Path of a Massive Particle in Curvilinear Space Using QED

8/3/2019 Path of a Massive Particle in Curvilinear Space Using QED

1/23

Author : Sid Senadheera [email protected]

Gravitational effects on a nanoparticle has been combined with the Hamiltonian for second order

perturbations in the Schrodinger wave equation in quantum mechanics. The stress energy tensor

in the Einstein field equations is added as a perturbation on top of the well known Newtonian

gravitational potential function. The experimental proof for the theory can be showing the

masses of the particles are quantized according to De Broglie hypothesis. And the magnitude of

the energy in quantization change according to the theoretically derived result in this work.

First let us look at the formation of magnetic moments in atoms. Let us consider

an atom in a homogeneous magnetic field H described by a vector

potential A [where A = -1/2 (r H)]. The Hamiltonian for the Z electrons, each

having an intrinsic spin magnetic moment m = 2B s and a momentum l, is :

Where all the terms containing the magnetic field have been grouped in H (1), that can be treated

as a perturbation of H (0). The nanoparticles in this case deviated from their paths due to para-

Theory :

Quantum Mechanical and Gravitational Effects on a Nanoparticle

*All rights reserved. No part of the contents of this draft may be reproduced in any form or by anymeans, electronically or otherwise, without the prior written permission from author, - Sid Senadheera .
http://sidath.senadheera.net/thesis.pdfhttp://sidath.senadheera.net/thesis.pdf


2/23

*All rights reserved. No part of the contents of this draft may be reproduced in any form or by anymeans, electronically or otherwise, without the prior written permission from author, Sid Senadheera .

magnetism. Therefore the paramagnetic term is isolated E(p) excluding all other terms. The

analysis of particle in a field in the Stern Gerlarch experiment can be similarly carried out.

The energy and force on a particle by a gravitational field will have a similar derivation. In this

case the Newtonian gravitational potential (used as U = mgz) has a perturbation term in the

energy by the Einsteins stress energy tensor Tt,x,y,z.


3/23


Particle in a Gravitational Field.

Pages (1-6 with topics that mention [REVISED]) gives a brief introduction to

Tensors, Gravitation and Stress Energy Tensor that gives the energy in a unit

volume of space.Particle Energy derived by Geodesic Deviations [REVISED]

The Einstein field equations (EFE) or Einstein's equations are a set of ten equations in Einstein's

theory of general relativity which describe the fundamental interaction of gravitation as a result

of spacetime being curved by matter and energy. First published by Albert Einstein in 1915 as

a tensor equation, the EFE equate spacetime curvature (expressed by the Einstein tensor -

G t,x,y,z.) with the energy and momentum within that spacetime (expressed by the stress-energy

tensor Tt,x,y,z ).

Here we are mostly interested in theTt,x,y,ztensor. For simplicity we are expressing this tensor in

Cartesian coordinates, although it has to be expressed in spherical coordinates. In gravitation,

energy is in the curvature of space-time.

The relationship between the metric tensor and the Einstein tensor allows the EFE to be written

as a set of non-linear partial differential equations when used in this way. The solutions of the

EFE are the components of the metric tensor. The trajectories of particles and radiation

(geodesics) in the resulting geometry are then calculated using the geodesic equation.

The metric tensor can be solved by the above equation, which would result in.


4/23

Physicists usually work in local coordinates (i.e. coordinates defined on some local patch ofM).

In local coordinatesx

(where is an index which runs from 0 to 3) the metric can be written in

the form

The factors dx

are one-form gradients of the scalar coordinate fieldsx. The metric is thus

a linear combination of tensor products of one-form gradients of coordinates. The

coefficients g are a set of 16 real-valued functions (since the tensor g is actually a tensor

fielddefined at all points of a space-time manifold). In order for the metric to be symmetric we

must have

giving 10 independent coefficients. If we denote the symmetric tensor product by juxtaposition

(so thatdxdx

=dx

dx

) we can write the metric in the form

If the local coordinates are specified, or understood from context, the metric can be written as a

44 symmetric matrix with entries g. The nondegeneracy ofg means that this matrix is non-

singular (i.e. has non-vanishing determinant), while the Lorentzian signature ofg implies that the

matrix has one negative and three positive eigenvalues.


