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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?

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  • $\begingroup$ You could do this, but why would you want to? It would have no correspondence with reality. All the electrons would be in the lowest s-state orbital. $\endgroup$ – Lewis Miller Feb 29 '16 at 16:40
  • $\begingroup$ @LewisMiller I don't think that's right; there may not be exchange effects but there would still be correlation. $\endgroup$ – lemon Feb 29 '16 at 16:44
  • $\begingroup$ @LewisMiller, thanks for reply, I agree with lemon in that correlation effects still exist, and therefore I can't see any reason to suspect that all the electrons would be in the lowest s-state orbital (even more, the coulombic repulsion would be very strong). $\endgroup$ – user1420303 Feb 29 '16 at 17:40
  • $\begingroup$ If you have access to vast computing power you could use QMC to find the solution :) $\endgroup$ – lemon Feb 29 '16 at 17:54
  • $\begingroup$ @lemon , I do not have an specific problem to solve, it is just a conceptual dude. $\endgroup$ – user1420303 Feb 29 '16 at 18:09
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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

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    $\begingroup$ -1 The questions (as I understand it) was: what do you get if you solve the TISE for a multi-fermion system without building antisymmetry into the functional form? $\endgroup$ – lemon Feb 29 '16 at 16:59
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    $\begingroup$ The importance of symmetrization of the wave function must be taken into consideration in solving TISE- as other solutions become unphysical in the sense that it can not represent particles found in nature-thats why a total picture was presented-if some things are extra info that can be edited. $\endgroup$ – drvrm Feb 29 '16 at 17:45
  • $\begingroup$ @drvrm Thank you very much. I agree with what you said, but I feel that your answer is not specific for the question. $\endgroup$ – user1420303 Mar 1 '16 at 13:24
  • $\begingroup$ Looking at the development of methods of solving TSE for systems having more than 2 particles -the solutions become complex as the potentials are with tails for large r and even separability of equations becomes difficult and effectively 'variational techniques' are adopted to get physical solutions. in nuclear case its better as the potentials are short range. the H-F method is a way out for treating such multi -electron systems ; a good description of its limitations is given in-www.springer.com/cda/content/.../cda.../9784431548249-c1.pdf?...0... by T Tsuneda - $\endgroup$ – drvrm Mar 1 '16 at 13:54

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