Content uploaded by Alexander Soiguine
Author content
All content in this area was uploaded by Alexander Soiguine on Jun 24, 2023
Content may be subject to copyright.
1
Hydrogen Atom Model in Geometric Algebra Terms
Alexander SOIGUINE
1
Abstract: Geometric Algebra formalism opens the door to developing a theory replacing conventional quantum theory
(Mathematics Subject Classification, item 81.) Generalizations, stemming from changing of complex numbers by geometrically
feasible objects in three dimensions, followed by unambiguous definition of states, observables, measurements, bring into reality
clear explanations of weird quantum mechanical features, for example primitively considering atoms as a kind of planetary
system, very familiar from macroscopic experience but recklessly used in a physically very different situation. In the current
work the three-sphere becomes the playground of the torsion kind states eliminating abstract Hilbert space vectors. The states
as points evolve, governed by updated Schrodinger equation, and act as operators on observables in measurements.
Key words: geometric algebra, states, observables, measurements
1. Introduction. Experimental reasons of the planetary model of
atoms
The Rutherford scattering experiments [1] demonstrated that positively charged part of
atoms was very small, less than 10-14 m, and only one of hundred thousand positively
charged alpha particles got deflected by angles greater than 90 degrees when
bombarding gold leaf of only a few atoms thick. The logically reasonable conclusion was
that positively charged nucleus concentrating majority of mass occupied very small
regions.
Scattering results were explained using the retractive force between positively charged
alpha particles and nucleus. Calculation of the force followed from Coulomb potential
that was only experimentally justified for macroscopic point like objects but recklessly
used in a physically very different situation.
So it happened that the planetary model appeared from viewing atoms as something
very familiar from macroscopic experience: planets (electrons) rotating around the Sun
(nucleus.) Mathematical formalism was also taken from available mathematics at hand:
mostly the Hilbert spaces [2].
The approach suggested in this article uses different point of view and assumes that
actual weirdness of all conventional quantum mechanics comes from logical
inconsistence of what is meant by basic quantum mechanical definitions and has
nothing to do with the phenomena scale. The theory should speak about proper
1
Website: https://soiguine.com; E-mail: alex@soiguine.com
2
separation of measurement process arrangement into operator, three-sphere
element, acting on observable, and operand, measured observable.
Unambiguous definition of states and observables, does not matter are we in “classical”
or “quantum” frame, should follow the general paradigm, [3], [4], [5], [6]:
- Measurement of observable by state
2
is a map:
,
where is an element of the set of observables, is element of,
generally though not necessarily, another set, set of states.
- The result (value) of a measurement of observable by state is the result of
a map sequence:
,
where is a set of (Boolean) algebra subsets identifying possible results of
measurements.
State and observable are different things. Evolution of a state should be considered
separately, and then action of modified state will be applied to an observable in
measurements.
The option to expand, to lift the space where physical processes are considered, may
have critical consequences to a theory. A kind of expanding is the core of the suggested
formulation aimed at the theory deeper than conventional quantum mechanics. States
as Hilbert space complex valued vectors with formal imaginary unit are lifted to a torsion
kind object identified by points on sphere .
2. Working with g-qubits instead of qubits.
A theory that is an alternative to conventional quantum mechanics has been under
development for a while, see, [3], [4], [7], [6], [8].
Its novel features are:
- Replacing complex numbers by elements of even subalgebra of geometric
algebra in three dimensions, that’s by elements of the form “scalar plus bivector”.
- Elementary physical objects bear the structure: position in space plus explicitly
defined object as the , geometric algebra in three dimensions, elements.
- Operators acting on those objects are identified as direct sums of position
translation and points on the three-sphere . All those points are connected, due
to hedgehog theorem, by parallel (Clifford) translations.
- Evolution of the part of operators by Clifford translations is governed by
generalization of the Schrodinger equation with unit bivectors in three dimensions
instead of formal imaginary unit.
2
One should say “by a state”. State is operator acting on observable.
3
In the following the part of the operators will only be considered.
The even subalgebra
is subalgebra of elements of the form where
and are (real)
3
scalars and is some unit bivector arbitrary placed in three-
dimensional space. Elements of
can be depict as in Fig. 2.1.
Fig.2.1. An element of
In the
multiplication is more complicated than in Hilbert space . It reads:
It is not commutative due to the not commutative product of bivectors . Indeed,
taking vectors to which and are dual: , , we have:
, is oriented unit value volume.
Then:
and
We see that if then , , so
that is the same, up to replacing by , as for complex numbers.
3
In the current formalism scalars can only be real numbers. “Complex” scalars make no sense anymore,
see, for example, [4], [8].
4
Unit value elements of
, when , will be called g-qubits. The wave
functions, states, implemented as g-qubits store much more information than qubits,
see Fig 2.2.
Fig. 2.2. Geomectrically picted qubits and g-qubits
3. Implementation of the definitions from the Introduction in the
g-qubit state case
General definition of measurement in the suggested approach is based on:
- the set of observables, particularly elements of
,
- the set of states, normalized elements of
, g-qubits,
- special case of measurement of a
observable by g-
qubit (wave function) is defined as
with the result:
(3.1)
Since g-qubit (state, wave function) is normalized, the measurement can be written in
exponential form:
where . The above is updated variant of quantum mechanical formula
5
The lift from to
needs a reference frame of unit value bivectors. This
frame, as a solid, can be arbitrary rotated in three dimensions. In that sense we have
principal fiber bundle
with the standard fiber as group of rotations which is also
effectively identified by elements of
. Probabilities of the results of measurements are
measures of the states giving considered results.