5/23


Curvature [Revised]

The metricg completely determines the curvature of spacetime. According to the fundamental

theorem of Riemannian geometry, there is a unique connection on any Lorentzianmanifold that is compatible with the metric and torsion-free. This connection is called theLevi-

Civita connection. The Christoffel symbols of this connection are given in terms of partial

derivatives of the metric in local coordinatesx

by the formula

.

The curvature of spacetime is then given by the Riemann curvature tensor which is defined in

terms of the Levi-Civita connection . In local coordinates this tensor is given by:

The curvature is then expressible purely in terms of the metric g and its derivatives.

Einstein's equations [Revised]

One of the core ideas of general relativity is that the metric (and the associated geometry of

spacetime) is determined by the matter and energy content of spacetime. Einstein's field

equations:

Where :


6/23


relate the metric (and the associated curvature tensors) to the stress-energy tensor T.

This tensor equation is a complicated set of nonlinear partial differential equations for the metric

components. Exact solutions of Einstein's field equations are very difficult to find.

Introductory Quantum Mechanics : In Three Dimensions and Four Dimensions

To extend the discussion to more than one dimension is easy. The potential energy becomes

U(x,y,z) and we add partial derivatives with respect to y and z. The S-E then becomes

There is a corresponding change in the TISE.

The set of derivatives in ( ) above is usually denoted by a special symbol:

The normalization condition becomes

Of course, coordinate systems other than Cartesian can be used. For central force problems (such

as the hydrogen atom) it is better to use spherical coordinates.


7/23


Evidence for Quantized Gravitational States of the Neutron

Main Reference : T. J. Bowles, "Quantum effects of gravity,"Nature 415 267-268 (2002).

Particle confinement and the wave-like properties of matter lead, according to quantum

mechanical principles, to self-interference which is the origin of energy quantization. The

particle-in-a-box problem is used in introductory quantum mechanics courses to illustrate this

fundamental quantum effect.

Terrestrial objects are confined by the Earth's gravitational field, but the quantum effects of

gravity are not observed in the macro-world because the gravitational interaction is weak. Thus,

the gravitational energy levels are very closely spaced and for all practical purposes form a

continuum.

In spite of the lack of evidence for quantized gravitational energy levels, the "quantum bouncer"

has been a favorite example in the repertoire of solvable one-dimensional problems for those

who teach quantum chemistry and quantum physics. Schrdinger's equation for the quantum

bouncer near the surface of the Earth is,

where the particle is confined by the impenetrable potential barrier of the Earth's surface (V = )

and the attractive gravitational interaction (V = mgz for z > 0).


8/23


The energy eigenvalues for the quantum bouncer are (1,2,3),

where ai are the roots of the Airy function. The first five roots are 2.33810, 4.08794, 5.52055,

6.78670, and 7.94413.

The associated eigenfunctions are,

where

Because there is no analytical expression for the eigenfunctions each one must be normalized

using a numeric algorithm.

Very recently an international team at the Institute Laue-Langevin in Grenoble France lead by

V. Nesvizhevsky (4) published evidence for the quantized gravitational states of the neutron.

To read a short summary of this experiment in Nature Magazineby Thomas Bowles (5)

click here. Another summary has just been published in Physics Today. (6)

To appreciate the significance of this accomplishment we calculate the neutron's ground state

energy and wave function in the Earth's gravitational field using the equations above. The mass

of the neutron is 1.675x10-27

kg which yields a ground-state energy of E1 = 2.254x10-31

J. This
http://www.nature.com/cgi-taf/DynaPage.taf?file=/nature/journal/v415/n6869/full/415267a_fs.htmlhttp://www.nature.com/cgi-taf/DynaPage.taf?file=/nature/journal/v415/n6869/full/415267a_fs.htmlhttp://www.nature.com/cgi-taf/DynaPage.taf?file=/nature/journal/v415/n6869/full/415267a_fs.html


9/23


corresponds to a classical vertical velocity of 1.6 cm/s. Thus gravitational confinement requires a

source of ultra-cold neutrons (UCNs).

Furthermore, the energy of the first excited state is 3.941x10-31 J, so the energy difference

between the ground state and the first excited state is equivalent to photon with a wavelength of

1.2x106

m. Clearly traditional spectroscopic methods cannot be used to establish the existence of

quantized gravitational states for the neutron.

