Solve the Maxwell’s equations and Schrodinger’s equation but avoiding the Sommer
Abstract::
Solving Maxwell’s equation and Schrodinger’s equation is usually completed in the frequency
domain. In the frequency domain, the two equations can be solved by Green’s function method.
Maxwell equation and Schrodinger equation can be simplified into the Helmholtz equation, and then
the Helmholtz equation is solved by Green’s function method. Both Green’s function methods
require Sommerfeld radiation conditions. Sommerfeld radiation condition requires that the
solution of the equation is equivalent to an attenuated plane wave on an infinite boundary.
However, when we solve the Helmholtz equation, we often want to obtain the asymptotic
behaviour of the far-field. The Sommerfeld radiation condition directly specifies the far-field
asymptotic behaviour of the field, rather than the asymptotic behaviour obtained by calculation. In
this way, the Sommerfeld radiation condition makes the process of solving differential equations
unreasonable. If there are no other better methods, then we must use Sommerfeld radiation
conditions. In addition, the Sommerfeld radiation condition is a strict frequency domain
condition. As long as the frequency of the solution deviates from the frequency of Green’s
function, the effectiveness of the solution is greatly reduced. The electromagnetic field signals we
encounter are usually signalled with specific frequency bands. Green’s function is a function
with a fixed frequency. In this way, the frequency of the electromagnetic field signal may deviate
from the frequency of Green’s function, so it can not fully meet the Sommerfeld radiation
condition. So we can’t prove that we got the right solution. In this paper, Maxwell’s equations
and Schrodinger’s equations are solved directly in the time domain. A new method similar to Green’s
function method is introduced in the time domain. In this method, the new Green’s function and
electromagnetic field adopt different waves. For example, if the electromagnetic signal is a
retarded wave, Green’s function uses an advanced wave. If the electromagnetic signal is an
advanced wave, Green’s function uses a retarded wave. Therefore, at the infinite boundary,
the retarded wave and the advanced wave do not arrive at the same time, so they are not zero at
the same time, so their inner product is zero. That is, the surface integral is zero, which avoids
Sommerfeld and similar radiation conditions. Another advantage of this new method is that it is
insensitive to frequency, which means that the effectiveness of the solution is guaranteed even if
the frequency of the solution is different from that of Green’s function.
General metadata
Submitted 01/10/2021
Revised 16/10/2021
Accepted 25/03/2022
Shuangren Zhao “Solve the Maxwell’s equations and Schrodinger’s equation but avoiding the Sommerfeld radiation condition” Theoretical Physics Letters, vol. 10, no. 05.
DOI - 10.1490/5012578.454ptl
Maxwell’s equation is usually converted into the wave equation, which can be the wave equation of electric field, magnetic field, magnetic vector potential and scalar potential. Then the wave equation is transformed into the Helmholtz equation in frequency domain. The Schrodinger equation can also be transformed into the Helmholtz equation in frequency domain. The Green’s function method can be used to find the unique solution of Helmholtz equation. The solution of the Helmholtz equation by Green’s function method needs Sommerfeld radiation condition. The Lorentz reciprocity theorem[3, 4] can also be used to solve Maxwell’s equation. Lorentz reciprocity theorem is also a frequency domain theorem. The uniqueness of the solution also needs the radiation condition similar to Sommerfeld’s, that is, the Silver-Muller radiation condition. But this brings a problem. If we know the form of the solution, we can judge whether it satisfies the Sommerfeld or Silver-Muller radiation conditions. But before solving the equation we don’t know the form of the solution. It seems unreasonable to require the solution to satisfy such radiation conditions. This boundary condition is too demanding. In fact, it specifies the asymptotic form of the solution at infinity. This asymptotic form, that is, the far-field normality, is often the solution we require. We are often concerned with the radiation directivity pattern of the antenna. The pattern is determined by the far field of radiation. Any regulations and conditions of far-field reduce the correctness and accuracy of our calculation results. The Sommerfeld radiation condition tells us that the solution of the Helmholtz equation is unique only if the Sommerfeld radiation condition is satisfied. That means that If Sommerfeld radiation condition is not satisfied, the solution of the Helmholtz equation obtained by Green’s function method can not guarantee its uniqueness. The method of Lorentz reciprocity theorem has similar problems.
Conclusion
In order to solve the Maxwell’s equations and Schrodinger equation, Green’s function method and Sommerfeld’s and Silver-Muller’s radiation conditions are required to obtain a unique solution. However since Schrodinger equation, Sommerfeld’s and Silver-Muller’s radiation conditions restrict the solution properties in infinity, the correctness of the method is reduced largely.
In this article, this author introduced a new Green’s function method which can make the surface integral term vanish. When the surface integral term vanishes, one of the fields on the surface is retarded wave and the other is advanced wave. The inner product of the retarded wave and the advanced wave at the boundary will naturally be zero. Physically, it shows that the mutual energy flow generated by the advanced wave and the retarded wave will not flow out of the universe. It has a strong physical meaning. Therefore, no matter whether these waves meet the radiation conditions or not, as long as one of them is retarded wave and the other is advanced wave, the surface integral will be zero without any problem. The new method is also stable with frequency, that means even if the solution deviates with the frequency of Green’s function, the surface integral still can be guaranteed to vanish. Hence offers a more correct and accurate solution. According to all these, Sommerfeld’s and Silver-Muller’s radiation conditions are avoided.