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Mass Density and Energy Densities in Schrodinger’s Equation

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Adegbite Inioluwa Precious^*

This version is not peer-reviewed

Abstract

Defining integrals over volume element in curved spacetime together with the mass and energy densities within that region of spacetime is apparently equivalent to the given energy and mass of that body or particle. With this in mind equations involving energies and mass can be rewritten in terms of their densities and the integral over the volume element. The idea find its place in the Hamiltonian formulations, and can be carried over into quantum mechanical frameworks systematically accounting for the effect of spacetime volume structures and gravity on dynamics of quantum systems. This article briefly explores this concept and runs through various frameworks and equations relevant to this idea, particularly the schrodinger-Newton equation and discusses the cosmological constant as a vacuum energy density.

Keywords:

Subject: Physical Sciences - Other

Introduction

The energy density refers to the amount of energy stored in a given volume of space, it involves the distribution of energy within that volume. These energies may be in various forms, be it radiant energies from mater sources, or heat energies etc.

Mathematically this energy would be an integral of the energy density over a volume element. In curved spacetime the volume element is factored or weighted by the determinant of the metric tensor corresponding to the given geometry which the spacetime continuum is currently in, encoding geometric properties. The relationship between the volume element and the determinant of metric tensor is useful for describing the geometry of curved spacetime for that given volume of spacetime. This energy is given as;

E = ε_{0} \int d v .

ε_{0} i s t h e e n e r g y d e n s i t y w h e r e \int d v = \int \sqrt{- g} d^{3} X d t, d^{3} X = d x d y d z,

l e a d i n g t o E = \int \sqrt{- g} ε_{0} d^{3} X d t

Mass density similar to the energy density, characterizes the density of matter content in a given volume of space. This provides insight on the distribution of matter in that region of spacetime. Just like with energy densities, integrals are used to calculate total quantity of mass over given volume of space.

For the mass density we define the following;

m = \int \sqrt{- g} ρ d^{3} X d t, w i t h ρ b e i n g t h e m a s s d e n s i t y

The mass density and energy density contributes to the stress energy on spacetime leading to the curvature of spacetime around that region or volume. The energy content in a region or volume of space may be of various forms coming from astronomical sources and bodies in space.

One can incorporate these energy densities and mass densities to quantum mechanical systems by introducing them as additional terms to the Hamiltonian, and the Schrodinger’s equation. This is analogous to the Schrodinger-Newton equation, where the effect of gravity is being considered in the framework.

Schrodinger-Newton Equation

The Schrodinger-Newton equation being of the form;

i ℏ \frac{\partial ψ}{\partial t} = (\frac{i ℏ^{2}}{2 m} \frac{\partial^{2}}{\partial x^{2}} + V (x) + m ϕ) ψ

Is a modification of the Schrodinger’s equation in terms of Newtonian gravitational potential [19], where a gravitational term is added to describe the interaction of a particle with the gravitational field, it is sometimes referred to as the Schrodinger-Poisson equation.

The Schrodinger-Newton equation assumes that matter waves and particles remain quantum mechanical and gravity remain classical, leading to the equation being considered a semi classical equation.

If the potential term were to be replaced with the solution to Poisson’s equation the schrodinger-newton equation takes the following form;

i ℏ \frac{\partial ψ}{\partial t} = (\frac{i ℏ^{2}}{2 m} \frac{\partial^{2}}{\partial x^{2}} + V (x) - G m^{2} \int \frac{1}{| y - y_{0} |} d y) ψ

The equation as stated earlier are semi-classical in nature, relating classical gravity, to quantum matter.

The gravitational potential

ϕ

incorporates the effect of gravity specifically Newtonian gravity into the Schrodinger’s equation, but it does not account for the curvature of spacetime as defined by general relativity [20,21].

