A New Approach in Quantum Mechanics

A paper of mine was recently published in the journal "Foundations of Physics". In this paper I introduced an equation from which the Schrodinger equation can be derived. This is a new approach with which known problems in quantum mechanics (such as scattering problems) can be solved with new insights. For example, the equation introduced in this paper takes into account the spin of the particle in scattering problems and makes specific predictions on how a particle with spin can scatter from a step potential. The equation can be used to solve other problems in quantum mechanics as well. This equation can have notable implications for physics in the non-relativistic limit.

Following are the comments of one of the reviewers of this paper:

" This article is an important work, and should be published as soon as possible. Dirac deduced his equation ---the Dirac equation by introducing 4*4 matrices, which is first order both for time and space and can deduces the Klein-Gordon equation. The Schrödinger equation is first order for time and second order for space, it has been used for a century to treat non-relativistic quantum mechanical problems. No one was aware of looking for a Dirac equation analogue of the Schrödinger equation. The author the first time proposed an equation with first order time and first order space for the Schrödinger equation, it is brand new idea and work."

Here is the link to a translated version of a blog post on this article originally written in Chinese.

Following is the 3D version of the equation in momentum space


 

The matrices η are 4 dimensional matrices and \gamma_i are the Dirac gamma matrices. (Note that η can also be 2x2 matrices but in that case the 3D version of the equation is not possible because there is no fourth matrix that anticommutes with the Pauli matrix. The 1d version is possible and agrees with standard quantum mechanics.)

In another recently published paper I have also shown that this first order equation is the non-relativistic limit of the Dirac equation and the Pauli equation can be derived from it:

http://www.ijqf.org/archives/3480

I have therefore proposed that this equation is the non-relativistic equation for fermions and Schrodinger equation more appropriately describes scalars in the non-relativistic limit. This equation allows for the description of fermions and boson in the non-relativistic limit. The Lagrangian for free fermions and scalars in the non-relativistic limit can therefore be written as





Note that the scalar and fermion wave functions have dimensions of Energy^(3/2) in the above Lagrangian density. In the presence of interactions, for example, for scalars we can have a phi^4 like interaction term which would lead to the non-linear Schrodinger equation which is employed in the description of the Bose-Einstein condensate.

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[1] http://arxiv.org/pdf/1502.04274v2.pdf