Mathemat

===Changing basis for a spin-1/2 particle===
A stationary spin- particle has a two-dimensional Hilbert space. One orthonormal basis is:
|{\uparrow}_z \rangle \,, \; |{\downarrow}_z \rangle
where is the state with a definite value of the spin operator equal to + and is the state with a definite value of the spin operator equal to −.
It has to be determined by convention if the complex numbers \{\psi_n\} are inside or outside of the inner product, and each convention gives different results.
\langle\psi| \equiv (\boldsymbol\psi, \cdot ) = \sum_n (\boldsymbol e_n, \cdot ) \, \psi_n
\langle\psi| \equiv (\boldsymbol\psi, \cdot ) = \sum_n (\boldsymbol e_n \psi_n, \cdot ) = \sum_n (\boldsymbol e_n, \cdot ) \, \psi_n^*