However we can rewrite this in terms of the mass density and the volume-element

m = \int \sqrt{- g} ρ d^{3} X d t

Thus the Schrodinger Newton equation becomes;

i ℏ \frac{\partial ψ}{\partial t} = (\frac{i ℏ^{2}}{2 m} \frac{\partial^{2}}{\partial x^{2}} + V (x) + \int \sqrt{- g} ρ ϕ d^{3} X d t) ψ

This brings in the concept of general relativity through the Integral over curved spacetime volume. Much similar to the Schrodinger-Newton Equation the effects of external radiant energies in spacetime particularly curved spacetime can be incorporated to the quantum mechanical framework with the energy defined as;

E = \int \sqrt{- g} ε_{0} d^{3} X d t

The Hamiltonian operator can then be defined as;

\hat{H} = \frac{i ℏ^{2}}{2 m} \frac{\partial^{2}}{\partial x^{2}} + V (x) + \int \sqrt{- g} ε_{0} d^{3} X d t

And the Schrodinger’s equation becomes;

i ℏ \frac{\partial ψ}{\partial t} = (\frac{i ℏ^{2}}{2 m} \frac{\partial^{2}}{\partial x^{2}} + V (x) + \int \sqrt{- g} ε_{0} d^{3} X d t) ψ

This can include external effects such as astronomical radiations from matter sources in space, even at high energies to the quantum mechanical framework.

The Cosmological Constant

The cosmological constant sometimes interpreted as the energy density of space or vacuum, is the energy density associated with the expansion of the universe and dark energy. In quantum field theory empty space or vacuum is said to have underlying quantum fields, which can have fluctuations in their ground states at the lowest energy sometimes called zero point energy which is present at all regions of spacetime.

One can also include the cosmological constant to the framework, accounting for dark energy and the universal expansion. To do this the energy is then defined as;

E = \int \sqrt{- g} (ε_{0} - λ) d^{3} X d t,

w h e r e l a m b a (λ) i s t h e c o s m o l o g i c a l c o n s t a n t

Then the Schrodinger’s equation accounting for the external radiations, matter sources and universal expansion is then of the form;

i ℏ \frac{\partial ψ}{\partial t} = (\frac{i ℏ^{2}}{2 m} \frac{\partial^{2}}{\partial x^{2}} + V (x) + \int \sqrt{- g} (ε_{0} - λ) d^{3} X d t) ψ

In the absence of external matter source or radiations and energy densities

ε_{0} \to 0

, the equation becomes

i ℏ \frac{\partial ψ}{\partial t} = (\frac{i ℏ^{2}}{2 m} \frac{\partial^{2}}{\partial x^{2}} + V (x) - \int \sqrt{- g} (λ) d^{3} X d t) ψ

This equation, and its corresponding Hamiltonian relates the overall energy of the system to the cosmological constant.

Although the Zero point energy from quantum field theory is said to contribute to the energy defined for the cosmological constant, however the observed value of the cosmological constant is yet so small compared to the zero point energy said to be contributing to it. This is the cosmological constant problem [16].

In quantum field theory the vacuum energy is defined in terms of zero point energy, in astrophysics and cosmology the vacuum energy is defined in terms of the cosmological constant. From a possibly unifying perspective it is believed that the zero point energy contributes to the cosmological constant’s value of energy hence the vacuum energy with the contributions of zero point energy can be known from the observed cosmological vacuum energy. But this contribution seems annulled in observation because the observed vacuum energy from cosmology appears to be too small to have contributions from the more enormous value of zero point energy.

One can hypothesize for this observation may be that Zero-point energy is a short-lived energy fluctuation only appearing as brief fluctuations hence their contributions would also be short-lived making them harder to observe, just as with the virtual particles. Hence only the small energy from the cosmological constant is more easily observed.

One can also hypothesize that the zero point energy is a virtual energy as with virtual particles, in that they may not be real or actually observable energies. So their contributions to the real energy from the cosmological constant might be a virtual contributions even with such enormous value. Although these are mostly hypothetical but it is possible that an answer to the cosmological constant problem may be found when the true nature of zero point energy becomes clearer.

Nuclear Energies in Volume of Spacetime

For a nuclear reactions the mass energy equivalence equation

E = m c^{2}

is necessary for calculating the energy of that reaction, however if the mass density of the nuclear system in a volume of curved spacetime is specified, then the mass is given to be

m = \int \sqrt{- g} ρ d^{3} X d t

, and the equation becomes;

E = \int \sqrt{- g} ρ c^{2} d^{3} X d t

For nuclear fission and radioactive decay processes it can be given as;

E = \int \sqrt{- g} ∆ ρ c^{2} d^{3} X d t

This way we get to relate geometry of spacetime to the concept through the determinant of the metric tensor to the energy of the nuclear system.

Negative Energy

Given the energy content in a volume of curved spacetime

E = \int \sqrt{- g} (ε_{0} - λ) d^{3} X d t

, we can consider the absence of other forms of energies and radiations setting

ε_{0} \to 0

this gives the equation;

E = \int \sqrt{- g} (- λ) d^{3} X d t

It is seen that it leads to a negative valued energy {9,10] which drives inflations or expansion of the universe. It is the presence of positive energies, from matter sources and radiations that counteracts this inflationary negative energy.