4. Evolution of g-qubit states
Measurement of an observable by a state is defined as . Evolution of a
state is its movement on surface of .
Consider necessary formalism.
Multiplication of two geometric algebra exponents reads, see Sec.1.2 of [8]:
It follows from the formula for bivector multiplication:
with vectors to which the unit bivectors and are duals: , .
In the current case
, , , ,
and we get above formula for .
The product of two exponents is again an exponent, because generally
and , see Sec.1.3 of [8].
Multiplication of an exponent by another exponent is often called Clifford translation.
Using the term translation follows from the fact that Clifford translation does not change
distances between the exponents it acts upon when we identify exponents as points on
unit sphere :
This result follows again from :
Assume the angle in Clifford translation is a variable one. Then in the case
:
6
If is dual to some unit vector , (this is the case of the matrix Hamiltonian
map to
, see [5] ), then and
that is obviously Geometric Algebra generalization of the Schrodinger equation.
5. Model of the alpha particle deflection by nucleus
An observable will be from the set of duals, bivectors, of parallel vectors coming from a
plane parallel to a plane passing through the nucleus. Section of the incoming particles
by an arbitrary plane passing through the central line looking at the nucleus and parallel
to incoming vectors is shown in Fig. 5.1:
Fig.5.1. Incoming alpha particles
An alpha particle moving along the vector passing exactly through the nucleus should
deflect, bounce, in the opposite direction. Assuming that deflection happens as torsion
in plane , defined by bivector parallel to incoming vectors, and denoting an arbitrary
incoming vector as we get dual bivector of deflection of (looking in the
opposite direction):
,
or deflecting vector itself:
If we denote maximum value of at which deflection takes place by then for
arbitrary the deflection vector is, see Fig. 5.2:
7
Fig.5.2. Results of deflection
If we know the value of
(assuming ) for particles deflected by angle equal to
the nucleus effective radius can be calculated, see Fig.5.3:
Fig.5.3. For the calculation of the effective radius
From this figure, angle of deflection is:
Then
8
and
(5.1)
So, whilst the nucleus occupies very small volume, it is extended up to an effective
radius (5.1).
6. Movement of an electron relative to nucleus
Let a state is of Hamiltonian type
4
:
(6.1)
where is vector in three dimensions.
An observable it will act upon is something of a torsion kind, . Thus, at instant
of time we have the following result of action of state (6.1):
(6.2)
The Hamiltonian type of wave function (6.1) bears its origin from proton, while the
observable represents electron.
The geometric algebra existence of the hydrogen atom can only follow from stable
sequence of measurement results (6.2) with appropriate combination(s) of and .
Let
Then
, bivector part of (6.1) is
, and the scalar part of the wave function (6.1) is .
If initial bivector plane of observable is ,
, the scalar
part is then , thus .
4
Not critical, just for resembling traditional form of states.
9
Let us denote the plane
as ,
.
Assuming , that’s the torsion of electron instantly follows the action of wave
function, then we get the following result of action of the state (6.1) on observable
:
That means that at every single instant of time the torsion angle is additionally
increased by . It follows particularly that, for synchronization, after rotation by
, , when
, the addition by
should be
, ,
Thus, the equation connecting and is:
, , ,
The numbers and correspond to common quantum mechanics numbers, angular
quantum number and principal quantum number.
7. Conclusions
It was demonstrated that the geometric algebra formalism along with generalization of
complex numbers and subsequent lift of the two-dimensional Hilbert space valued
qubits to geometrically feasible elements of even subalgebra [6] of geometric algebra in
three dimensions allows, particularly, to explain what actually means senseless “find
system in state”. The approach particularly allows to eliminate primitive Bohr’s planetary
model of the hydrogen atom and explains the atom structure in pure geometrical terms.
Some other highly impressive perspectives of the approach comprise, particularly,
explanations of the double-slit experiment, collapse of wave functions [9], possibility to
modify blockchain information back and forth in time, see Sec.6.3 of [8]. All that
supports feasibility of the suggested approach to replace formalism of conventional
quantum theory.
10
References
[1]
E. Rutherford, "The Scattering of α and β Particles by Matter and the Structure of the
Atom," Philosophical Magazine, vol. 21, pp. 669-688, 1911.
[2]
J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton, New
Jersey: Princeton University Press, 1955.
[3]
A. Soiguine, "What quantum "state" really is?," June 2014. [Online]. Available:
http://arxiv.org/abs/1406.3751.
[4]
A. Soiguine, "Geometric Algebra, Qubits, Geometric Evolution, and All That," January
2015. [Online]. Available: http://arxiv.org/abs/1502.02169.
[5]
A. Soiguine, Geometric Phase in Geometric Algebra Qubit Formalism, Saarbrucken:
LAMBERT Academic Publishing, 2015.
[6]
A. Soiguine, The Torsion Mechanics, LAMBERT Academic Publishing, 2023.
[7]
A. M. Soiguine, "Complex Conjugation - Relative to What?," in Clifford Algebras with
Numeric and Symbolic Computations, Boston, Birkhauser, 1996, pp. 284-294.
[8]
A. Soiguine, The Geometric Algebra Lift of Qubits and Beyond, LAMBERT Academic
Publishing, 2020.
[9]
A. Soiguine, "Scattering of Geometric Algebra Wave Functions and Collapse in
Measurements," Journal of Applied Matrhematics and Physics, vol. 8, pp. 1838-1844,
2020.