Quantum-Systems at Escape Velocities

Adegbite Inioluwa Precious

doi:10.20944/preprints202406.0960.v2

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Quantum-Systems at Escape Velocities

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Adegbite Inioluwa Precious^*

This version is not peer-reviewed.

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15 June 2024

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18 June 2024

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Abstract

This article briefly shows how particle’s moving at escape velocity can be described in a quantum mechanical frameworks, this is achieved by deriving quantum mechanical equations that puts escape velocity into consideration. The equations are derived from Hamiltonians and Hamiltonian operators formulated from classical kinetic and potential terms which are then quantized. Equations were also presented for specific situations of the particle at escape velocity. Situations such as free-particles at escape velocity, Self-gravitating situations of a charged or neutral particle even while free of the gravitational influence of the large massive body due to its escape velocity, leaving only the self-gravitation as the only gravitational influence on the particle which in the case of a massless, charge less particle reduces back to the equation of free-particle at escape velocity. Other very brief expositions on trajectories and more are also present in this brief article.

Keywords:

quantum mechanics

;

gravity

;

Quantum-gravity

;

escape velocity

Subject:

Physical Sciences - Theoretical Physics

Introduction

While various quantum theories of gravity exists [3,16,17,18,19,20] an accepted quantum theory of gravity remains unavailable. However one can only make attempts to model specific situations where quantum mechanical activities are in play in the presence of gravity. Models such as the semi-classical gravity and QFT in curved spacetime [1,2] have been useful for such scenarios and further developments on them are ongoing.

As an attempt to complement the existing efforts the study of quantum mechanical particles in the presence of gravity is being made in this article, with the famed Schrodinger’s equation being the basis of the study.

However this brief study of quantum mechanical systems within a gravitational field is limited to motions at escape velocity, as the entire formulations and equations only focuses on escape velocity.

Escape velocity is normally studied classically, but it would be of interest to also study it within the frameworks of quantum mechanics, giving insights on the state of a particle or matter-wave as it becomes free of the influence of gravity.

The Schrodinger’s equation accurately describes the behavior of particles but does not directly associate it with gravitational influence and a reformulation seems necessary. Starting with the development of the kinetic term for particles in a gravitational field, a gravity associated Hamiltonian can be formulated, quantized and applied a suitable term for the Schrodinger’s equation. But as stated earlier this kinetic term is only applies for a particle moving at escape velocity.

While the particle may be free from the gravitational influence of another body it may still experience self-gravitation, provided it is a massive particle. Such case will also be accounted for in this study. Likewise a case of gravitational red shifting of the particle moving away from the primary body at escape velocity is of interest as the red shifting would influence the state of the particle this will also be described by the equation.

At the end of the study it is expected that a fair amount of insights is provided as to how one can model a particle’s behavior as it moves in a gravitational field at escape velocity, both in self-gravitating conditions and in absence of other external influence apart from gravity.

Schrodinger’s-Equation

From the quantization of energy and radiation, asserting that they come in discrete wave-packets referred to as quanta, to the postulate that these quantized energies are proportional to the frequency of the matter-wave, quantum mechanics [4,5,6] began to find its place in physics. Following this was the de-Broglie relation making us understand that the wave-particle duality is true for all matter wave and that the momentum of a matter-wave is inversely proportional to its wavelength.

Motivated by the fact that a differential wave-equation would be necessary in describing this wave nature of matter, Erwin-Schrodinger went on to formulate the popular wave-equation now known as Schrodinger’s equation.

The full time-dependent equation is written as;

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

With

" ψ "

being the wave-function representing a quantum state. Following the superposition principle this state would be a linear combination of possible quantum states.

The wave function being a solution to the wave-equation implies that the equation satisfies the conditions of linearity.