Conclusions

The Schrodinger-Newton equation has briefly been discussed as a semi classical equation relating Gravity from Newtonian theory of gravity to quantum mechanics, describing hoe particle behave in the presence of gravitational field. Gravity is treated classically and matter is treated quantum mechanically, with gravity introduced through an additional term

m ϕ

to the Schrodinger’s equation with

ϕ

being the gravitational potential of the gravitational field.

Re-expressing the mass in terms of the mass density and the integral of the volume element in curved spacetime, the schrodinger-newton equation is then defined in terms of the mass density and accounts for curved spacetime through the determinant of the metric tensor in the integral of the volume element.

The equation is further being extended in terms of external energy densities coming from radiation or matter sources in curved spacetime. This is done by taking the additional term to be an energy term written as the integral of the volume element multiplied by the energy density of the external radiation or matter sources.

The use of the cosmological constant as the vacuum energy density in the Schrodinger’s equation is also shown, and the cosmological constant problem was also discussed with possible hypothesis being raised. One of which entailed the possibility of the zero-point energy having only temporal contributions to the vacuum energy from the cosmological constant, hence only the small consistent value from the cosmological constant is easily observed. Another hypothesis raised is that Zero-point energy is basically a virtual form of energy just as with the virtual particle, so even if their values are enormous their contributions to the real energy from the cosmological constant would not be directly observable. And finally nuclear energies and negative energies were discussed in terms of energy densities and mass densities.

These discussions are fundamentally exploratory, however they show some simple ways which particle interacts with external energy, Gravity and mass sources in curved volume of spacetime.

Declarations

I hereby declare that this article, titled; “Mass and energy densities in Schrodinger’s equation”.

Is written with no conflicting interest, neither is there any existing or pre-existing affiliation with any institution.
No prior funds is received by the author from any organization, individual or institution.
The content of the article is written with respect to ethics.
The content of the article does not involve experimentation with human and/or animal subjects.
Data-availability; no data, table or software prepared by an external body or institution is directly applicable to this article.

References

Jose W Maluf, Localization of energy in general relativity, Journal of Mathematical Physics, 36(8), 4242-4247, (1995).
Theodore Frankel et-al, Energy density and spatial curvature in general relativity, Journal of Mathematical Physics, 22(4), 813-817, (1981).
Richard Harrison et-al, a numerical study of the Schrodinger-Newton equations, Nonlinearity, 16(1), 101, (2002).
Giovanni Manfredi, The Schrodinger-Newton equation beyond Newton, General relativity and Gravitation, 47(2), 1, (2015).
Mohammad Bahrami et-al, The Schrodinger-Newton equation and its foundations, New Journal of Physics, 16(11), 115007, (2014).
Nouredine Zettili, Quantum mechanics: Concepts and applications, John Wiley & Sons, (2009).
Barry Simon, Schrodinger Operators in the twentieth century, Journal of Mathematical Physics, 41(6), 3523-3555, (2000).
Feliks Aleksandrovich Berezin et-al, the Schrodinger Equation, Springer Science & Business Media, (2012).
Robert J Nemiroff et-al, an exposition on Friedman Cosmology with Negative energy densities, Journal of cosmology and astroparticle physics, 2015(06), 006, (2015).
Tomislav Prokopec, Negative energy cosmology and the cosmological constant, arXiv preprint, arXiv: 1105.0078, (2011).
William Simpson & Ulf Leonhardt, Forces of the quantum vacuum: an introduction to Casmir physics, world scientific, (2015).
Simon-Saunders, Is the zero-point energy real? Ontological aspects of quantum field theory, 313-343, (2002).
Kimball A Milton, The Casmir effect: Physical manifestation of Zero-point energy, world scientific, (2001).
Thomas Banks, The cosmological constant problem, Physics today, 57(3), 46-51, (2004).
Lucas Lombriser, on the cosmological problem, Physics letters B, 797, 134804, (2019).
Steven Weinberg, the cosmological constant problem, Review of modern physics, 61(1), 1, (1989).
Eric Poisson & Clifford M Will, Gravity, Gravity, (2014).
Martin Counthan, Presenting Newtonian gravity, European journal of physics, 28(6), 1189, (2007).
RA Capuzzo Dolcetta, Classical Newtonian gravity, Springer, berlin, (2019).
Michael Paul Hobson et-al, General relativity: an introduction for Physicists, Cambridge University Press, (2006).
Robert M Wald, General relativity, University of Chicago press, (2010).

