... relation1
Assuming that the quantum covariance $<(\hat{A} \hat{B} + \hat{B} \hat{A})/2 - <\hat{A}> <\hat{B}> >$ is zero. 
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... 1.2
In some cases, such as the free-particle, one must use special tricks to normalize the wavefunction. See Merzbacher [2], section 8.1. 
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... with.3
It is not completely necessary to do this, however; for example, the Hylleras treatment of the Helium atom uses two-particle basis functions which are not further expanded in terms of single-particle functions. 
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... spin.4
This is the procedure of the Unrestricted Hartree Fock (UHF) method. 
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