved

Is the work clearly and accurately presented and does it cite the current literature?

No
Is the study design appropriate and is the work technically sound?

No
Are sufficient details of methods and analysis provided to allow replication by others?

Yes
If applicable, is the statistical analysis and its interpretation appropriate?

Not applicable
Are all the source data underlying the results available to ensure full reproducibility?

No source data required
Are the conclusions drawn adequately supported by the results?

No

Competing Interests

No competing interests were disclosed.

Reviewer Expertise

Theoretical nuclear physics, magnetised dense nuclear matter

Respond to this report

Responses (1)

Author Response

16 Nov 2023

Luis Grave de Peralta, Nano Tech Center, Texas Tech University, Lubbock, 79410, USA

The author appreciates the reviewer opinion, cite "the effort of the author to present alternative approaches to what could be considered a standard problem is appreciated." The author also agrees with the reviewer in that a direct comparison between approaches would have been appreciated. However, the author consider that it is outside of the scope of this article. There are references in the manuscript (Ref. 3-4) where a reader can find the statistical approach often used for describing a relativistic Fermi gas.

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Competing Interests

No competing interests were disclosed.

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Reviewer Report

48 Views

11 Aug 2022 | for Version 1

Niels Benedikter, Department of Mathematics, University of Milan, Milan, Italy

48 Views Cite this report Responses(2)

Not Approved

Is the work clearly and accurately presented and does it cite the current literature?

No
Is the study design appropriate and is the work technically sound?

Yes
Are sufficient details of methods and analysis provided to allow replication by others?

Yes
If applicable, is the statistical analysis and its interpretation appropriate?

Not applicable
Are all the source data underlying the results available to ensure full reproducibility?

No source data required
Are the conclusions drawn adequately supported by the results?

No

References

1. Kothari DS, Singh BN: Thermodynamics of a relativistic Fermi-Dirac gas. Proc. R. Soc. Lond. A. 1942; 180 (983). Publisher Full Text

Competing Interests

No competing interests were disclosed.

Reviewer Expertise

mathematical and theoretical physics, many-body quantum systems, Fermi gas

Respond to this report

Responses (2)

Author Response

25 Aug 2022

Luis Grave de Peralta, Nano Tech Center, Texas Tech University, Lubbock, 79410, USA

The author appreciates the “Yes” response of the reviewer to the question “Is the study design appropriate and is the work technically sound? This is because as stated in the Conclusions, the author does not claim that the results presented in this work are originals.

The author also appreciates the reviewer opinion that “the problem considered is … on the level of a student exercise in an advanced quantum mechanics course.” This is because, as stated in the Conclusions, the author considers that “the relative simplicity of the approach presented in this work allows for teaching the theory of a relativistic Fermi gas in introductory quantum mechanics courses,” which is not a common practice due to the relative complexity of relativistic quantum mechanics.

The author respectfully disagrees with the reviewer in the following: Eqs. (4) and (5) give the energies of a relativistic spin-0 particle completely confined in a cubic box. These formulas are a three-dimensional generalization of the one-dimensional infinite well case. The formulas corresponding to the one-dimensional case were recently published for the first time (see Refs. (6) to (8)). Certainly, Eqs. (4) and (5) are simple; thus, it could be surprising that they were reported recently. The reason for this is that there are a lot of technical difficulties for solving the Klein-Gordon and the Dirac equations for a one-dimensional infinite well (see for instance, P. Alberto, C. Fiolhais, and V. M. S. Gil, Eur. J. Phys, 17, 19 (1996)). The originality of the approach presented in this work is that starting from Eqs. (4) and (5), we can arrive to well-known relativistic results using the same approach commonly followed in Introductory Quantum Mechanics and Solid-State courses for studying a non-relativistic Fermi gas (see for instance Refs. (1) and (2)).

The author also respectfully disagrees with the reviewer in the following: at a first look, Eq. (3) could be seemed like a trivial relativistic extension of the Schrödinger equation, but Eq. (3) is a very interesting and reach equation that only recently has been intensively studied (see for instance Refs. (6), (7), and (12)). Note that in many interesting cases (for instance, the relativistic harmonic oscillator and the Coulomb potential) the eigenfunctions of the energy operator are not eigenfunctions of the γ operator. Fortunately, as the reviewer cleverly pointed out, the eigenfunctions of the energy operator are eigenfunctions of the γ operator for a relativistic spin-0 particle completely confined in a cube. This simplifies the solution of Eq. (3) for this specific case (see Refs. (6) to (8)). Also note that Eq. (3) describes a constant number (one) of spin-0 particles; therefore, in contrast with the Klein-Gordon equation, there are not paradoxes associated to Eq. (3) for an infinite well.

Equation (2) is as least as interesting and reach as Eq. (3). The author suggests to the readers to go carefully over Refs. (6) and (12). There is a lot of original work there.

View more View less

Competing Interests

No competing interests were disclosed.

Reviewer Response

09 Sep 2022

Niels Benedikter, Department of Mathematics, University of Milan, Milan, Italy

Equation (1) and its trivial rewriting (2) have already been given by Dirac, Klein, and Fock. As early as 1964, this equation has been given in the textbook on relativistic quantum mechanics by Bjorken and Drell.

Mathematically it does not present any difficulty, as may be learned for example from the textbook "Analysis" by Lieb and Loss. In fact, the equation has been widely studied in mathematical physics, and its solution as discussed in the paper is obvious to any researcher with some basic knowledge of Fourier analysis.

Equation (3) constitutes a naive and uncontrolled approximation, that, except for the treated trivial case, cannot be expected to yield a correct result.

View more View less

Competing Interests

No competing interests were disclosed.

Alongside their report, reviewers assign a status to the article:

Approved - the paper is scientifically sound in its current form and only minor, if any, improvements are suggested

Approved with reservations - A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit.

Not approved - fundamental flaws in the paper seriously undermine the findings and conclusions

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A simple approach for extending up to the ultra-relativistic limit the theory of a non-relativistic Fermi gas

Abstract

Keywords

I. Introduction

II. Positive energies of a spin-0 particle in a box

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III. Relativistic Fermi gas at T = 0 K

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IV. Electronic heat capacity of a relativistic Fermi gas

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V. Conclusions

Data availability statement

References

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