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© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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Submitted:

26 April 2024

Posted:

26 April 2024

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Abstract

Keywords:

Subject: Physical Sciences - Other

Introduction

E = ε_{0} \int d v .

ε_{0} i s t h e e n e r g y d e n s i t y w h e r e \int d v = \int \sqrt{- g} d^{3} X d t, d^{3} X = d x d y d z,

l e a d i n g t o E = \int \sqrt{- g} ε_{0} d^{3} X d t

For the mass density we define the following;

m = \int \sqrt{- g} ρ d^{3} X d t, w i t h ρ b e i n g t h e m a s s d e n s i t y

Schrodinger-Newton Equation

The Schrodinger-Newton equation being of the form;

i ℏ \frac{\partial ψ}{\partial t} = (\frac{i ℏ^{2}}{2 m} \frac{\partial^{2}}{\partial x^{2}} + V (x) + m ϕ) ψ

The Schrodinger-Newton equation assumes that matter waves and particles remain quantum mechanical and gravity remain classical, leading to the equation being considered a semi classical equation.

If the potential term were to be replaced with the solution to Poisson’s equation the schrodinger-newton equation takes the following form;

i ℏ \frac{\partial ψ}{\partial t} = (\frac{i ℏ^{2}}{2 m} \frac{\partial^{2}}{\partial x^{2}} + V (x) - G m^{2} \int \frac{1}{| y - y_{0} |} d y) ψ

The equation as stated earlier are semi-classical in nature, relating classical gravity, to quantum matter.

The gravitational potential

ϕ

incorporates the effect of gravity specifically Newtonian gravity into the Schrodinger’s equation, but it does not account for the curvature of spacetime as defined by general relativity [20,21].

However we can rewrite this in terms of the mass density and the volume-element

m = \int \sqrt{- g} ρ d^{3} X d t

Thus the Schrodinger Newton equation becomes;

i ℏ \frac{\partial ψ}{\partial t} = (\frac{i ℏ^{2}}{2 m} \frac{\partial^{2}}{\partial x^{2}} + V (x) + \int \sqrt{- g} ρ ϕ d^{3} X d t) ψ

E = \int \sqrt{- g} ε_{0} d^{3} X d t

The Hamiltonian operator can then be defined as;

\hat{H} = \frac{i ℏ^{2}}{2 m} \frac{\partial^{2}}{\partial x^{2}} + V (x) + \int \sqrt{- g} ε_{0} d^{3} X d t

And the Schrodinger’s equation becomes;

i ℏ \frac{\partial ψ}{\partial t} = (\frac{i ℏ^{2}}{2 m} \frac{\partial^{2}}{\partial x^{2}} + V (x) + \int \sqrt{- g} ε_{0} d^{3} X d t) ψ

This can include external effects such as astronomical radiations from matter sources in space, even at high energies to the quantum mechanical framework.

The Cosmological Constant

One can also include the cosmological constant to the framework, accounting for dark energy and the universal expansion. To do this the energy is then defined as;

E = \int \sqrt{- g} (ε_{0} - λ) d^{3} X d t,

w h e r e l a m b a (λ) i s t h e c o s m o l o g i c a l c o n s t a n t

Then the Schrodinger’s equation accounting for the external radiations, matter sources and universal expansion is then of the form;

i ℏ \frac{\partial ψ}{\partial t} = (\frac{i ℏ^{2}}{2 m} \frac{\partial^{2}}{\partial x^{2}} + V (x) + \int \sqrt{- g} (ε_{0} - λ) d^{3} X d t) ψ

In the absence of external matter source or radiations and energy densities

ε_{0} \to 0

, the equation becomes

i ℏ \frac{\partial ψ}{\partial t} = (\frac{i ℏ^{2}}{2 m} \frac{\partial^{2}}{\partial x^{2}} + V (x) - \int \sqrt{- g} (λ) d^{3} X d t) ψ

This equation, and its corresponding Hamiltonian relates the overall energy of the system to the cosmological constant.