The probability distributions, Y(z)2, for the ground and first excited states are shown in Figures 1

and 4. They hold the key to the experimental design that Nesvizhevsky's and his group used to

establish that the neutron's gravitational states are quantized. To down-load a Mathcad file that

will generate the neutron eigenstates numericallyclick here.

Figure 1

A schematic of the experiment that Nesvizhevsky's team used to gather evidence for quantized

neutron gravitational states is shown in Figure 2. The neutron is confined by the attractive
http://www.users.csbsju.edu/~frioux/neutron/neutron.mcdhttp://www.users.csbsju.edu/~frioux/neutron/neutron.mcdhttp://www.users.csbsju.edu/~frioux/neutron/neutron.mcdhttp://www.users.csbsju.edu/~frioux/neutron/neutron.mcd


10/23


gravitational field and the repulsive reflecting mirror surface. Ultra-cold neutrons (UCNs) with

total velocities less than 8 m/s are "lobbed" into the apparatus. In the vertical direction the

neutrons are subject to the gravitational interaction with the Earth, but there are no forces in the

horizontal direction. The vertical and horizontal degrees of freedom are independent of one

another in the design of this experiment, because care has been taken to eliminate vibrations, and

extraneous electric and magnetic fields.

Figure 2 [Nature(Volume 415 page 267) copyright 2002 Macmillan Publishers Ltd.]

The apparatus shown in Figure 2 records neutron throughput as a function of absorber height.

The data collected are shown in Figure 3. The shaded circles are the actual data points. We will

not be concerned with the solid, dashed, or dotted lines in the figure. The most important feature

of the data for this analysis is the sharp increase in neutron throughput at about 20 mm.
http://www.nature.com/http://www.nature.com/http://www.nature.com/http://www.nature.com/


11/23


Figure 3 [Nature(Volume 415 page 299) copyright 2002 Macmillan Publishers Ltd.]

The argument will be made that the neutron wave function shown in Figure 1 is consistent with

the data presented in Figure 3. To demonstrate this we calculate the probability that the ground-

state neutron will be found in the absorber for a variety of absorber heights. This requires

numerical evaluation of

where az is the absorber height. These calculations are presented in the table given below.

Absorber

height/mm

Probability

in

Absorber

10 0.380

15 0.089

20 0.012

25 0.001

It is clear from these calculations and Figure 1 that the probability of finding the neutron in the

absorber falls off sharply at about 20 mm. This analysis, therefore, is consistent with the sharp

increase in neutron throughput at this absorber height.
http://www.nature.com/http://www.nature.com/http://www.nature.com/http://www.nature.com/


12/23


Many neutron gravitational states besides the ground state are occupied and it is therefore

necessary to explore the experimental implications of this fact. The first excited state is, as

mentioned previously, shown in Figure 4. This wave function extends further in the z-direction

than the ground state function, going to zero around 35 mm. The experimental significance of

this is that the neutron throughput should show another abrupt increase in the neighborhood of

30 mm, an absorber height for which, the first excited state neutrons have a low probability

of being absorbed. This phenomena should be repeated for all other occupied excited states as

the absorber reaches the spatial extent of each excited state wave function.

Figure 4

With regard to this expected effect Bowles has commented (5)

The data show some hint of stepped increases at the values corresponding to higher energy

states, consistent with the existence of these states, but they are not yet conclusive. Nonetheless,

the evidence for the existence of the first energy state is convincing and confirms that a quantum


13/23


effect occurs in the gravitational trap. The difficulty of this measurement should not be

underestimated. The researchers are measuring a quantum effect caused by gravity that requires a

resolution of 10-15

eV. Interactions of the neutrons with other fields would normally obscure such

a tiny effect, but the neutron's lack of electric charge and the low kinetic energy of the UCNs

make such observations possible.

In summary, thanks to Nesvizhevsky and his team, we now have some direct evidence for

quantized gravitational states. The "quantum bouncer", previously a purely academic exercise,

can now be applied to a real-life example.

