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# Solve the Maxwell’s equations and Schrodinger’s equation but avoiding the Sommer

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`Abstract:: Solving Maxwell’s equation and Schrodinger’s equation is usually completed in the frequencydomain. 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 thenthe Helmholtz equation is solved by Green’s function method. Both Green’s function methodsrequire Sommerfeld radiation conditions. Sommerfeld radiation condition requires that thesolution 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 asymptoticbehaviour of the far-field. The Sommerfeld radiation condition directly specifies the far-fieldasymptotic behaviour of the field, rather than the asymptotic behaviour obtained by calculation. Inthis way, the Sommerfeld radiation condition makes the process of solving differential equationsunreasonable. If there are no other better methods, then we must use Sommerfeld radiationconditions. In addition, the Sommerfeld radiation condition is a strict frequency domaincondition. As long as the frequency of the solution deviates from the frequency of Green’sfunction, the effectiveness of the solution is greatly reduced. The electromagnetic field signals weencounter are usually signalled with specific frequency bands. Green’s function is a functionwith a fixed frequency. In this way, the frequency of the electromagnetic field signal may deviatefrom the frequency of Green’s function, so it can not fully meet the Sommerfeld radiationcondition. So we can’t prove that we got the right solution. In this paper, Maxwell’s equationsand Schrodinger’s equations are solved directly in the time domain. A new method similar to Green’sfunction method is introduced in the time domain. In this method, the new Green’s function andelectromagnetic field adopt different waves. For example, if the electromagnetic signal is aretarded wave, Green’s function uses an advanced wave. If the electromagnetic signal is anadvanced 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 atthe same time, so their inner product is zero. That is, the surface integral is zero, which avoidsSommerfeld and similar radiation conditions. Another advantage of this new method is that it isinsensitive to frequency, which means that the effectiveness of the solution is guaranteed even ifthe frequency of the solution is different from that of Green’s function.`

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.

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