Nuclear Energies in Volume of Spacetime

For a nuclear reactions the mass energy equivalence equation

E = m c^{2}

is necessary for calculating the energy of that reaction, however if the mass density of the nuclear system in a volume of curved spacetime is specified, then the mass is given to be

m = \int \sqrt{- g} ρ d^{3} X d t

, and the equation becomes;

E = \int \sqrt{- g} ρ c^{2} d^{3} X d t

For nuclear fission and radioactive decay processes it can be given as;

E = \int \sqrt{- g} ∆ ρ c^{2} d^{3} X d t

This way we get to relate geometry of spacetime to the concept through the determinant of the metric tensor to the energy of the nuclear system.

Negative Energy

Given the energy content in a volume of curved spacetime

E = \int \sqrt{- g} (ε_{0} - λ) d^{3} X d t

, we can consider the absence of other forms of energies and radiations setting

ε_{0} \to 0

this gives the equation;

E = \int \sqrt{- g} (- λ) d^{3} X d t

Conclusions

m ϕ

to the Schrodinger’s equation with

ϕ

being the gravitational potential of the gravitational field.

These discussions are fundamentally exploratory, however they show some simple ways which particle interacts with external energy, Gravity and mass sources in curved volume of spacetime.

Declarations

I hereby declare that this article, titled; “Mass and energy densities in Schrodinger’s equation”.

Is written with no conflicting interest, neither is there any existing or pre-existing affiliation with any institution.
No prior funds is received by the author from any organization, individual or institution.
The content of the article is written with respect to ethics.
The content of the article does not involve experimentation with human and/or animal subjects.
Data-availability; no data, table or software prepared by an external body or institution is directly applicable to this article.

References

Jose W Maluf, Localization of energy in general relativity, Journal of Mathematical Physics, 36(8), 4242-4247, (1995).
Theodore Frankel et-al, Energy density and spatial curvature in general relativity, Journal of Mathematical Physics, 22(4), 813-817, (1981).
Richard Harrison et-al, a numerical study of the Schrodinger-Newton equations, Nonlinearity, 16(1), 101, (2002).
Giovanni Manfredi, The Schrodinger-Newton equation beyond Newton, General relativity and Gravitation, 47(2), 1, (2015).
Mohammad Bahrami et-al, The Schrodinger-Newton equation and its foundations, New Journal of Physics, 16(11), 115007, (2014).
Nouredine Zettili, Quantum mechanics: Concepts and applications, John Wiley & Sons, (2009).
Barry Simon, Schrodinger Operators in the twentieth century, Journal of Mathematical Physics, 41(6), 3523-3555, (2000).
Feliks Aleksandrovich Berezin et-al, the Schrodinger Equation, Springer Science & Business Media, (2012).
Robert J Nemiroff et-al, an exposition on Friedman Cosmology with Negative energy densities, Journal of cosmology and astroparticle physics, 2015(06), 006, (2015).
Tomislav Prokopec, Negative energy cosmology and the cosmological constant, arXiv preprint, arXiv: 1105.0078, (2011).
William Simpson & Ulf Leonhardt, Forces of the quantum vacuum: an introduction to Casmir physics, world scientific, (2015).
Simon-Saunders, Is the zero-point energy real? Ontological aspects of quantum field theory, 313-343, (2002).
Kimball A Milton, The Casmir effect: Physical manifestation of Zero-point energy, world scientific, (2001).
Thomas Banks, The cosmological constant problem, Physics today, 57(3), 46-51, (2004).
Lucas Lombriser, on the cosmological problem, Physics letters B, 797, 134804, (2019).
Steven Weinberg, the cosmological constant problem, Review of modern physics, 61(1), 1, (1989).
Eric Poisson & Clifford M Will, Gravity, Gravity, (2014).
Martin Counthan, Presenting Newtonian gravity, European journal of physics, 28(6), 1189, (2007).
RA Capuzzo Dolcetta, Classical Newtonian gravity, Springer, berlin, (2019).
Michael Paul Hobson et-al, General relativity: an introduction for Physicists, Cambridge University Press, (2006).
Robert M Wald, General relativity, University of Chicago press, (2010).

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Mass Density and Energy Densities in Schrodinger’s Equation

Abstract

Introduction

Schrodinger-Newton Equation

The Cosmological Constant

Nuclear Energies in Volume of Spacetime

Negative Energy

Conclusions

Declarations

References

Abstract

Introduction

Schrodinger-Newton Equation

The Cosmological Constant

Nuclear Energies in Volume of Spacetime

Negative Energy

Conclusions

Declarations

References

MDPI Initiatives

Important Links

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