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Time independent Schrödinger's equation neglects spin. What is expected to get by solving that equation for atoms without forcing antisymmetry of wave function with the permutation of coordinates of electrons?
There are methods that impose antisymmetry of wave equation by forcing its functional form, e.g. Hartree-Fock method. Is this equivalent to solve the Pauli's equation without magnetic field? Or, are they truly solutions to time independent Schrödinger's equation?
I think that antisymmetric or symmetric property of the wave functions as solutions of schrodinger equations does not neglect spin as spin part of the wave function is a space to be used when you describe a real system having spin dependent hamiltonian operator and the symmetry property is an essential feature of representation..
The symmetric/anti-symmetric nature of wave function becomes important to distinguish what type of physical system is being represented-
Identical particles are particles that cannot be distinguished from one another, even in principle. Species of identical particles include, but are not limited to elementary particles such as electrons only , composite such as atomic nuclei, as well as atoms and molecules. There are two main categories of identical particles: bosons, which can share quantum states, and fermions, which do not share quantum states due to the pauli exclusion principle.
There are two ways in which one might distinguish between particles. The first method relies on differences in the particles' intrinsic physical properties, such as mass, charge and spin.
Even if the particles have equivalent physical properties,they can be distinguished if we can track their trajectory.
The problem with above approach is that it contradicts the principles of QM . the particles do not possess definite positions during the periods between measurements. Instead, they are governed by Psi Wave functions that give the probability of finding a particle at each position.
As time passes, the wave functions tend to spread out and overlap. Once this happens, it becomes impossible to determine, in a subsequent measurement, which of the particle positions correspond to those measured earlier. The particles are then said to be indistinguishable.
The choice of symmetry or antisymmetry is determined by the species of particle. For example, we must always use symmetric states when describing photons or He-4 atoms, and antisymmetric states when describing e or p.
antisymmetry gives rise to the Pauli exclusion principle which forbids identical fermions from sharing the same quantum state. Systems of many identical fermions are described by F-D statistics
Parastatistics are also possible.
In certain two-dimensional systems, mixed symmetry can occur. These exotic particles are known as anyons, and they obey other statistics .
Experimental evidence for the existence of anyons exists in the Fractional Q. Hall effect, a phenomenon observed in the two-dimensional electron gases that form the inversion layer of MOSFETs.
There is another type of statistic, known as braid statistics which are associated with particles known as plektons.
The spin statistics relates the exchange symmetry of identical particles to their spin. It states that bosons have integer spin, and fermions have half-integer spin. Anyons possess fractional spin.
https://en.wikipedia.org/wiki/Identical_particles#Symmetrical_and_antisymmetrical_states
