Figure 1. SARS-CoV-2 and perpetual immunity: A significant
number of high-affinity antibodies are created, which outwit the virus.
QUANTUM PERTURBATION MODEL OF COVID-19 ATTACK AND IMMUNITY
RESPONSE
Based on the published data regarding COVID-19 cases, a person who has
recovered from a COVID-19 may be re-infected by the virus several times.
However, the body’s immunity system comes into play and develops
resistance to the virus by producing high-affinity antibodies. When we
have concurrent infections, our immune system adapts to the virus and
produces a diverse repertoire of immunoglobulins (Ig) that have a higher
affinity to take down the virus as described in Figure 1 (Gazumyan et
al., 2012; Kumar et al., 2014). Furthermore, the strength of the body’s
immunity grows with additional factors such as vaccination and the
number of times the virus infects. One of the most effective methods for
predicting the strength and impact of SARS-CoV-2 attacks is to create a
model and explain it in terms of some known parameters.
We have developed a model based on the Quantum Perturbation Theory. We
call this model the ‘Quantum Perturbation Model’. The model relates the
strength of the COVID-19 attack to a wave function that contains
information about the system (a person infected with SARS-CoV-2) and
quantized energy levels that indicate the likelihood of illness
recurrence.
Every measurable quantity is represented by an operator say Ồ and
the state of a system are represented by a wave function ψ in
Quantum Mechanics. When the operator operates on the wave function, it
gives us the same wave function multiplied by certain numbers. The
numbers are called Eigen Values and provide the measurable value
of the parameter; one is interested to calculate. For example, the total
energy operator is represented by Hamiltonian H. This operator operates
on wave function ψ as given below (Griffith, 1994; Liboff, 2002).
HΨ = EΨ (1)
where E gives us the energy of the system.
When a quantum system perturbs the energy of the system also changes
slightly. The change in energy depends upon the nature and strength of
perturbation.
The rules and equations of Quantum Mechanics are applied to a system
with size in nanometer (nm) or smaller. The size of coronavirus is from
20 nm up to as big as 500 nm (Williamson et al., 2020; Sud et al.,
2020). This size range is very appropriate to be discovered and
investigated by the laws and equations of Quantum Mechanics. Therefore,
we have applied Quantum Perturbation Theory to the attack and effect of
SARS-CoV-2 on humans.
Let the energy operator be given by Hamiltonian H of the Quantum
Mechanical system is compatible to the full strength of the attack by
coronavirus on a person then the energy eigenvalue E will give the
effect of COVID-19 on the health of the person. Let us further assume
that the first attack in full strength by coronavirus on a person is
represented by an un-perturbed system then mathematically it can be
expressed as (Griffith, 1994; Liboff, 2002; Reed, 2007),
H0 Ψn(0) =
En(0)Ψn(0) (2)
Where H0 is the Hamiltonian (total strength of the
coronavirus) and Ψn(0) is the
unperturbed wave function of the system in its nth state. The energy of
the unperturbed system is also called the zeroth-order correction to
energy. Equation (2) gives the unperturbed zeroth-order correction in
energy further simplified in equation (3).
En(0)= <
Ψn(0) │H0│Ψn(0)> (3)
Then small perturbation is applied to find the approximate solution for
the same system under a slight perturbation.
H Ψn = En Ψn(4)
To see the effect of perturbation, we expand the Hamiltonian into a
modified form,
H = H 0 + λH p (5)
Where λ is a dimensionless parameter meant to keep track of the degree
of “smallness”. Hp is the perturbed Hamiltonian. When
λ → 0, H0 → H.
Incorporating equation (4) into equation (3), we get,
(H 0 + λH p) Ψn = EnΨn (6)
We can easily the perturbed wave function in terms of the unperturbed
wave function and the strength of perturbation λ. And the energy of the
perturbed system in terms of the energy of the unperturbed system and λ.
Therefore,
Ψn = Ψn(0) + λ
Ψn(1) + λ2Ψn(2) + λ3Ψn(3) (7)
And En = En(0)+ λ En(1) + λ2En(2) + λ3En(3) (8)
Putting equation (7) and equation (8) in equation (4) and collecting
terms with the same exponent of λ, we can get variation or correction to
the energy and wave function of the system.