Restrictions on the Wavefunction

We ve noted that is interpreted as the probability density for the particle(s) it describes. Therefore, we require that y be normalized-, if we integrate over all space, 

Exploring Chemistry with Electronic Structure Methods 

We can do this because the Schrodinger equation is an eigenvalue equation, and in general, if/is a solution to an eigenvalue equation, then cfis also, for any value of r. For the Schrodinger equation, it is easy to show that H(ct(/) = cH(t(/) and that (ct(/) = c( t(/) thus, if y is a solution to the Schrodinger equation, then cy is as well. 

Secondly, y must also be antisymmetric, meaning that it must change sign when two identical particles are interchanged. For a simple function, antisymmetry means that the following relation holds  

For an electronic wavefunction, antisymmetry is a physical requirement following from the fact that electrons are fermions. It is essentially a requirement that y agree with the results of experimental physics. More specifically, this requirement means that any valid wavefunction must satisfy the following condition  

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