The time independent version of the Schrodinger’s equation is written as;

\hat{E ψ} = \frac{ℏ^{2}}{2 m} \frac{\partial^{2} ψ}{\partial x^{2}} + V (x) ψ

To derive the Schrodinger’s equation, the concepts of Hamiltonians and operators should first be put forward. Starting with the Hamiltonian

H = T + V

where “T” is the kinetic energy term and “V” is the potential energy term. When we have the kinetic energy for a particle of mass “m” given as

T = \frac{p^{2}}{2 m}

the Hamiltonian for such particle is then;

H = \frac{p^{2}}{2 m} + V

Since the Hamiltonian gives the total amount of energy the Hamiltonian is thus equated to the energy

E

such that

E = \frac{p^{2}}{2 m} + V

However this is still classical and must be quantized for us to have a true quantum mechanical description, to achieve this the energy and momentum are replaced with their respective quantum operators which would act on the wave-function

" ψ "

. The operators are presented in the form;

\hat{E} = i ℏ \frac{\partial}{\partial t}, \hat{P} = i ℏ \frac{\partial}{\partial x}

With the replacement, we then have;

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

The resulting equation satisfies the necessary conditions of a typical wave-equation, thus the Schrodinger’s equation has been derived.

The primary purpose for which the Schrodinger equation is being derived in this section of this article is the reference to similar reasoning’s that would be applied to the equation that would be derived, describing the behavior of quantum particles at escape velocities in a gravitational field.

Quantum Mechanical Systems at Escape Velocity

Escape velocity is known to be the minimum speed at which a secondary object becomes free of the gravitational influence of a primary body, more accurately regarded as escape speed considering it is scalar when the particular direction of the secondary object is not defined. However when a moving body with defined direction reaches the escape speed the term escape velocity becomes applicable for the moving body. The Escape speed is varied with the distance separating the primary body and secondary body likewise with the radius of the secondary body (if spherical). With the escape velocity given as [14,15]

v_{e} = \sqrt{\frac{2 G M}{r}} = \sqrt{2 g r}

and alternatively as

v_{e} = \sqrt{2 ϕ}

where “r” is the distance or the radius,

ϕ

is the gravitational potential equivalent to half the squared escape velocity

(\frac{1}{2} {v_{e}}^{2})

we consider a moving mass

m

of a particle at escape velocity, the energy equation is then given as follows:

m ϕ = \frac{1}{2} m {v_{e}}^{2} w h e r e p = m v_{e}, m = \frac{p}{v_{e}}

With the substitution for “m” being taken, the equation assumes the form

\frac{ϕ}{v_{e}} p = \frac{1}{2} p v_{e}

The momentum “p” is the momentum at escape velocity, promoting the momentum and energy to operators the following arises

\hat{E} = \frac{ϕ}{v_{e}} \hat{P} w h e r e \hat{P} = i ℏ \frac{\partial}{\partial x} \hat{E} = i ℏ \frac{ϕ}{v_{e}} \frac{\partial}{\partial x}

Given that

\frac{ϕ}{v_{e}} p = \frac{1}{2} m {v_{e}}^{2}

we can say a kinetic term should exist with the form:

T = i ℏ \frac{ϕ}{v_{e}} \frac{\partial}{\partial x}

In the classical Hamiltonian, the kinetic term can be applied with which a Hamiltonian operator and equation is then formulated;

H = T + V, H = \frac{ϕ}{v_{e}} p + V (x)

\hat{H} = i ℏ \frac{ϕ}{v_{e}} \frac{\partial}{\partial x} + V (x)

i ℏ \frac{\partial ψ}{\partial t} = \hat{H} ψ, i ℏ \frac{\partial ψ}{\partial t} = i ℏ \frac{ϕ}{v_{e}} \frac{\partial ψ}{\partial x} + V (x) ψ

This is an equation for a particle moving at escape velocity escaping the influence of gravity sourced from the primary body where V(x) is the potential term. Taking the form another step further we consider the following relation;

