Suppose we have two electrons in some spin-independent potential V r for example in an atom. We know the two-electron wave function is antisymmetric. Now, the Hamiltonian has no spin-dependence, so we must be able to construct a set of common eigenstates of the Hamiltonian, the total spin, and the z - component of the total spin.
For two electrons, there are four basis states in the spin space. The eigenstates of S and S z are the singlet state. It is evident by inspection that the singlet spin wave function is antisymmetric in the two particles, the triplet symmetric. This overall antisymmetry requirement actually determines the magnetic properties of atoms. This arises from the electrostatic repulsion energy between the electrons. In the spatially antisymmetric state, the two electrons have zero probability of being at the same place, and are on average further apart than in the spatially symmetric state.
Therefore, the electrostatic repulsion raises the energy of the spatially symmetric state above that of the spatially antisymmetric state. It follows that the lower energy state has the spins pointing in the same direction. This is the first step in understanding ferromagnetism. Another example of the importance of overall wave function antisymmetry for fermions is provided by the specific heat of hydrogen gas.
This turns out to be heavily dependent on whether the two protons spin one-half in the H 2 molecule have their spins parallel or antiparallel, even though that alignment involves only a very tiny interaction energy. If the proton spins are antiparallel, that is to say in the singlet state, the molecule is called parahydrogen. The triplet state is called orthohydrogen. These two distinct gases are remarkably stable — in the absence of magnetic impurities, para — ortho transitions take weeks.
This are three ways that came to mind, I'm pretty sure that there are other since it's really a fundamental property of our equation. This leads to the two different statistics. Sign up to join this community.
The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Why must fermions be antisymmetric?
Asked 6 years, 1 month ago. Active 6 years, 1 month ago. Viewed 4k times. Improve this question. It is not a result one can prove in QM, one has to use relativity. If we consider the orbital approximation that uses a product wavefunction.
This approximation allows us to separate the two particle equation into two one-electron equations:. Unfortunately, by doing this we have introduced unphysical labels to the indistinguishable particles. And this is wrong: the effect of it is that the particles do not interfere with each other because they are in different dimensions six dimensional space - remember? We can deepen our understanding of the quantum mechanical description of multi-electron atoms by examining the concepts of electron indistinguishability and the Pauli Exclusion Principle in detail.
A subtle, but important part of the conceptual picture, is that the electrons in a multi-electron system are not distinguishable from one another by any experimental means. Since the electrons are indistinguishable, the probability density we calculate by squaring the modulus of our multi-electron wavefunction also cannot change when the electrons are interchanged permuted between different orbitals.
In general, if we interchange two identical particles, the world does not change. As we will see below, this requirement leads to the idea that the world can be divided into two types of particles based on their behavior with respect to permutation or interchange.
This could be, for example, a two-electron wavefunction for helium. This new wavefunction must have the property that. If we exchange or permute two identical particles twice, we are by definition back to the original situation. Since we then are back to the original state, the effect of the double permutation must equal 1; i.
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