Literature cited:

1. P. W. Langhoff, "Schrdinger particle in a gravitational well,"Am. J. Phys.39, 954-957(1971).

2. R. L. Gibbs, "The quantum bouncer,"Am. J. Phys.43, 25-28 (1975).3. J. Gea-Banacloche, "A quantum bouncing ball,"Am. J. Phys.67, 776-782 (1999).4. V. Nesvizhevsky, et al., "Quantum states of neutrons in the Earth's gravitational

field,"Nature415 297-299 (2002).

5. T. J. Bowles, "Quantum effects of gravity,"Nature415 267-268 (2002).6. B. Schwartzchild, "Ultracold Neutrons Exhibit Quantum States in the Earth's

Gravitational Field," Physics Today55 (3) 20-23 (2002).

7. Experimental Evidence of particle - gravitational interactionhttp://physicsworld.com/cws/article/news/3525
http://physicsworld.com/cws/article/news/3525http://physicsworld.com/cws/article/news/3525http://physicsworld.com/cws/article/news/3525


14/23


In the 4D case : The Schrodinger wave equation can be written as :

gis the metric tensor in spherical coordinates, usually takes the form around a massive object

e.g. Sun as Schwarzschild space-time geometry (below). The previous S-E equation, in 3D

space treated space-time as uniform (with no curvature in space) and only the distance from the

center of the earth to the particle was taken as the variable in the magnitude of gravitational

potential energy. In the new derivation in 4D space a new variable is introduced to the

gravitational potential as a perturbation. That is the amount of energy in a unit volume of space

due to the curvature of space-time as below.


15/23


Gives the path of an object (a neutron or a nanoparticle) in 4-D spacetime geometry related to the

Einsteins Stress Energy TensorTt,x,y,z


16/23


First and second order corrections for energy can be written as :

Only the following two components change if the mgz primary energy term is kept constant.

In vacuum, curvature due to mass (or energy) :

At lower energies quantization levels should come close together. The Force on a Nanoparticle

in 4-D space can be written as : {where is defined here as the 4-D gradient}


17/23


Taking the equation that gives energy from the perturbation, derived earlier :

If

As a first approximation

Where

In Schwarzschild geometry, which is appropriate for the sun, the above equation gives the

variation of energy in a (nano)particle or a neutron due to the space-time curvature.

Metric Tensor is given by...


18/23


Experimental Predictions

(1) Under highly controlled conditions measure the masses of a random sample of nanoparticles.

The sample should have a discrete distribution of energies from the derived theory. Since the

energy and mass are related by De Broglie equation, the sample of particles have discrete masses

as the perturbation term varies in gravitational potential energy.

(2) If the masses could be deviated as a mass spectrometer, a fringe pattern should result taking

into account the following equation for deflecting forces. Since the energy of the particle is

quantized, the deflecting forces on them are quantized as well.

Conclusion:

It can be shown using general theory of relativity [GR] and 1st

and 2nd

order perturbation theory

in Quantum mechanics, that nanoparticles have quantized energy levels while travelling on a

geodesic [from GR] that has minimum gravitational potential energy. This would prove that

nanoparticles cannot come in all sizes by the De Broglie hypothesis. Their sizes are restricted by

quantization. Their increase of energy is not continuous but, discrete too. This also proves that

the gravitation is quantized as well, where the particle is immersed. And space-time curvature is

quantized too. This is an indirect proof for quantum-gravity. Several methods have been

suggested to experimentally to prove the above theory.


19/23

The final equation from my PhD thesis is simplified to show that the shortest distance

measurable in space is the PlankWavelength and the corresponding energy is Plank

Energy This is a correct result that can be derived from the Heisenberg Uncertainty

Principle as well. If the theoretical work done in my thesis has any inconsistency this result

cannot be obtained.


20/23

Schwarzschild Metric Tensor can be written

.

------------------------------------------------------------------------------------------------------------------


21/23

..

+


22/23

To simplify consider only the r coordinate and t coordinate

and the first order perturbation

= ;

d =

d =

When the wavelength [ Limit n 1 ] in discrete steps


23/23

=

=

For a particle [with diameter d] falling under a gravitational field,

this is the Energy required to measure when

Date post:	07-Apr-2018
Category:	Documents
Upload:	sidsenadheera
View:	224 times
Download:	0 times

Path of a Massive Particle in Curvilinear Space Using QED

Documents