\frac{2 ϕ}{c^{2}} = \frac{r_{s}}{r}, ϕ = \frac{c^{2}}{2} (\frac{r_{s}}{r})

Working with this relation where

r_{s}

is the Schwarzschild radius and “c” the speed of light, the energy operator, Hamiltonian and equation then assumes the following form:

E = \frac{c^{2}}{2 v_{e}} (\frac{r_{s}}{r}) p, \hat{E} = \frac{c^{2}}{2 v_{e}} (\frac{r_{s}}{r}) i ℏ \frac{\partial}{\partial x}

H = \frac{c^{2}}{2 v_{e}} (\frac{r_{s}}{r}) p + V (x)

\hat{H} ψ = \frac{c^{2}}{2 v_{e}} (\frac{r_{s}}{r}) i ℏ \frac{\partial ψ}{\partial x} + V (x) ψ, i ℏ \frac{\partial ψ}{\partial t} = + \frac{c^{2}}{2 v_{e}} (\frac{r_{s}}{r}) i ℏ \frac{\partial ψ}{\partial x} + V (x) ψ

Gravitational Redshift [9,10,11]

A relationship exists between the gravitational redshift parameter

z

and the gravitational Lorentz factor

γ

which is expressed as;

γ = (1 + z) = {(1 - \frac{r_{s}}{r})}^{- \frac{1}{2}}

In such case the following expression holds

\frac{r_{s}}{r} = (1 - \frac{1}{{(1 + z)}^{2}}) w h e r e z = \frac{Δ f}{f_{0}} = \frac{Δ λ}{λ_{0}} = \frac{Δ ϕ}{c^{2}}

If one were to interpret this in the context of escape velocity, then this red-shift would be the shift in the frequency or wavelength of the particle as it moves further away from the primary body at escape velocity (this time not at the speed of light and is non-relativistic) described by the equation;

i ℏ \frac{\partial ψ}{\partial t} = \frac{c^{2}}{2 v_{e}} i ℏ \frac{\partial ψ}{\partial x} (1 - \frac{1}{{(1 + z)}^{2}}) + V (x) ψ

In a personally written article, gravitational redshift was pictured in terms of the relativistic energy momentum relation and the Feynman momentum space propagators defined as integrals over gravitationally red shifted momenta.

Freely Moving Particles

When we consider a situation where the potential energy is V(X) = 0 the particle is moving without being influenced by any other external force apart from gravity which it is already free from for such case the equation is written as.

i ℏ \frac{\partial ψ}{\partial t} = \frac{c^{2}}{2 v_{e}} (\frac{r_{s}}{r}) i ℏ \frac{\partial ψ}{\partial x}

However it still remains at the region of the gravitational influence, even though it is at escape velocity and is free from any other external force.

But when it is moving further away at escape velocity from the gravitational well without any external influence and its potential energy is now approaching zero. It begins to experience a gravitational shift in its frequency or wavelength. Therefore it is required that the gravitational redshift parameters is considered in the equation.

i ℏ \frac{\partial ψ}{\partial t} = \frac{c^{2}}{2 v_{e}} i ℏ \frac{\partial ψ}{\partial x} (1 - \frac{1}{{(1 + z)}^{2}})

One challenge in this case of freely moving particle is that it becomes too generic as the equation does not include properties such as mass or charge to specify the exact particle in consideration.

Self-Gravitating Particle

This is a special case as it describes a situation in which the particle is moving free from the Gravitational influence of another body and is now experiencing only its own gravity acting on itself in such condition the potential energy is best given by

V (X) = G \int \frac{m^{2}}{| y - y_{0} |} d^{3} y

i ℏ \frac{\partial ψ}{\partial t} = \frac{c^{2}}{2 v_{e}} (\frac{r_{s}}{r}) i ℏ \frac{\partial ψ}{\partial x} + G \int \frac{m^{2}}{| y - y_{0} |} d^{3} y

Where “m” is the mass of the particle in question. But for massless particles which does not gravitate the equation reduces to that of a freely moving particle.

Charged Self-Gravitating Particle

If the particle is known to carry a charge and is self-gravitating the equation would then include the electrical potential energy of the particle

V (X) = Q v_{p}

call

v_{p}

the electric potential and “Q” the electric charge. This yields;

i ℏ \frac{\partial ψ}{\partial t} = \frac{c^{2}}{2 v_{e}} (\frac{r_{s}}{r}) i ℏ \frac{\partial ψ}{\partial x} + Q v_{p} + G \int \frac{m^{2}}{| y - y_{0} |} d^{3} y

Escape Velocity for Spherically Symmetric Mass Distribution

For a spherically symmetric mass distribution of primary body, it can be said that the escape velocity is proportional to the square-root of the mass density and also proportional to the radius the above statement has the mathematical expression given by

v_{e} = K r \sqrt{ρ} w h e r e K = \sqrt{π G \frac{8}{3}}

This can as well be applied in our equations both when the particle is at the region of the gravitational field and when moving away from it.

Trajectories

In a situation where the particle is moving vertically at escape velocity is still within the gravitational well and is not moving away from the primary body as with gravitational redshift, then the particle’s motion is expected to follow a curved path. When the particle is moving at escape velocity its trajectory is expected to be parabolic. However if the particle attains a velocity greater than the escape velocity then it follows a hyperbolic trajectory.

Schwarzchild Radius

The Schwarzschild radius is the radial distance between the centers of a body (such as a spherically symmetric body or other kinds) to its event horizon, one typical body with spherical symmetry is a black hole.

The Schwarzschild radius is in fact a physical parameter in the Schwarzschild metric being an exact solution to Einstein’s field equation in general relativity [7,8,12] useful in modelling black holes, a condition where the escape velocity has to be faster than the speed of light then we are indeed dealing with a black hole.

If it were truly possible for any particle to move faster than the speed of light then the equations we have derived would have been very useful in describing a particle at escape velocity in a black hole’s gravitational field or when it is moving away from it. But as far as it is known the speed of light is regarded as the universal speed limit.

It should also be said of the Schwarzschild radius of an astronomical object that it is proportional to the mass or mass density of that astronomical object.

Conclusions

Quantum system moving at escape velocities in a gravitational field has been presented with well-motivated equations describing various situations and cases. It is seen that the dynamics of quantum particles in a region of gravitational influence can be studied at a basic level even without a full quantum theory of gravity and escape velocity no longer has to remain a completely classical concept.

The special case of a charged self-gravitating particle that is free from gravitational influence of another larger mass is shown to have a possible equation describing the situation by applying the self-gravitating potential energy term and the electrical potential energy term in the equation, provided the particle is a massive particle and has a charge.

If the particle possesses no mass nor a charge the equation reduces to the equation of a freely moving particle in the gravitational field at escape velocity, which is shown to be quite generic as it would not include any property that specifies the exact particle that is being studied.

The use of the gravitational redshift parameter

z = Δ f {f_{0}}^{- 1} = Δ λ {λ_{0}}^{- 1} = Δ ϕ c^{- 2}

also helps to account for the departure of the particle further away from the primary body at escape velocity leading to an eventual shift in the frequency, wavelength and energy of the particle/ matter wave.

Declarations

I hereby declare that this article, titled; “Quantum systems at escape velocity”.

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.

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Quantum-Systems at Escape Velocities

Abstract

Keywords:

Subject:

Introduction

Schrodinger’s-Equation

Quantum Mechanical Systems at Escape Velocity

Gravitational Redshift [9,10,11]

Freely Moving Particles

Self-Gravitating Particle

Charged Self-Gravitating Particle

Escape Velocity for Spherically Symmetric Mass Distribution

Trajectories

Schwarzchild Radius

Conclusions

Declarations

References

MDPI Initiatives

Important Links

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